:: COMPL_SP semantic presentation
begin
definition
let M be non empty MetrStruct ;
let S be SetSequence of M;
attrS is pointwise_bounded means :Def1: :: COMPL_SP:def 1
for i being Nat holds S . i is bounded ;
end;
:: deftheorem Def1 defines pointwise_bounded COMPL_SP:def_1_:_
for M being non empty MetrStruct
for S being SetSequence of M holds
( S is pointwise_bounded iff for i being Nat holds S . i is bounded );
registration
let M be non empty Reflexive MetrStruct ;
cluster non empty Relation-like non-empty NAT -defined bool the carrier of M -valued Function-like total V27( NAT , bool the carrier of M) pointwise_bounded for Element of bool [:NAT,(bool the carrier of M):];
existence
ex b1 being SetSequence of M st
( b1 is pointwise_bounded & b1 is non-empty )
proof
consider x being set such that
A1: x in the carrier of M by XBOOLE_0:def_1;
reconsider x = x as Point of M by A1;
reconsider X = {x} as Subset of M ;
take S = NAT --> X; ::_thesis: ( S is pointwise_bounded & S is non-empty )
A2: now__::_thesis:_for_x1,_x2_being_Point_of_M_st_x1_in_X_&_x2_in_X_holds_
dist_(x1,x2)_<=_1
let x1, x2 be Point of M; ::_thesis: ( x1 in X & x2 in X implies dist (x1,x2) <= 1 )
assume that
A3: x1 in X and
A4: x2 in X ; ::_thesis: dist (x1,x2) <= 1
A5: x2 = x by A4, TARSKI:def_1;
x1 = x by A3, TARSKI:def_1;
hence dist (x1,x2) <= 1 by A5, METRIC_1:1; ::_thesis: verum
end;
A6: now__::_thesis:_for_i_being_Nat_holds_S_._i_is_bounded
let i be Nat; ::_thesis: S . i is bounded
reconsider i9 = i as Element of NAT by ORDINAL1:def_12;
S . i9 = X by FUNCOP_1:7;
hence S . i is bounded by A2, TBSP_1:def_7; ::_thesis: verum
end;
for x being set st x in dom S holds
not S . x is empty by FUNCOP_1:7;
hence ( S is pointwise_bounded & S is non-empty ) by A6, Def1, FUNCT_1:def_9; ::_thesis: verum
end;
end;
definition
let M be non empty Reflexive MetrStruct ;
let S be SetSequence of M;
func diameter S -> Real_Sequence means :Def2: :: COMPL_SP:def 2
for i being Nat holds it . i = diameter (S . i);
existence
ex b1 being Real_Sequence st
for i being Nat holds b1 . i = diameter (S . i)
proof
defpred S1[ set , set ] means for i being Nat st i = $1 holds
$2 = diameter (S . i);
A1: for x being set st x in NAT holds
ex y being set st
( y in REAL & S1[x,y] )
proof
let x be set ; ::_thesis: ( x in NAT implies ex y being set st
( y in REAL & S1[x,y] ) )
assume x in NAT ; ::_thesis: ex y being set st
( y in REAL & S1[x,y] )
then reconsider i = x as Element of NAT ;
take diameter (S . i) ; ::_thesis: ( diameter (S . i) in REAL & S1[x, diameter (S . i)] )
thus ( diameter (S . i) in REAL & S1[x, diameter (S . i)] ) ; ::_thesis: verum
end;
consider f being Function of NAT,REAL such that
A2: for x being set st x in NAT holds
S1[x,f . x] from FUNCT_2:sch_1(A1);
take f ; ::_thesis: for i being Nat holds f . i = diameter (S . i)
let i be Nat; ::_thesis: f . i = diameter (S . i)
i in NAT by ORDINAL1:def_12;
hence f . i = diameter (S . i) by A2; ::_thesis: verum
end;
uniqueness
for b1, b2 being Real_Sequence st ( for i being Nat holds b1 . i = diameter (S . i) ) & ( for i being Nat holds b2 . i = diameter (S . i) ) holds
b1 = b2
proof
let D1, D2 be Real_Sequence; ::_thesis: ( ( for i being Nat holds D1 . i = diameter (S . i) ) & ( for i being Nat holds D2 . i = diameter (S . i) ) implies D1 = D2 )
assume that
A3: for i being Nat holds D1 . i = diameter (S . i) and
A4: for i being Nat holds D2 . i = diameter (S . i) ; ::_thesis: D1 = D2
now__::_thesis:_for_x_being_Element_of_NAT_holds_D1_._x_=_D2_._x
let x be Element of NAT ; ::_thesis: D1 . x = D2 . x
thus D1 . x = diameter (S . x) by A3
.= D2 . x by A4 ; ::_thesis: verum
end;
hence D1 = D2 by FUNCT_2:63; ::_thesis: verum
end;
end;
:: deftheorem Def2 defines diameter COMPL_SP:def_2_:_
for M being non empty Reflexive MetrStruct
for S being SetSequence of M
for b3 being Real_Sequence holds
( b3 = diameter S iff for i being Nat holds b3 . i = diameter (S . i) );
theorem Th1: :: COMPL_SP:1
for M being non empty Reflexive MetrStruct
for S being pointwise_bounded SetSequence of M holds diameter S is bounded_below
proof
let M be non empty Reflexive MetrStruct ; ::_thesis: for S being pointwise_bounded SetSequence of M holds diameter S is bounded_below
let S be pointwise_bounded SetSequence of M; ::_thesis: diameter S is bounded_below
set d = diameter S;
now__::_thesis:_for_n_being_Element_of_NAT_holds_-_1_<_(diameter_S)_._n
let n be Element of NAT ; ::_thesis: - 1 < (diameter S) . n
A1: diameter (S . n) = (diameter S) . n by Def2;
S . n is bounded by Def1;
then 0 <= (diameter S) . n by A1, TBSP_1:21;
hence - 1 < (diameter S) . n by XXREAL_0:2; ::_thesis: verum
end;
hence diameter S is bounded_below by SEQ_2:def_4; ::_thesis: verum
end;
theorem Th2: :: COMPL_SP:2
for M being non empty Reflexive MetrStruct
for S being pointwise_bounded SetSequence of M st S is V172() holds
( diameter S is bounded_above & diameter S is V103() )
proof
let M be non empty Reflexive MetrStruct ; ::_thesis: for S being pointwise_bounded SetSequence of M st S is V172() holds
( diameter S is bounded_above & diameter S is V103() )
let S be pointwise_bounded SetSequence of M; ::_thesis: ( S is V172() implies ( diameter S is bounded_above & diameter S is V103() ) )
assume A1: S is V172() ; ::_thesis: ( diameter S is bounded_above & diameter S is V103() )
set d = diameter S;
A2: now__::_thesis:_for_n_being_Element_of_NAT_holds_(diameter_S)_._n_<_((diameter_S)_._0)_+_1
let n be Element of NAT ; ::_thesis: (diameter S) . n < ((diameter S) . 0) + 1
A3: ((diameter S) . 0) + 0 < ((diameter S) . 0) + 1 by XREAL_1:8;
A4: diameter (S . n) = (diameter S) . n by Def2;
A5: diameter (S . 0) = (diameter S) . 0 by Def2;
A6: S . 0 is bounded by Def1;
S . n c= S . 0 by A1, PROB_1:def_4;
then (diameter S) . n <= (diameter S) . 0 by A6, A4, A5, TBSP_1:24;
hence (diameter S) . n < ((diameter S) . 0) + 1 by A3, XXREAL_0:2; ::_thesis: verum
end;
now__::_thesis:_for_m,_n_being_Element_of_NAT_st_m_in_dom_(diameter_S)_&_n_in_dom_(diameter_S)_&_m_<=_n_holds_
(diameter_S)_._n_<=_(diameter_S)_._m
let m, n be Element of NAT ; ::_thesis: ( m in dom (diameter S) & n in dom (diameter S) & m <= n implies (diameter S) . n <= (diameter S) . m )
assume that
m in dom (diameter S) and
n in dom (diameter S) and
A7: m <= n ; ::_thesis: (diameter S) . n <= (diameter S) . m
A8: S . m is bounded by Def1;
A9: diameter (S . m) = (diameter S) . m by Def2;
A10: diameter (S . n) = (diameter S) . n by Def2;
S . n c= S . m by A1, A7, PROB_1:def_4;
hence (diameter S) . n <= (diameter S) . m by A8, A10, A9, TBSP_1:24; ::_thesis: verum
end;
hence ( diameter S is bounded_above & diameter S is V103() ) by A2, SEQM_3:def_4, SEQ_2:def_3; ::_thesis: verum
end;
theorem :: COMPL_SP:3
for M being non empty Reflexive MetrStruct
for S being pointwise_bounded SetSequence of M st S is V173() holds
diameter S is V102()
proof
let M be non empty Reflexive MetrStruct ; ::_thesis: for S being pointwise_bounded SetSequence of M st S is V173() holds
diameter S is V102()
let S be pointwise_bounded SetSequence of M; ::_thesis: ( S is V173() implies diameter S is V102() )
assume A1: S is V173() ; ::_thesis: diameter S is V102()
set d = diameter S;
now__::_thesis:_for_m,_n_being_Element_of_NAT_st_m_in_dom_(diameter_S)_&_n_in_dom_(diameter_S)_&_m_<=_n_holds_
(diameter_S)_._n_>=_(diameter_S)_._m
let m, n be Element of NAT ; ::_thesis: ( m in dom (diameter S) & n in dom (diameter S) & m <= n implies (diameter S) . n >= (diameter S) . m )
assume that
m in dom (diameter S) and
n in dom (diameter S) and
A2: m <= n ; ::_thesis: (diameter S) . n >= (diameter S) . m
A3: S . n is bounded by Def1;
A4: diameter (S . m) = (diameter S) . m by Def2;
A5: diameter (S . n) = (diameter S) . n by Def2;
S . m c= S . n by A1, A2, PROB_1:def_5;
hence (diameter S) . n >= (diameter S) . m by A3, A5, A4, TBSP_1:24; ::_thesis: verum
end;
hence diameter S is V102() by SEQM_3:def_3; ::_thesis: verum
end;
theorem Th4: :: COMPL_SP:4
for M being non empty Reflexive MetrStruct
for S being pointwise_bounded SetSequence of M st S is V172() & lim (diameter S) = 0 holds
for F being sequence of M st ( for i being Nat holds F . i in S . i ) holds
F is Cauchy
proof
let M be non empty Reflexive MetrStruct ; ::_thesis: for S being pointwise_bounded SetSequence of M st S is V172() & lim (diameter S) = 0 holds
for F being sequence of M st ( for i being Nat holds F . i in S . i ) holds
F is Cauchy
let S be pointwise_bounded SetSequence of M; ::_thesis: ( S is V172() & lim (diameter S) = 0 implies for F being sequence of M st ( for i being Nat holds F . i in S . i ) holds
F is Cauchy )
assume that
A1: S is V172() and
A2: lim (diameter S) = 0 ; ::_thesis: for F being sequence of M st ( for i being Nat holds F . i in S . i ) holds
F is Cauchy
set d = diameter S;
A3: diameter S is V103() by A1, Th2;
A4: diameter S is bounded_below by Th1;
let F be sequence of M; ::_thesis: ( ( for i being Nat holds F . i in S . i ) implies F is Cauchy )
assume A5: for i being Nat holds F . i in S . i ; ::_thesis: F is Cauchy
let r be Real; :: according to TBSP_1:def_4 ::_thesis: ( r <= 0 or ex b1 being Element of NAT st
for b2, b3 being Element of NAT holds
( not b1 <= b2 or not b1 <= b3 or not r <= dist ((F . b2),(F . b3)) ) )
assume r > 0 ; ::_thesis: ex b1 being Element of NAT st
for b2, b3 being Element of NAT holds
( not b1 <= b2 or not b1 <= b3 or not r <= dist ((F . b2),(F . b3)) )
then consider n being Element of NAT such that
A6: for m being Element of NAT st n <= m holds
abs (((diameter S) . m) - 0) < r by A2, A4, A3, SEQ_2:def_7;
take n ; ::_thesis: for b1, b2 being Element of NAT holds
( not n <= b1 or not n <= b2 or not r <= dist ((F . b1),(F . b2)) )
let m1, m2 be Element of NAT ; ::_thesis: ( not n <= m1 or not n <= m2 or not r <= dist ((F . m1),(F . m2)) )
assume that
A7: n <= m1 and
A8: n <= m2 ; ::_thesis: not r <= dist ((F . m1),(F . m2))
A9: S . m2 c= S . n by A1, A8, PROB_1:def_4;
A10: F . m2 in S . m2 by A5;
A11: F . m1 in S . m1 by A5;
A12: abs (((diameter S) . n) - 0) < r by A6;
A13: diameter (S . n) = (diameter S) . n by Def2;
A14: S . n is bounded by Def1;
then 0 <= (diameter S) . n by A13, TBSP_1:21;
then A15: (diameter S) . n < r by A12, ABSVALUE:def_1;
S . m1 c= S . n by A1, A7, PROB_1:def_4;
then dist ((F . m1),(F . m2)) <= (diameter S) . n by A9, A11, A10, A14, A13, TBSP_1:def_8;
hence not r <= dist ((F . m1),(F . m2)) by A15, XXREAL_0:2; ::_thesis: verum
end;
theorem Th5: :: COMPL_SP:5
for r being Real
for M being non empty Reflexive symmetric triangle MetrStruct
for p being Point of M st 0 <= r holds
diameter (cl_Ball (p,r)) <= 2 * r
proof
let r be Real; ::_thesis: for M being non empty Reflexive symmetric triangle MetrStruct
for p being Point of M st 0 <= r holds
diameter (cl_Ball (p,r)) <= 2 * r
let M be non empty Reflexive symmetric triangle MetrStruct ; ::_thesis: for p being Point of M st 0 <= r holds
diameter (cl_Ball (p,r)) <= 2 * r
let p be Point of M; ::_thesis: ( 0 <= r implies diameter (cl_Ball (p,r)) <= 2 * r )
A1: dist (p,p) = 0 by METRIC_1:1;
assume 0 <= r ; ::_thesis: diameter (cl_Ball (p,r)) <= 2 * r
then A2: p in cl_Ball (p,r) by A1, METRIC_1:12;
A3: now__::_thesis:_for_x,_y_being_Point_of_M_st_x_in_cl_Ball_(p,r)_&_y_in_cl_Ball_(p,r)_holds_
dist_(x,y)_<=_2_*_r
let x, y be Point of M; ::_thesis: ( x in cl_Ball (p,r) & y in cl_Ball (p,r) implies dist (x,y) <= 2 * r )
assume that
A4: x in cl_Ball (p,r) and
A5: y in cl_Ball (p,r) ; ::_thesis: dist (x,y) <= 2 * r
A6: dist (x,p) <= r by A4, METRIC_1:12;
A7: dist (x,y) <= (dist (x,p)) + (dist (p,y)) by METRIC_1:4;
dist (p,y) <= r by A5, METRIC_1:12;
then (dist (x,p)) + (dist (p,y)) <= r + r by A6, XREAL_1:7;
hence dist (x,y) <= 2 * r by A7, XXREAL_0:2; ::_thesis: verum
end;
cl_Ball (p,r) is bounded by TOPREAL6:59;
hence diameter (cl_Ball (p,r)) <= 2 * r by A2, A3, TBSP_1:def_8; ::_thesis: verum
end;
definition
let M be MetrStruct ;
let U be Subset of M;
attrU is open means :Def3: :: COMPL_SP:def 3
U in Family_open_set M;
end;
:: deftheorem Def3 defines open COMPL_SP:def_3_:_
for M being MetrStruct
for U being Subset of M holds
( U is open iff U in Family_open_set M );
definition
let M be MetrStruct ;
let A be Subset of M;
attrA is closed means :Def4: :: COMPL_SP:def 4
A ` is open ;
end;
:: deftheorem Def4 defines closed COMPL_SP:def_4_:_
for M being MetrStruct
for A being Subset of M holds
( A is closed iff A ` is open );
registration
let M be MetrStruct ;
cluster empty -> open for Element of bool the carrier of M;
coherence
for b1 being Subset of M st b1 is empty holds
b1 is open
proof
let S be Subset of M; ::_thesis: ( S is empty implies S is open )
A1: TopSpaceMetr M = TopStruct(# the carrier of M,(Family_open_set M) #) ;
assume S is empty ; ::_thesis: S is open
then S in Family_open_set M by A1, PRE_TOPC:1;
hence S is open by Def3; ::_thesis: verum
end;
cluster empty -> closed for Element of bool the carrier of M;
coherence
for b1 being Subset of M st b1 is empty holds
b1 is closed
proof
let S be Subset of M; ::_thesis: ( S is empty implies S is closed )
assume S is empty ; ::_thesis: S is closed
then A2: [#] M = S ` ;
[#] M in Family_open_set M by PCOMPS_1:30;
then [#] M is open by Def3;
hence S is closed by A2, Def4; ::_thesis: verum
end;
end;
registration
let M be non empty MetrStruct ;
cluster non empty open for Element of bool the carrier of M;
existence
ex b1 being Subset of M st
( b1 is open & not b1 is empty )
proof
[#] M in Family_open_set M by PCOMPS_1:30;
then [#] M is open by Def3;
hence ex b1 being Subset of M st
( b1 is open & not b1 is empty ) ; ::_thesis: verum
end;
cluster non empty closed for Element of bool the carrier of M;
existence
ex b1 being Subset of M st
( b1 is closed & not b1 is empty )
proof
(({} M) `) ` = {} M ;
then ({} M) ` is closed by Def4;
hence ex b1 being Subset of M st
( b1 is closed & not b1 is empty ) ; ::_thesis: verum
end;
end;
theorem Th6: :: COMPL_SP:6
for M being MetrStruct
for A being Subset of M
for A9 being Subset of (TopSpaceMetr M) st A9 = A holds
( ( A is open implies A9 is open ) & ( A9 is open implies A is open ) & ( A is closed implies A9 is closed ) & ( A9 is closed implies A is closed ) )
proof
let M be MetrStruct ; ::_thesis: for A being Subset of M
for A9 being Subset of (TopSpaceMetr M) st A9 = A holds
( ( A is open implies A9 is open ) & ( A9 is open implies A is open ) & ( A is closed implies A9 is closed ) & ( A9 is closed implies A is closed ) )
let A be Subset of M; ::_thesis: for A9 being Subset of (TopSpaceMetr M) st A9 = A holds
( ( A is open implies A9 is open ) & ( A9 is open implies A is open ) & ( A is closed implies A9 is closed ) & ( A9 is closed implies A is closed ) )
let A9 be Subset of (TopSpaceMetr M); ::_thesis: ( A9 = A implies ( ( A is open implies A9 is open ) & ( A9 is open implies A is open ) & ( A is closed implies A9 is closed ) & ( A9 is closed implies A is closed ) ) )
assume A1: A9 = A ; ::_thesis: ( ( A is open implies A9 is open ) & ( A9 is open implies A is open ) & ( A is closed implies A9 is closed ) & ( A9 is closed implies A is closed ) )
thus ( A is open implies A9 is open ) ::_thesis: ( ( A9 is open implies A is open ) & ( A is closed implies A9 is closed ) & ( A9 is closed implies A is closed ) )
proof
assume A is open ; ::_thesis: A9 is open
then A in Family_open_set M by Def3;
hence A9 is open by A1, PRE_TOPC:def_2; ::_thesis: verum
end;
thus ( A9 is open implies A is open ) ::_thesis: ( A is closed iff A9 is closed )
proof
assume A9 is open ; ::_thesis: A is open
then A9 in Family_open_set M by PRE_TOPC:def_2;
hence A is open by A1, Def3; ::_thesis: verum
end;
thus ( A is closed implies A9 is closed ) ::_thesis: ( A9 is closed implies A is closed )
proof
assume A is closed ; ::_thesis: A9 is closed
then A ` is open by Def4;
then A ` in Family_open_set M by Def3;
then A9 ` is open by A1, PRE_TOPC:def_2;
hence A9 is closed by TOPS_1:3; ::_thesis: verum
end;
assume A9 is closed ; ::_thesis: A is closed
then A ` in Family_open_set M by A1, PRE_TOPC:def_2;
then A ` is open by Def3;
hence A is closed by Def4; ::_thesis: verum
end;
definition
let T be TopStruct ;
let S be SetSequence of T;
attrS is open means :Def5: :: COMPL_SP:def 5
for i being Nat holds S . i is open ;
attrS is closed means :Def6: :: COMPL_SP:def 6
for i being Nat holds S . i is closed ;
end;
:: deftheorem Def5 defines open COMPL_SP:def_5_:_
for T being TopStruct
for S being SetSequence of T holds
( S is open iff for i being Nat holds S . i is open );
:: deftheorem Def6 defines closed COMPL_SP:def_6_:_
for T being TopStruct
for S being SetSequence of T holds
( S is closed iff for i being Nat holds S . i is closed );
Lm1: for T being TopSpace ex S being SetSequence of T st
( S is open & S is closed & ( not T is empty implies S is non-empty ) )
proof
let T be TopSpace; ::_thesis: ex S being SetSequence of T st
( S is open & S is closed & ( not T is empty implies S is non-empty ) )
take S = NAT --> ([#] T); ::_thesis: ( S is open & S is closed & ( not T is empty implies S is non-empty ) )
A1: now__::_thesis:_for_i_being_Nat_holds_S_._i_is_closed
let i be Nat; ::_thesis: S . i is closed
i in NAT by ORDINAL1:def_12;
hence S . i is closed by FUNCOP_1:7; ::_thesis: verum
end;
now__::_thesis:_for_i_being_Nat_holds_S_._i_is_open
let i be Nat; ::_thesis: S . i is open
i in NAT by ORDINAL1:def_12;
hence S . i is open by FUNCOP_1:7; ::_thesis: verum
end;
hence ( S is open & S is closed ) by A1, Def5, Def6; ::_thesis: ( not T is empty implies S is non-empty )
assume not T is empty ; ::_thesis: S is non-empty
then for x being set st x in dom S holds
not S . x is empty by FUNCOP_1:7;
hence S is non-empty by FUNCT_1:def_9; ::_thesis: verum
end;
registration
let T be TopSpace;
cluster non empty Relation-like NAT -defined bool the carrier of T -valued Function-like total V27( NAT , bool the carrier of T) open for Element of bool [:NAT,(bool the carrier of T):];
existence
ex b1 being SetSequence of T st b1 is open
proof
ex S being SetSequence of T st
( S is open & S is closed & ( not T is empty implies S is non-empty ) ) by Lm1;
hence ex b1 being SetSequence of T st b1 is open ; ::_thesis: verum
end;
cluster non empty Relation-like NAT -defined bool the carrier of T -valued Function-like total V27( NAT , bool the carrier of T) closed for Element of bool [:NAT,(bool the carrier of T):];
existence
ex b1 being SetSequence of T st b1 is closed
proof
ex S being SetSequence of T st
( S is open & S is closed & ( not T is empty implies S is non-empty ) ) by Lm1;
hence ex b1 being SetSequence of T st b1 is closed ; ::_thesis: verum
end;
end;
registration
let T be non empty TopSpace;
cluster non empty Relation-like non-empty NAT -defined bool the carrier of T -valued Function-like total V27( NAT , bool the carrier of T) open for Element of bool [:NAT,(bool the carrier of T):];
existence
ex b1 being SetSequence of T st
( b1 is open & b1 is non-empty )
proof
ex S being SetSequence of T st
( S is open & S is closed & ( not T is empty implies S is non-empty ) ) by Lm1;
hence ex b1 being SetSequence of T st
( b1 is open & b1 is non-empty ) ; ::_thesis: verum
end;
cluster non empty Relation-like non-empty NAT -defined bool the carrier of T -valued Function-like total V27( NAT , bool the carrier of T) closed for Element of bool [:NAT,(bool the carrier of T):];
existence
ex b1 being SetSequence of T st
( b1 is closed & b1 is non-empty )
proof
ex S being SetSequence of T st
( S is open & S is closed & ( not T is empty implies S is non-empty ) ) by Lm1;
hence ex b1 being SetSequence of T st
( b1 is closed & b1 is non-empty ) ; ::_thesis: verum
end;
end;
definition
let M be MetrStruct ;
let S be SetSequence of M;
attrS is open means :Def7: :: COMPL_SP:def 7
for i being Nat holds S . i is open ;
attrS is closed means :Def8: :: COMPL_SP:def 8
for i being Nat holds S . i is closed ;
end;
:: deftheorem Def7 defines open COMPL_SP:def_7_:_
for M being MetrStruct
for S being SetSequence of M holds
( S is open iff for i being Nat holds S . i is open );
:: deftheorem Def8 defines closed COMPL_SP:def_8_:_
for M being MetrStruct
for S being SetSequence of M holds
( S is closed iff for i being Nat holds S . i is closed );
registration
let M be non empty MetrSpace;
cluster non empty Relation-like non-empty NAT -defined bool the carrier of M -valued Function-like total V27( NAT , bool the carrier of M) pointwise_bounded open for Element of bool [:NAT,(bool the carrier of M):];
existence
ex b1 being SetSequence of M st
( b1 is non-empty & b1 is pointwise_bounded & b1 is open )
proof
consider x being set such that
A1: x in the carrier of M by XBOOLE_0:def_1;
reconsider x = x as Point of M by A1;
set B = Ball (x,1);
take S = NAT --> (Ball (x,1)); ::_thesis: ( S is non-empty & S is pointwise_bounded & S is open )
A2: now__::_thesis:_for_y_being_set_st_y_in_dom_S_holds_
not_S_._y_is_empty
let y be set ; ::_thesis: ( y in dom S implies not S . y is empty )
assume y in dom S ; ::_thesis: not S . y is empty
then reconsider n = y as Element of NAT ;
A3: Ball (x,1) = S . n by FUNCOP_1:7;
dist (x,x) = 0 by METRIC_1:1;
hence not S . y is empty by A3, METRIC_1:11; ::_thesis: verum
end;
A4: now__::_thesis:_for_i_being_Nat_holds_S_._i_is_open
let i be Nat; ::_thesis: S . i is open
i in NAT by ORDINAL1:def_12;
then A5: S . i = Ball (x,1) by FUNCOP_1:7;
Ball (x,1) in Family_open_set M by PCOMPS_1:29;
hence S . i is open by A5, Def3; ::_thesis: verum
end;
now__::_thesis:_for_i_being_Nat_holds_S_._i_is_bounded
let i be Nat; ::_thesis: S . i is bounded
i in NAT by ORDINAL1:def_12;
hence S . i is bounded by FUNCOP_1:7; ::_thesis: verum
end;
hence ( S is non-empty & S is pointwise_bounded & S is open ) by A2, A4, Def1, Def7, FUNCT_1:def_9; ::_thesis: verum
end;
cluster non empty Relation-like non-empty NAT -defined bool the carrier of M -valued Function-like total V27( NAT , bool the carrier of M) pointwise_bounded closed for Element of bool [:NAT,(bool the carrier of M):];
existence
ex b1 being SetSequence of M st
( b1 is non-empty & b1 is pointwise_bounded & b1 is closed )
proof
consider x being set such that
A6: x in the carrier of M by XBOOLE_0:def_1;
reconsider x = x as Point of M by A6;
set B = cl_Ball (x,1);
take S = NAT --> (cl_Ball (x,1)); ::_thesis: ( S is non-empty & S is pointwise_bounded & S is closed )
A7: now__::_thesis:_for_y_being_set_st_y_in_dom_S_holds_
not_S_._y_is_empty
let y be set ; ::_thesis: ( y in dom S implies not S . y is empty )
assume y in dom S ; ::_thesis: not S . y is empty
then reconsider n = y as Element of NAT ;
A8: cl_Ball (x,1) = S . n by FUNCOP_1:7;
dist (x,x) = 0 by METRIC_1:1;
hence not S . y is empty by A8, METRIC_1:12; ::_thesis: verum
end;
A9: now__::_thesis:_for_i_being_Nat_holds_S_._i_is_closed
let i be Nat; ::_thesis: S . i is closed
i in NAT by ORDINAL1:def_12;
then A10: S . i = cl_Ball (x,1) by FUNCOP_1:7;
([#] M) \ (cl_Ball (x,1)) in Family_open_set M by NAGATA_1:14;
then (cl_Ball (x,1)) ` is open by Def3;
hence S . i is closed by A10, Def4; ::_thesis: verum
end;
now__::_thesis:_for_i_being_Nat_holds_S_._i_is_bounded
let i be Nat; ::_thesis: S . i is bounded
A11: i in NAT by ORDINAL1:def_12;
cl_Ball (x,1) is bounded by TOPREAL6:59;
hence S . i is bounded by A11, FUNCOP_1:7; ::_thesis: verum
end;
hence ( S is non-empty & S is pointwise_bounded & S is closed ) by A7, A9, Def1, Def8, FUNCT_1:def_9; ::_thesis: verum
end;
end;
theorem Th7: :: COMPL_SP:7
for M being MetrStruct
for S being SetSequence of M
for S9 being SetSequence of (TopSpaceMetr M) st S9 = S holds
( ( S is open implies S9 is open ) & ( S9 is open implies S is open ) & ( S is closed implies S9 is closed ) & ( S9 is closed implies S is closed ) )
proof
let M be MetrStruct ; ::_thesis: for S being SetSequence of M
for S9 being SetSequence of (TopSpaceMetr M) st S9 = S holds
( ( S is open implies S9 is open ) & ( S9 is open implies S is open ) & ( S is closed implies S9 is closed ) & ( S9 is closed implies S is closed ) )
let S be SetSequence of M; ::_thesis: for S9 being SetSequence of (TopSpaceMetr M) st S9 = S holds
( ( S is open implies S9 is open ) & ( S9 is open implies S is open ) & ( S is closed implies S9 is closed ) & ( S9 is closed implies S is closed ) )
let S9 be SetSequence of (TopSpaceMetr M); ::_thesis: ( S9 = S implies ( ( S is open implies S9 is open ) & ( S9 is open implies S is open ) & ( S is closed implies S9 is closed ) & ( S9 is closed implies S is closed ) ) )
assume A1: S9 = S ; ::_thesis: ( ( S is open implies S9 is open ) & ( S9 is open implies S is open ) & ( S is closed implies S9 is closed ) & ( S9 is closed implies S is closed ) )
thus ( S is open implies S9 is open ) ::_thesis: ( ( S9 is open implies S is open ) & ( S is closed implies S9 is closed ) & ( S9 is closed implies S is closed ) )
proof
assume A2: S is open ; ::_thesis: S9 is open
let i be Nat; :: according to COMPL_SP:def_5 ::_thesis: S9 . i is open
S . i is open by A2, Def7;
hence S9 . i is open by A1, Th6; ::_thesis: verum
end;
thus ( S9 is open implies S is open ) ::_thesis: ( S is closed iff S9 is closed )
proof
assume A3: S9 is open ; ::_thesis: S is open
let i be Nat; :: according to COMPL_SP:def_7 ::_thesis: S . i is open
S9 . i is open by A3, Def5;
hence S . i is open by A1, Th6; ::_thesis: verum
end;
thus ( S is closed implies S9 is closed ) ::_thesis: ( S9 is closed implies S is closed )
proof
assume A4: S is closed ; ::_thesis: S9 is closed
let i be Nat; :: according to COMPL_SP:def_6 ::_thesis: S9 . i is closed
S . i is closed by A4, Def8;
hence S9 . i is closed by A1, Th6; ::_thesis: verum
end;
assume A5: S9 is closed ; ::_thesis: S is closed
let i be Nat; :: according to COMPL_SP:def_8 ::_thesis: S . i is closed
S9 . i is closed by A5, Def6;
hence S . i is closed by A1, Th6; ::_thesis: verum
end;
theorem Th8: :: COMPL_SP:8
for M being non empty Reflexive symmetric triangle MetrStruct
for S, CL being Subset of M st S is bounded holds
for S9 being Subset of (TopSpaceMetr M) st S = S9 & CL = Cl S9 holds
( CL is bounded & diameter S = diameter CL )
proof
let M be non empty Reflexive symmetric triangle MetrStruct ; ::_thesis: for S, CL being Subset of M st S is bounded holds
for S9 being Subset of (TopSpaceMetr M) st S = S9 & CL = Cl S9 holds
( CL is bounded & diameter S = diameter CL )
let S, CL be Subset of M; ::_thesis: ( S is bounded implies for S9 being Subset of (TopSpaceMetr M) st S = S9 & CL = Cl S9 holds
( CL is bounded & diameter S = diameter CL ) )
assume A1: S is bounded ; ::_thesis: for S9 being Subset of (TopSpaceMetr M) st S = S9 & CL = Cl S9 holds
( CL is bounded & diameter S = diameter CL )
set d = diameter S;
set T = TopSpaceMetr M;
let S9 be Subset of (TopSpaceMetr M); ::_thesis: ( S = S9 & CL = Cl S9 implies ( CL is bounded & diameter S = diameter CL ) )
assume that
A2: S = S9 and
A3: CL = Cl S9 ; ::_thesis: ( CL is bounded & diameter S = diameter CL )
percases ( S <> {} or S = {} ) ;
supposeA4: S <> {} ; ::_thesis: ( CL is bounded & diameter S = diameter CL )
A5: now__::_thesis:_for_x,_y_being_Point_of_M_st_x_in_CL_&_y_in_CL_holds_
not_dist_(x,y)_>_diameter_S
let x, y be Point of M; ::_thesis: ( x in CL & y in CL implies not dist (x,y) > diameter S )
assume that
A6: x in CL and
A7: y in CL ; ::_thesis: not dist (x,y) > diameter S
reconsider X = x, Y = y as Point of (TopSpaceMetr M) ;
set dxy = dist (x,y);
set dd = (dist (x,y)) - (diameter S);
set dd2 = ((dist (x,y)) - (diameter S)) / 2;
set Bx = Ball (x,(((dist (x,y)) - (diameter S)) / 2));
set By = Ball (y,(((dist (x,y)) - (diameter S)) / 2));
reconsider BX = Ball (x,(((dist (x,y)) - (diameter S)) / 2)), BY = Ball (y,(((dist (x,y)) - (diameter S)) / 2)) as Subset of (TopSpaceMetr M) ;
assume dist (x,y) > diameter S ; ::_thesis: contradiction
then (dist (x,y)) - (diameter S) > (diameter S) - (diameter S) by XREAL_1:14;
then A8: ((dist (x,y)) - (diameter S)) / 2 > 0 / 2 by XREAL_1:74;
Ball (y,(((dist (x,y)) - (diameter S)) / 2)) in Family_open_set M by PCOMPS_1:29;
then A9: BY is open by PRE_TOPC:def_2;
Ball (x,(((dist (x,y)) - (diameter S)) / 2)) in Family_open_set M by PCOMPS_1:29;
then A10: BX is open by PRE_TOPC:def_2;
dist (y,y) = 0 by METRIC_1:1;
then Y in BY by A8, METRIC_1:11;
then BY meets S9 by A3, A7, A9, TOPS_1:12;
then consider y1 being set such that
A11: y1 in BY and
A12: y1 in S9 by XBOOLE_0:3;
dist (x,x) = 0 by METRIC_1:1;
then X in BX by A8, METRIC_1:11;
then BX meets S9 by A3, A6, A10, TOPS_1:12;
then consider x1 being set such that
A13: x1 in BX and
A14: x1 in S9 by XBOOLE_0:3;
reconsider x1 = x1, y1 = y1 as Point of M by A13, A11;
A15: dist (x,x1) < ((dist (x,y)) - (diameter S)) / 2 by A13, METRIC_1:11;
dist (x1,y1) <= diameter S by A1, A2, A14, A12, TBSP_1:def_8;
then A16: (dist (x,x1)) + (dist (x1,y1)) < (((dist (x,y)) - (diameter S)) / 2) + (diameter S) by A15, XREAL_1:8;
A17: dist (y,y1) < ((dist (x,y)) - (diameter S)) / 2 by A11, METRIC_1:11;
dist (x,y1) <= (dist (x,x1)) + (dist (x1,y1)) by METRIC_1:4;
then dist (x,y1) < (((dist (x,y)) - (diameter S)) / 2) + (diameter S) by A16, XXREAL_0:2;
then (dist (x,y1)) + (dist (y1,y)) < ((((dist (x,y)) - (diameter S)) / 2) + (diameter S)) + (((dist (x,y)) - (diameter S)) / 2) by A17, XREAL_1:8;
hence contradiction by METRIC_1:4; ::_thesis: verum
end;
A18: now__::_thesis:_for_x,_y_being_Point_of_M_st_x_in_CL_&_y_in_CL_holds_
dist_(x,y)_<=_(diameter_S)_+_1
A19: (diameter S) + 0 < (diameter S) + 1 by XREAL_1:8;
let x, y be Point of M; ::_thesis: ( x in CL & y in CL implies dist (x,y) <= (diameter S) + 1 )
assume that
A20: x in CL and
A21: y in CL ; ::_thesis: dist (x,y) <= (diameter S) + 1
dist (x,y) <= diameter S by A5, A20, A21;
hence dist (x,y) <= (diameter S) + 1 by A19, XXREAL_0:2; ::_thesis: verum
end;
A22: now__::_thesis:_for_s_being_Real_st_(_for_x,_y_being_Point_of_M_st_x_in_CL_&_y_in_CL_holds_
dist_(x,y)_<=_s_)_holds_
diameter_S_<=_s
let s be Real; ::_thesis: ( ( for x, y being Point of M st x in CL & y in CL holds
dist (x,y) <= s ) implies diameter S <= s )
assume A23: for x, y being Point of M st x in CL & y in CL holds
dist (x,y) <= s ; ::_thesis: diameter S <= s
now__::_thesis:_for_x,_y_being_Point_of_M_st_x_in_S_&_y_in_S_holds_
dist_(x,y)_<=_s
let x, y be Point of M; ::_thesis: ( x in S & y in S implies dist (x,y) <= s )
assume that
A24: x in S and
A25: y in S ; ::_thesis: dist (x,y) <= s
S c= CL by A2, A3, PRE_TOPC:18;
hence dist (x,y) <= s by A23, A24, A25; ::_thesis: verum
end;
hence diameter S <= s by A1, A4, TBSP_1:def_8; ::_thesis: verum
end;
A26: CL <> {} by A2, A3, A4, PCOMPS_1:2;
(diameter S) + 1 > 0 + 0 by A1, TBSP_1:21, XREAL_1:8;
then CL is bounded by A18, TBSP_1:def_7;
hence ( CL is bounded & diameter S = diameter CL ) by A26, A5, A22, TBSP_1:def_8; ::_thesis: verum
end;
suppose S = {} ; ::_thesis: ( CL is bounded & diameter S = diameter CL )
hence ( CL is bounded & diameter S = diameter CL ) by A2, A3, PCOMPS_1:2; ::_thesis: verum
end;
end;
end;
begin
theorem Th9: :: COMPL_SP:9
for M being non empty MetrSpace
for C being sequence of M ex S being non-empty closed SetSequence of M st
( S is V172() & ( C is Cauchy implies ( S is pointwise_bounded & lim (diameter S) = 0 ) ) & ( for i being Nat ex U being Subset of (TopSpaceMetr M) st
( U = { (C . j) where j is Element of NAT : j >= i } & S . i = Cl U ) ) )
proof
let M be non empty MetrSpace; ::_thesis: for C being sequence of M ex S being non-empty closed SetSequence of M st
( S is V172() & ( C is Cauchy implies ( S is pointwise_bounded & lim (diameter S) = 0 ) ) & ( for i being Nat ex U being Subset of (TopSpaceMetr M) st
( U = { (C . j) where j is Element of NAT : j >= i } & S . i = Cl U ) ) )
set T = TopSpaceMetr M;
let C be sequence of M; ::_thesis: ex S being non-empty closed SetSequence of M st
( S is V172() & ( C is Cauchy implies ( S is pointwise_bounded & lim (diameter S) = 0 ) ) & ( for i being Nat ex U being Subset of (TopSpaceMetr M) st
( U = { (C . j) where j is Element of NAT : j >= i } & S . i = Cl U ) ) )
defpred S1[ set , set ] means for i being Nat st i = $1 holds
ex S being Subset of (TopSpaceMetr M) st
( S = { (C . j) where j is Element of NAT : j >= i } & $2 = Cl S );
A1: for x being set st x in NAT holds
ex y being set st
( y in bool the carrier of M & S1[x,y] )
proof
let x be set ; ::_thesis: ( x in NAT implies ex y being set st
( y in bool the carrier of M & S1[x,y] ) )
assume x in NAT ; ::_thesis: ex y being set st
( y in bool the carrier of M & S1[x,y] )
then reconsider x9 = x as Element of NAT ;
set S = { (C . j) where j is Element of NAT : j >= x9 } ;
{ (C . j) where j is Element of NAT : j >= x9 } c= the carrier of (TopSpaceMetr M)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in { (C . j) where j is Element of NAT : j >= x9 } or y in the carrier of (TopSpaceMetr M) )
assume y in { (C . j) where j is Element of NAT : j >= x9 } ; ::_thesis: y in the carrier of (TopSpaceMetr M)
then ex j being Element of NAT st
( C . j = y & j >= x9 ) ;
hence y in the carrier of (TopSpaceMetr M) ; ::_thesis: verum
end;
then reconsider S = { (C . j) where j is Element of NAT : j >= x9 } as Subset of (TopSpaceMetr M) ;
take Cl S ; ::_thesis: ( Cl S in bool the carrier of M & S1[x, Cl S] )
thus ( Cl S in bool the carrier of M & S1[x, Cl S] ) ; ::_thesis: verum
end;
consider S being SetSequence of M such that
A2: for x being set st x in NAT holds
S1[x,S . x] from FUNCT_2:sch_1(A1);
A3: now__::_thesis:_for_x_being_set_st_x_in_dom_S_holds_
not_S_._x_is_empty
let x be set ; ::_thesis: ( x in dom S implies not S . x is empty )
assume x in dom S ; ::_thesis: not S . x is empty
then reconsider i = x as Element of NAT ;
consider U being Subset of (TopSpaceMetr M) such that
A4: U = { (C . j) where j is Element of NAT : j >= i } and
A5: S . i = Cl U by A2;
A6: U c= S . i by A5, PRE_TOPC:18;
C . i in U by A4;
hence not S . x is empty by A6; ::_thesis: verum
end;
now__::_thesis:_for_i_being_Nat_holds_S_._i_is_closed
let i be Nat; ::_thesis: S . i is closed
i in NAT by ORDINAL1:def_12;
then ex U being Subset of (TopSpaceMetr M) st
( U = { (C . j) where j is Element of NAT : j >= i } & S . i = Cl U ) by A2;
hence S . i is closed by Th6; ::_thesis: verum
end;
then reconsider S = S as non-empty closed SetSequence of M by A3, Def8, FUNCT_1:def_9;
take S ; ::_thesis: ( S is V172() & ( C is Cauchy implies ( S is pointwise_bounded & lim (diameter S) = 0 ) ) & ( for i being Nat ex U being Subset of (TopSpaceMetr M) st
( U = { (C . j) where j is Element of NAT : j >= i } & S . i = Cl U ) ) )
now__::_thesis:_for_i_being_Element_of_NAT_holds_S_._(i_+_1)_c=_S_._i
let i be Element of NAT ; ::_thesis: S . (i + 1) c= S . i
consider U being Subset of (TopSpaceMetr M) such that
A7: U = { (C . j) where j is Element of NAT : j >= i } and
A8: S . i = Cl U by A2;
consider U1 being Subset of (TopSpaceMetr M) such that
A9: U1 = { (C . j) where j is Element of NAT : j >= i + 1 } and
A10: S . (i + 1) = Cl U1 by A2;
U1 c= U
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in U1 or x in U )
assume x in U1 ; ::_thesis: x in U
then consider j being Element of NAT such that
A11: x = C . j and
A12: j >= i + 1 by A9;
j >= i by A12, NAT_1:13;
hence x in U by A7, A11; ::_thesis: verum
end;
hence S . (i + 1) c= S . i by A8, A10, PRE_TOPC:19; ::_thesis: verum
end;
hence A13: S is V172() by KURATO_0:def_3; ::_thesis: ( ( C is Cauchy implies ( S is pointwise_bounded & lim (diameter S) = 0 ) ) & ( for i being Nat ex U being Subset of (TopSpaceMetr M) st
( U = { (C . j) where j is Element of NAT : j >= i } & S . i = Cl U ) ) )
thus ( C is Cauchy implies ( S is pointwise_bounded & lim (diameter S) = 0 ) ) ::_thesis: for i being Nat ex U being Subset of (TopSpaceMetr M) st
( U = { (C . j) where j is Element of NAT : j >= i } & S . i = Cl U )
proof
assume A14: C is Cauchy ; ::_thesis: ( S is pointwise_bounded & lim (diameter S) = 0 )
A15: now__::_thesis:_for_i_being_Nat_holds_S_._i_is_bounded
let i be Nat; ::_thesis: S . i is bounded
i in NAT by ORDINAL1:def_12;
then consider U being Subset of (TopSpaceMetr M) such that
A16: U = { (C . j) where j is Element of NAT : j >= i } and
A17: S . i = Cl U by A2;
reconsider U9 = U as Subset of M ;
U c= rng C
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in U or x in rng C )
assume x in U ; ::_thesis: x in rng C
then A18: ex j being Element of NAT st
( x = C . j & j >= i ) by A16;
dom C = NAT by FUNCT_2:def_1;
hence x in rng C by A18, FUNCT_1:def_3; ::_thesis: verum
end;
then U9 is bounded by A14, TBSP_1:14, TBSP_1:26;
hence S . i is bounded by A17, Th8; ::_thesis: verum
end;
then reconsider S9 = S as non-empty pointwise_bounded closed SetSequence of M by Def1;
set d = diameter S9;
A19: for r being real number st 0 < r holds
ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs (((diameter S9) . m) - 0) < r
proof
let r be real number ; ::_thesis: ( 0 < r implies ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs (((diameter S9) . m) - 0) < r )
assume A20: 0 < r ; ::_thesis: ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs (((diameter S9) . m) - 0) < r
reconsider R = r as Real by XREAL_0:def_1;
set R2 = R / 2;
R / 2 > 0 by A20, XREAL_1:139;
then consider p being Element of NAT such that
A21: for n, m being Element of NAT st p <= n & p <= m holds
dist ((C . n),(C . m)) < R / 2 by A14, TBSP_1:def_4;
take p ; ::_thesis: for m being Element of NAT st p <= m holds
abs (((diameter S9) . m) - 0) < r
let m be Element of NAT ; ::_thesis: ( p <= m implies abs (((diameter S9) . m) - 0) < r )
assume A22: p <= m ; ::_thesis: abs (((diameter S9) . m) - 0) < r
consider U being Subset of (TopSpaceMetr M) such that
A23: U = { (C . j) where j is Element of NAT : j >= m } and
A24: S . m = Cl U by A2;
reconsider U9 = U as Subset of M ;
A25: now__::_thesis:_for_x,_y_being_Point_of_M_st_x_in_U9_&_y_in_U9_holds_
dist_(x,y)_<=_R_/_2
let x, y be Point of M; ::_thesis: ( x in U9 & y in U9 implies dist (x,y) <= R / 2 )
assume that
A26: x in U9 and
A27: y in U9 ; ::_thesis: dist (x,y) <= R / 2
consider j being Element of NAT such that
A28: y = C . j and
A29: j >= m by A23, A27;
A30: j >= p by A22, A29, XXREAL_0:2;
consider i being Element of NAT such that
A31: x = C . i and
A32: i >= m by A23, A26;
i >= p by A22, A32, XXREAL_0:2;
hence dist (x,y) <= R / 2 by A21, A31, A28, A30; ::_thesis: verum
end;
A33: U c= rng C
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in U or x in rng C )
assume x in U ; ::_thesis: x in rng C
then A34: ex j being Element of NAT st
( x = C . j & j >= m ) by A23;
dom C = NAT by FUNCT_2:def_1;
hence x in rng C by A34, FUNCT_1:def_3; ::_thesis: verum
end;
then A35: U9 is bounded by A14, TBSP_1:14, TBSP_1:26;
then A36: diameter U9 = diameter (S . m) by A24, Th8;
C . m in U by A23;
then A37: diameter U9 <= R / 2 by A35, A25, TBSP_1:def_8;
rng C is bounded by A14, TBSP_1:26;
then diameter (S . m) >= 0 by A33, A36, TBSP_1:14, TBSP_1:21;
then A38: abs (diameter (S . m)) <= R / 2 by A37, A36, ABSVALUE:def_1;
R / 2 < R by A20, XREAL_1:216;
then abs (diameter (S . m)) < R by A38, XXREAL_0:2;
hence abs (((diameter S9) . m) - 0) < r by Def2; ::_thesis: verum
end;
thus S is pointwise_bounded by A15, Def1; ::_thesis: lim (diameter S) = 0
A39: diameter S9 is bounded_below by Th1;
diameter S9 is V103() by A13, Th2;
hence lim (diameter S) = 0 by A39, A19, SEQ_2:def_7; ::_thesis: verum
end;
let i be Nat; ::_thesis: ex U being Subset of (TopSpaceMetr M) st
( U = { (C . j) where j is Element of NAT : j >= i } & S . i = Cl U )
i in NAT by ORDINAL1:def_12;
hence ex U being Subset of (TopSpaceMetr M) st
( U = { (C . j) where j is Element of NAT : j >= i } & S . i = Cl U ) by A2; ::_thesis: verum
end;
theorem Th10: :: COMPL_SP:10
for M being non empty MetrSpace holds
( M is complete iff for S being non-empty pointwise_bounded closed SetSequence of M st S is V172() & lim (diameter S) = 0 holds
not meet S is empty )
proof
let M be non empty MetrSpace; ::_thesis: ( M is complete iff for S being non-empty pointwise_bounded closed SetSequence of M st S is V172() & lim (diameter S) = 0 holds
not meet S is empty )
set T = TopSpaceMetr M;
thus ( M is complete implies for S being non-empty pointwise_bounded closed SetSequence of M st S is V172() & lim (diameter S) = 0 holds
not meet S is empty ) ::_thesis: ( ( for S being non-empty pointwise_bounded closed SetSequence of M st S is V172() & lim (diameter S) = 0 holds
not meet S is empty ) implies M is complete )
proof
assume A1: M is complete ; ::_thesis: for S being non-empty pointwise_bounded closed SetSequence of M st S is V172() & lim (diameter S) = 0 holds
not meet S is empty
let S be non-empty pointwise_bounded closed SetSequence of M; ::_thesis: ( S is V172() & lim (diameter S) = 0 implies not meet S is empty )
assume that
A2: S is V172() and
A3: lim (diameter S) = 0 ; ::_thesis: not meet S is empty
defpred S1[ set , set ] means $2 in S . $1;
A4: for x being set st x in NAT holds
ex y being set st
( y in the carrier of M & S1[x,y] )
proof
A5: dom S = NAT by FUNCT_2:def_1;
let x be set ; ::_thesis: ( x in NAT implies ex y being set st
( y in the carrier of M & S1[x,y] ) )
assume A6: x in NAT ; ::_thesis: ex y being set st
( y in the carrier of M & S1[x,y] )
not S . x is empty by A6, A5, FUNCT_1:def_9;
then A7: ex y being set st y in S . x by XBOOLE_0:def_1;
S . x in rng S by A6, A5, FUNCT_1:def_3;
hence ex y being set st
( y in the carrier of M & S1[x,y] ) by A7; ::_thesis: verum
end;
consider F being Function of NAT, the carrier of M such that
A8: for x being set st x in NAT holds
S1[x,F . x] from FUNCT_2:sch_1(A4);
now__::_thesis:_for_i_being_Nat_holds_F_._i_in_S_._i
let i be Nat; ::_thesis: F . i in S . i
i in NAT by ORDINAL1:def_12;
hence F . i in S . i by A8; ::_thesis: verum
end;
then F is Cauchy by A2, A3, Th4;
then F is convergent by A1, TBSP_1:def_5;
then consider x being Point of M such that
A9: F is_convergent_in_metrspace_to x by METRIC_6:10;
reconsider F9 = F as sequence of (TopSpaceMetr M) ;
reconsider x9 = x as Point of (TopSpaceMetr M) ;
now__::_thesis:_for_i_being_Element_of_NAT_holds_x_in_S_._i
let i be Element of NAT ; ::_thesis: x in S . i
set F1 = F9 ^\ i;
reconsider Si = S . i as Subset of (TopSpaceMetr M) ;
A10: rng (F9 ^\ i) c= Si
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng (F9 ^\ i) or x in Si )
assume x in rng (F9 ^\ i) ; ::_thesis: x in Si
then consider y being set such that
A11: y in dom (F9 ^\ i) and
A12: (F9 ^\ i) . y = x by FUNCT_1:def_3;
reconsider y = y as Element of NAT by A11;
i <= y + i by NAT_1:11;
then A13: S . (y + i) c= S . i by A2, PROB_1:def_4;
x = F . (y + i) by A12, NAT_1:def_3;
then x in S . (y + i) by A8;
hence x in Si by A13; ::_thesis: verum
end;
F9 is_convergent_to x9 by A9, FRECHET2:28;
then F9 ^\ i is_convergent_to x9 by FRECHET2:15;
then A14: x in Lim (F9 ^\ i) by FRECHET:def_5;
S . i is closed by Def8;
then Si is closed by Th6;
then Lim (F9 ^\ i) c= Si by A10, FRECHET2:9;
hence x in S . i by A14; ::_thesis: verum
end;
hence not meet S is empty by KURATO_0:3; ::_thesis: verum
end;
assume A15: for S being non-empty pointwise_bounded closed SetSequence of M st S is V172() & lim (diameter S) = 0 holds
not meet S is empty ; ::_thesis: M is complete
let F be sequence of M; :: according to TBSP_1:def_5 ::_thesis: ( not F is Cauchy or F is convergent )
assume A16: F is Cauchy ; ::_thesis: F is convergent
consider S being non-empty closed SetSequence of M such that
A17: S is V172() and
A18: ( F is Cauchy implies ( S is pointwise_bounded & lim (diameter S) = 0 ) ) and
A19: for i being Nat ex U being Subset of (TopSpaceMetr M) st
( U = { (F . j) where j is Element of NAT : j >= i } & S . i = Cl U ) by Th9;
set d = diameter S;
A20: diameter S is V103() by A16, A17, A18, Th2;
not meet S is empty by A15, A16, A17, A18;
then consider x being set such that
A21: x in meet S by XBOOLE_0:def_1;
A22: diameter S is bounded_below by A16, A18, Th1;
reconsider x = x as Point of M by A21;
take x ; :: according to TBSP_1:def_2 ::_thesis: for b1 being Element of REAL holds
( b1 <= 0 or ex b2 being Element of NAT st
for b3 being Element of NAT holds
( not b2 <= b3 or not b1 <= dist ((F . b3),x) ) )
let r be Real; ::_thesis: ( r <= 0 or ex b1 being Element of NAT st
for b2 being Element of NAT holds
( not b1 <= b2 or not r <= dist ((F . b2),x) ) )
assume r > 0 ; ::_thesis: ex b1 being Element of NAT st
for b2 being Element of NAT holds
( not b1 <= b2 or not r <= dist ((F . b2),x) )
then consider n being Element of NAT such that
A23: for m being Element of NAT st n <= m holds
abs (((diameter S) . m) - 0) < r by A16, A18, A22, A20, SEQ_2:def_7;
take n ; ::_thesis: for b1 being Element of NAT holds
( not n <= b1 or not r <= dist ((F . b1),x) )
let m be Element of NAT ; ::_thesis: ( not n <= m or not r <= dist ((F . m),x) )
assume n <= m ; ::_thesis: not r <= dist ((F . m),x)
then A24: abs (((diameter S) . m) - 0) < r by A23;
A25: S . m is bounded by A16, A18, Def1;
A26: x in S . m by A21, KURATO_0:3;
A27: diameter (S . m) = (diameter S) . m by Def2;
consider U being Subset of (TopSpaceMetr M) such that
A28: U = { (F . j) where j is Element of NAT : j >= m } and
A29: S . m = Cl U by A19;
A30: U c= Cl U by PRE_TOPC:18;
F . m in U by A28;
then A31: dist ((F . m),x) <= diameter (S . m) by A29, A30, A26, A25, TBSP_1:def_8;
diameter (S . m) >= 0 by A25, TBSP_1:21;
then (diameter S) . m < r by A27, A24, ABSVALUE:def_1;
hence not r <= dist ((F . m),x) by A31, A27, XXREAL_0:2; ::_thesis: verum
end;
theorem Th11: :: COMPL_SP:11
for T being non empty TopSpace
for S being non-empty SetSequence of T st S is V172() holds
for F being Subset-Family of T st F = rng S holds
F is centered
proof
let T be non empty TopSpace; ::_thesis: for S being non-empty SetSequence of T st S is V172() holds
for F being Subset-Family of T st F = rng S holds
F is centered
let S be non-empty SetSequence of T; ::_thesis: ( S is V172() implies for F being Subset-Family of T st F = rng S holds
F is centered )
assume A1: S is V172() ; ::_thesis: for F being Subset-Family of T st F = rng S holds
F is centered
let F be Subset-Family of T; ::_thesis: ( F = rng S implies F is centered )
assume A2: F = rng S ; ::_thesis: F is centered
A3: now__::_thesis:_for_G_being_set_st_G_<>_{}_&_G_c=_F_&_G_is_finite_holds_
meet_G_<>_{}
defpred S1[ set , set ] means $1 = S . $2;
let G be set ; ::_thesis: ( G <> {} & G c= F & G is finite implies meet G <> {} )
assume that
A4: G <> {} and
A5: G c= F and
A6: G is finite ; ::_thesis: meet G <> {}
A7: for x being set st x in G holds
ex y being set st
( y in NAT & S1[x,y] )
proof
let x be set ; ::_thesis: ( x in G implies ex y being set st
( y in NAT & S1[x,y] ) )
assume x in G ; ::_thesis: ex y being set st
( y in NAT & S1[x,y] )
then ex y being set st
( y in dom S & S . y = x ) by A2, A5, FUNCT_1:def_3;
hence ex y being set st
( y in NAT & S1[x,y] ) ; ::_thesis: verum
end;
consider f being Function of G,NAT such that
A8: for x being set st x in G holds
S1[x,f . x] from FUNCT_2:sch_1(A7);
consider i being Nat such that
A9: for j being Nat st j in rng f holds
j <= i by A6, STIRL2_1:56;
A10: i in NAT by ORDINAL1:def_12;
dom S = NAT by FUNCT_2:def_1;
then S . i <> {} by A10, FUNCT_1:def_9;
then consider x being set such that
A11: x in S . i by XBOOLE_0:def_1;
A12: dom f = G by FUNCT_2:def_1;
now__::_thesis:_for_Y_being_set_st_Y_in_G_holds_
x_in_Y
let Y be set ; ::_thesis: ( Y in G implies x in Y )
assume A13: Y in G ; ::_thesis: x in Y
then A14: f . Y in rng f by A12, FUNCT_1:def_3;
then reconsider fY = f . Y as Element of NAT ;
A15: fY <= i by A9, A14;
Y = S . fY by A8, A13;
then S . i c= Y by A10, A1, A15, PROB_1:def_4;
hence x in Y by A11; ::_thesis: verum
end;
hence meet G <> {} by A4, SETFAM_1:def_1; ::_thesis: verum
end;
dom S = NAT by FUNCT_2:def_1;
then F <> {} by A2, RELAT_1:42;
hence F is centered by A3, FINSET_1:def_3; ::_thesis: verum
end;
theorem Th12: :: COMPL_SP:12
for M being non empty MetrStruct
for S being SetSequence of M
for F being Subset-Family of (TopSpaceMetr M) st F = rng S holds
( ( S is open implies F is open ) & ( S is closed implies F is closed ) )
proof
let M be non empty MetrStruct ; ::_thesis: for S being SetSequence of M
for F being Subset-Family of (TopSpaceMetr M) st F = rng S holds
( ( S is open implies F is open ) & ( S is closed implies F is closed ) )
let S be SetSequence of M; ::_thesis: for F being Subset-Family of (TopSpaceMetr M) st F = rng S holds
( ( S is open implies F is open ) & ( S is closed implies F is closed ) )
set T = TopSpaceMetr M;
let F be Subset-Family of (TopSpaceMetr M); ::_thesis: ( F = rng S implies ( ( S is open implies F is open ) & ( S is closed implies F is closed ) ) )
assume A1: F = rng S ; ::_thesis: ( ( S is open implies F is open ) & ( S is closed implies F is closed ) )
thus ( S is open implies F is open ) ::_thesis: ( S is closed implies F is closed )
proof
assume A2: S is open ; ::_thesis: F is open
let P be Subset of (TopSpaceMetr M); :: according to TOPS_2:def_1 ::_thesis: ( not P in F or P is open )
assume P in F ; ::_thesis: P is open
then consider x being set such that
A3: x in dom S and
A4: S . x = P by A1, FUNCT_1:def_3;
reconsider x = x as Nat by A3;
S . x is open by A2, Def7;
hence P is open by A4, Th6; ::_thesis: verum
end;
assume A5: S is closed ; ::_thesis: F is closed
let P be Subset of (TopSpaceMetr M); :: according to TOPS_2:def_2 ::_thesis: ( not P in F or P is closed )
assume P in F ; ::_thesis: P is closed
then consider x being set such that
A6: x in dom S and
A7: S . x = P by A1, FUNCT_1:def_3;
reconsider x = x as Nat by A6;
S . x is closed by A5, Def8;
hence P is closed by A7, Th6; ::_thesis: verum
end;
theorem Th13: :: COMPL_SP:13
for T being non empty TopSpace
for F being Subset-Family of T
for S being SetSequence of T st rng S c= F holds
ex R being SetSequence of T st
( R is V172() & ( F is centered implies R is non-empty ) & ( F is open implies R is open ) & ( F is closed implies R is closed ) & ( for i being Nat holds R . i = meet { (S . j) where j is Element of NAT : j <= i } ) )
proof
let T be non empty TopSpace; ::_thesis: for F being Subset-Family of T
for S being SetSequence of T st rng S c= F holds
ex R being SetSequence of T st
( R is V172() & ( F is centered implies R is non-empty ) & ( F is open implies R is open ) & ( F is closed implies R is closed ) & ( for i being Nat holds R . i = meet { (S . j) where j is Element of NAT : j <= i } ) )
let F be Subset-Family of T; ::_thesis: for S being SetSequence of T st rng S c= F holds
ex R being SetSequence of T st
( R is V172() & ( F is centered implies R is non-empty ) & ( F is open implies R is open ) & ( F is closed implies R is closed ) & ( for i being Nat holds R . i = meet { (S . j) where j is Element of NAT : j <= i } ) )
let S be SetSequence of T; ::_thesis: ( rng S c= F implies ex R being SetSequence of T st
( R is V172() & ( F is centered implies R is non-empty ) & ( F is open implies R is open ) & ( F is closed implies R is closed ) & ( for i being Nat holds R . i = meet { (S . j) where j is Element of NAT : j <= i } ) ) )
assume A1: rng S c= F ; ::_thesis: ex R being SetSequence of T st
( R is V172() & ( F is centered implies R is non-empty ) & ( F is open implies R is open ) & ( F is closed implies R is closed ) & ( for i being Nat holds R . i = meet { (S . j) where j is Element of NAT : j <= i } ) )
A2: for i being Nat holds { (S . j) where j is Element of NAT : j <= i } c= F
proof
let i be Nat; ::_thesis: { (S . j) where j is Element of NAT : j <= i } c= F
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { (S . j) where j is Element of NAT : j <= i } or x in F )
assume x in { (S . j) where j is Element of NAT : j <= i } ; ::_thesis: x in F
then A3: ex j being Element of NAT st
( x = S . j & j <= i ) ;
dom S = NAT by FUNCT_2:def_1;
then x in rng S by A3, FUNCT_1:def_3;
hence x in F by A1; ::_thesis: verum
end;
defpred S1[ set , set ] means for i being Nat st i = $1 holds
$2 = meet { (S . j) where j is Element of NAT : j <= i } ;
A4: for x being set st x in NAT holds
ex y being set st
( y in bool the carrier of T & S1[x,y] )
proof
let x be set ; ::_thesis: ( x in NAT implies ex y being set st
( y in bool the carrier of T & S1[x,y] ) )
assume x in NAT ; ::_thesis: ex y being set st
( y in bool the carrier of T & S1[x,y] )
then reconsider i = x as Element of NAT ;
set SS = { (S . j) where j is Element of NAT : j <= i } ;
{ (S . j) where j is Element of NAT : j <= i } c= F by A2;
then reconsider SS = { (S . j) where j is Element of NAT : j <= i } as Subset-Family of T by XBOOLE_1:1;
take meet SS ; ::_thesis: ( meet SS in bool the carrier of T & S1[x, meet SS] )
thus ( meet SS in bool the carrier of T & S1[x, meet SS] ) ; ::_thesis: verum
end;
consider R being SetSequence of T such that
A5: for x being set st x in NAT holds
S1[x,R . x] from FUNCT_2:sch_1(A4);
take R ; ::_thesis: ( R is V172() & ( F is centered implies R is non-empty ) & ( F is open implies R is open ) & ( F is closed implies R is closed ) & ( for i being Nat holds R . i = meet { (S . j) where j is Element of NAT : j <= i } ) )
now__::_thesis:_for_i_being_Element_of_NAT_holds_R_._(i_+_1)_c=_R_._i
let i be Element of NAT ; ::_thesis: R . (i + 1) c= R . i
set SS = { (S . j) where j is Element of NAT : j <= i } ;
set S1 = { (S . j) where j is Element of NAT : j <= i + 1 } ;
A6: { (S . j) where j is Element of NAT : j <= i } c= { (S . j) where j is Element of NAT : j <= i + 1 }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { (S . j) where j is Element of NAT : j <= i } or x in { (S . j) where j is Element of NAT : j <= i + 1 } )
assume x in { (S . j) where j is Element of NAT : j <= i } ; ::_thesis: x in { (S . j) where j is Element of NAT : j <= i + 1 }
then consider j being Element of NAT such that
A7: x = S . j and
A8: j <= i ;
j <= i + 1 by A8, NAT_1:13;
hence x in { (S . j) where j is Element of NAT : j <= i + 1 } by A7; ::_thesis: verum
end;
A9: meet { (S . j) where j is Element of NAT : j <= i } = R . i by A5;
S . 0 in { (S . j) where j is Element of NAT : j <= i } ;
then meet { (S . j) where j is Element of NAT : j <= i + 1 } c= meet { (S . j) where j is Element of NAT : j <= i } by A6, SETFAM_1:6;
hence R . (i + 1) c= R . i by A5, A9; ::_thesis: verum
end;
hence R is V172() by KURATO_0:def_3; ::_thesis: ( ( F is centered implies R is non-empty ) & ( F is open implies R is open ) & ( F is closed implies R is closed ) & ( for i being Nat holds R . i = meet { (S . j) where j is Element of NAT : j <= i } ) )
A10: for i being Nat holds { (S . j) where j is Element of NAT : j <= i } is finite
proof
deffunc H1( set ) -> set = S . $1;
let i be Nat; ::_thesis: { (S . j) where j is Element of NAT : j <= i } is finite
set SS = { (S . j) where j is Element of NAT : j <= i } ;
A11: { (S . j) where j is Element of NAT : j <= i } c= { H1(j) where j is Element of NAT : j in i + 1 }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { (S . j) where j is Element of NAT : j <= i } or x in { H1(j) where j is Element of NAT : j in i + 1 } )
assume x in { (S . j) where j is Element of NAT : j <= i } ; ::_thesis: x in { H1(j) where j is Element of NAT : j in i + 1 }
then consider j being Element of NAT such that
A12: x = S . j and
A13: j <= i ;
j < i + 1 by A13, NAT_1:13;
then j in i + 1 by NAT_1:44;
hence x in { H1(j) where j is Element of NAT : j in i + 1 } by A12; ::_thesis: verum
end;
A14: i + 1 is finite ;
{ H1(j) where j is Element of NAT : j in i + 1 } is finite from FRAENKEL:sch_21(A14);
hence { (S . j) where j is Element of NAT : j <= i } is finite by A11; ::_thesis: verum
end;
thus ( F is centered implies R is non-empty ) ::_thesis: ( ( F is open implies R is open ) & ( F is closed implies R is closed ) & ( for i being Nat holds R . i = meet { (S . j) where j is Element of NAT : j <= i } ) )
proof
assume A15: F is centered ; ::_thesis: R is non-empty
now__::_thesis:_for_x_being_set_st_x_in_dom_R_holds_
not_R_._x_is_empty
let x be set ; ::_thesis: ( x in dom R implies not R . x is empty )
assume x in dom R ; ::_thesis: not R . x is empty
then reconsider i = x as Element of NAT ;
set SS = { (S . j) where j is Element of NAT : j <= i } ;
A16: S . 0 in { (S . j) where j is Element of NAT : j <= i } ;
A17: { (S . j) where j is Element of NAT : j <= i } c= F by A2;
{ (S . j) where j is Element of NAT : j <= i } is finite by A10;
then meet { (S . j) where j is Element of NAT : j <= i } <> {} by A15, A16, A17, FINSET_1:def_3;
hence not R . x is empty by A5; ::_thesis: verum
end;
hence R is non-empty by FUNCT_1:def_9; ::_thesis: verum
end;
thus ( F is open implies R is open ) ::_thesis: ( ( F is closed implies R is closed ) & ( for i being Nat holds R . i = meet { (S . j) where j is Element of NAT : j <= i } ) )
proof
assume A18: F is open ; ::_thesis: R is open
let i be Nat; :: according to COMPL_SP:def_5 ::_thesis: R . i is open
set SS = { (S . j) where j is Element of NAT : j <= i } ;
A19: { (S . j) where j is Element of NAT : j <= i } c= F by A2;
then reconsider SS = { (S . j) where j is Element of NAT : j <= i } as Subset-Family of T by XBOOLE_1:1;
SS is finite by A10;
then A20: meet SS is open by A18, A19, TOPS_2:11, TOPS_2:20;
i in NAT by ORDINAL1:def_12;
hence R . i is open by A5, A20; ::_thesis: verum
end;
thus ( F is closed implies R is closed ) ::_thesis: for i being Nat holds R . i = meet { (S . j) where j is Element of NAT : j <= i }
proof
assume A21: F is closed ; ::_thesis: R is closed
let i be Nat; :: according to COMPL_SP:def_6 ::_thesis: R . i is closed
set SS = { (S . j) where j is Element of NAT : j <= i } ;
A22: i in NAT by ORDINAL1:def_12;
A23: { (S . j) where j is Element of NAT : j <= i } c= F by A2;
then reconsider SS = { (S . j) where j is Element of NAT : j <= i } as Subset-Family of T by XBOOLE_1:1;
meet SS is closed by A21, A23, TOPS_2:12, TOPS_2:22;
hence R . i is closed by A5, A22; ::_thesis: verum
end;
let i be Nat; ::_thesis: R . i = meet { (S . j) where j is Element of NAT : j <= i }
i in NAT by ORDINAL1:def_12;
hence R . i = meet { (S . j) where j is Element of NAT : j <= i } by A5; ::_thesis: verum
end;
theorem :: COMPL_SP:14
for M being non empty MetrSpace holds
( M is complete iff for F being Subset-Family of (TopSpaceMetr M) st F is closed & F is centered & ( for r being Real st r > 0 holds
ex A being Subset of M st
( A in F & A is bounded & diameter A < r ) ) holds
not meet F is empty )
proof
let M be non empty MetrSpace; ::_thesis: ( M is complete iff for F being Subset-Family of (TopSpaceMetr M) st F is closed & F is centered & ( for r being Real st r > 0 holds
ex A being Subset of M st
( A in F & A is bounded & diameter A < r ) ) holds
not meet F is empty )
set T = TopSpaceMetr M;
thus ( M is complete implies for F being Subset-Family of (TopSpaceMetr M) st F is closed & F is centered & ( for r being Real st r > 0 holds
ex A being Subset of M st
( A in F & A is bounded & diameter A < r ) ) holds
not meet F is empty ) ::_thesis: ( ( for F being Subset-Family of (TopSpaceMetr M) st F is closed & F is centered & ( for r being Real st r > 0 holds
ex A being Subset of M st
( A in F & A is bounded & diameter A < r ) ) holds
not meet F is empty ) implies M is complete )
proof
reconsider NULL = 0 as Real ;
deffunc H1( Element of NAT ) -> Element of REAL = 1 / (1 + $1);
assume A1: M is complete ; ::_thesis: for F being Subset-Family of (TopSpaceMetr M) st F is closed & F is centered & ( for r being Real st r > 0 holds
ex A being Subset of M st
( A in F & A is bounded & diameter A < r ) ) holds
not meet F is empty
consider seq being Real_Sequence such that
A2: for n being Element of NAT holds seq . n = H1(n) from SEQ_1:sch_1();
set Ns = NULL (#) seq;
let F be Subset-Family of (TopSpaceMetr M); ::_thesis: ( F is closed & F is centered & ( for r being Real st r > 0 holds
ex A being Subset of M st
( A in F & A is bounded & diameter A < r ) ) implies not meet F is empty )
assume that
A3: F is closed and
A4: F is centered and
A5: for r being Real st r > 0 holds
ex A being Subset of M st
( A in F & A is bounded & diameter A < r ) ; ::_thesis: not meet F is empty
A6: for n being Element of NAT holds seq . n = 1 / (n + 1) by A2;
then A7: NULL (#) seq is convergent by SEQ_2:7, SEQ_4:30;
defpred S1[ set , set ] means for i being Nat st i = $1 holds
for A being Subset of M st A = $2 holds
( A in F & A is bounded & diameter A < 1 / (i + 1) );
A8: for x being set st x in NAT holds
ex y being set st
( y in bool the carrier of M & S1[x,y] )
proof
let x be set ; ::_thesis: ( x in NAT implies ex y being set st
( y in bool the carrier of M & S1[x,y] ) )
assume x in NAT ; ::_thesis: ex y being set st
( y in bool the carrier of M & S1[x,y] )
then reconsider i = x as Element of NAT ;
consider A being Subset of M such that
A9: A in F and
A10: A is bounded and
A11: diameter A < 1 / (i + 1) by A5, XREAL_1:139;
take A ; ::_thesis: ( A in bool the carrier of M & S1[x,A] )
thus ( A in bool the carrier of M & S1[x,A] ) by A9, A10, A11; ::_thesis: verum
end;
consider f being SetSequence of M such that
A12: for x being set st x in NAT holds
S1[x,f . x] from FUNCT_2:sch_1(A8);
rng f c= F
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng f or x in F )
assume x in rng f ; ::_thesis: x in F
then consider y being set such that
A13: y in dom f and
A14: f . y = x by FUNCT_1:def_3;
reconsider y = y as Element of NAT by A13;
f . y in F by A12;
hence x in F by A14; ::_thesis: verum
end;
then consider R being SetSequence of (TopSpaceMetr M) such that
A15: R is V172() and
A16: ( F is centered implies R is non-empty ) and
( F is open implies R is open ) and
A17: ( F is closed implies R is closed ) and
A18: for i being Nat holds R . i = meet { (f . j) where j is Element of NAT : j <= i } by Th13;
reconsider R9 = R as non-empty SetSequence of M by A4, A16;
now__::_thesis:_for_i_being_Nat_holds_R9_._i_is_bounded
let i be Nat; ::_thesis: R9 . i is bounded
f . 0 in { (f . j) where j is Element of NAT : j <= i } ;
then meet { (f . j) where j is Element of NAT : j <= i } c= f . 0 by SETFAM_1:3;
then R . i c= f . 0 by A18;
hence R9 . i is bounded by A12, TBSP_1:14; ::_thesis: verum
end;
then reconsider R9 = R9 as non-empty pointwise_bounded SetSequence of M by Def1;
set dR = diameter R9;
A19: now__::_thesis:_for_n_being_Element_of_NAT_holds_
(_(NULL_(#)_seq)_._n_<=_(diameter_R9)_._n_&_(diameter_R9)_._n_<=_seq_._n_)
let n be Element of NAT ; ::_thesis: ( (NULL (#) seq) . n <= (diameter R9) . n & (diameter R9) . n <= seq . n )
set Sn = { (f . j) where j is Element of NAT : j <= n } ;
A20: f . n in { (f . j) where j is Element of NAT : j <= n } ;
R . n = meet { (f . j) where j is Element of NAT : j <= n } by A18;
then A21: R . n c= f . n by A20, SETFAM_1:3;
then diameter (R9 . n) <= diameter (f . n) by A12, TBSP_1:24;
then A22: diameter (R9 . n) <= H1(n) by A12, XXREAL_0:2;
f . n is bounded by A12;
then A23: 0 <= diameter (R9 . n) by A21, TBSP_1:14, TBSP_1:21;
A24: (NULL (#) seq) . n = NULL * (seq . n) by SEQ_1:9;
H1(n) = seq . n by A2;
hence ( (NULL (#) seq) . n <= (diameter R9) . n & (diameter R9) . n <= seq . n ) by A23, A22, A24, Def2; ::_thesis: verum
end;
A25: lim seq = 0 by A6, SEQ_4:30;
then A26: lim (NULL (#) seq) = NULL * 0 by A6, SEQ_2:8, SEQ_4:30;
A27: seq is convergent by A6, SEQ_4:30;
then A28: lim (diameter R9) = 0 by A25, A7, A26, A19, SEQ_2:20;
A29: R9 is closed by A3, A17, Th7;
then meet R9 <> {} by A1, A15, A28, Th10;
then consider x0 being set such that
A30: x0 in meet R9 by XBOOLE_0:def_1;
reconsider x0 = x0 as Point of M by A30;
A31: diameter R9 is convergent by A27, A25, A7, A26, A19, SEQ_2:19;
A32: now__::_thesis:_for_y_being_set_st_y_in_F_holds_
x0_in_y
let y be set ; ::_thesis: ( y in F implies x0 in y )
assume A33: y in F ; ::_thesis: x0 in y
then reconsider Y = y as Subset of (TopSpaceMetr M) ;
defpred S2[ set , set ] means for i being Nat st i = $1 holds
$2 = (R . i) /\ Y;
A34: for x being set st x in NAT holds
ex z being set st
( z in bool the carrier of M & S2[x,z] )
proof
let x be set ; ::_thesis: ( x in NAT implies ex z being set st
( z in bool the carrier of M & S2[x,z] ) )
assume x in NAT ; ::_thesis: ex z being set st
( z in bool the carrier of M & S2[x,z] )
then reconsider i = x as Element of NAT ;
take (R . i) /\ Y ; ::_thesis: ( (R . i) /\ Y in bool the carrier of M & S2[x,(R . i) /\ Y] )
thus ( (R . i) /\ Y in bool the carrier of M & S2[x,(R . i) /\ Y] ) ; ::_thesis: verum
end;
consider f9 being SetSequence of M such that
A35: for x being set st x in NAT holds
S2[x,f9 . x] from FUNCT_2:sch_1(A34);
A36: now__::_thesis:_for_x_being_set_st_x_in_dom_f9_holds_
not_f9_._x_is_empty
deffunc H2( set ) -> set = f . $1;
let x be set ; ::_thesis: ( x in dom f9 implies not f9 . x is empty )
assume x in dom f9 ; ::_thesis: not f9 . x is empty
then reconsider i = x as Element of NAT ;
set SS = { (f . j) where j is Element of NAT : j <= i } ;
A37: f . i in { (f . j) where j is Element of NAT : j <= i } ;
A38: { (f . j) where j is Element of NAT : j <= i } c= { H2(j) where j is Element of NAT : j in i + 1 }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { (f . j) where j is Element of NAT : j <= i } or x in { H2(j) where j is Element of NAT : j in i + 1 } )
assume x in { (f . j) where j is Element of NAT : j <= i } ; ::_thesis: x in { H2(j) where j is Element of NAT : j in i + 1 }
then consider j being Element of NAT such that
A39: x = f . j and
A40: j <= i ;
j < i + 1 by A40, NAT_1:13;
then j in i + 1 by NAT_1:44;
hence x in { H2(j) where j is Element of NAT : j in i + 1 } by A39; ::_thesis: verum
end;
A41: {Y} \/ { (f . j) where j is Element of NAT : j <= i } c= F
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in {Y} \/ { (f . j) where j is Element of NAT : j <= i } or z in F )
assume A42: z in {Y} \/ { (f . j) where j is Element of NAT : j <= i } ; ::_thesis: z in F
percases ( z in {Y} or z in { (f . j) where j is Element of NAT : j <= i } ) by A42, XBOOLE_0:def_3;
suppose z in {Y} ; ::_thesis: z in F
hence z in F by A33, TARSKI:def_1; ::_thesis: verum
end;
suppose z in { (f . j) where j is Element of NAT : j <= i } ; ::_thesis: z in F
then ex j being Element of NAT st
( z = f . j & j <= i ) ;
hence z in F by A12; ::_thesis: verum
end;
end;
end;
A43: i + 1 is finite ;
{ H2(j) where j is Element of NAT : j in i + 1 } is finite from FRAENKEL:sch_21(A43);
then meet ({Y} \/ { (f . j) where j is Element of NAT : j <= i } ) <> {} by A4, A41, A38, FINSET_1:def_3;
then (meet {Y}) /\ (meet { (f . j) where j is Element of NAT : j <= i } ) <> {} by A37, SETFAM_1:9;
then Y /\ (meet { (f . j) where j is Element of NAT : j <= i } ) <> {} by SETFAM_1:10;
then Y /\ (R . i) <> {} by A18;
hence not f9 . x is empty by A35; ::_thesis: verum
end;
A44: now__::_thesis:_for_i_being_Nat_holds_f9_._i_is_closed
let i be Nat; ::_thesis: f9 . i is closed
reconsider Ri = R . i as Subset of (TopSpaceMetr M) ;
i in NAT by ORDINAL1:def_12;
then A45: f9 . i = Ri /\ Y by A35;
R9 . i is closed by A29, Def8;
then A46: Ri is closed by Th6;
Y is closed by A3, A33, TOPS_2:def_2;
hence f9 . i is closed by A46, A45, Th6; ::_thesis: verum
end;
now__::_thesis:_for_i_being_Nat_holds_f9_._i_is_bounded
let i be Nat; ::_thesis: f9 . i is bounded
i in NAT by ORDINAL1:def_12;
then A47: f9 . i = (R9 . i) /\ Y by A35;
R9 . i is bounded by Def1;
hence f9 . i is bounded by A47, TBSP_1:14, XBOOLE_1:17; ::_thesis: verum
end;
then reconsider f9 = f9 as non-empty pointwise_bounded closed SetSequence of M by A36, A44, Def1, Def8, FUNCT_1:def_9;
A48: f9 . 0 = (R . 0) /\ Y by A35;
set df = diameter f9;
now__::_thesis:_for_n_being_Element_of_NAT_holds_
(_(NULL_(#)_seq)_._n_<=_(diameter_f9)_._n_&_(diameter_f9)_._n_<=_(diameter_R9)_._n_)
reconsider Y9 = Y as Subset of M ;
let n be Element of NAT ; ::_thesis: ( (NULL (#) seq) . n <= (diameter f9) . n & (diameter f9) . n <= (diameter R9) . n )
A49: (NULL (#) seq) . n = NULL * (seq . n) by SEQ_1:9;
A50: (R . n) /\ Y9 = f9 . n by A35;
A51: R9 . n is bounded by Def1;
then diameter (f9 . n) <= diameter (R9 . n) by A50, TBSP_1:24, XBOOLE_1:17;
then A52: diameter (f9 . n) <= (diameter R9) . n by Def2;
(R . n) /\ Y c= R . n by XBOOLE_1:17;
then 0 <= diameter (f9 . n) by A51, A50, TBSP_1:14, TBSP_1:21;
hence ( (NULL (#) seq) . n <= (diameter f9) . n & (diameter f9) . n <= (diameter R9) . n ) by A52, A49, Def2; ::_thesis: verum
end;
then A53: lim (diameter f9) = 0 by A7, A26, A31, A28, SEQ_2:20;
now__::_thesis:_for_i_being_Element_of_NAT_holds_f9_._(i_+_1)_c=_f9_._i
let i be Element of NAT ; ::_thesis: f9 . (i + 1) c= f9 . i
A54: f9 . i = (R . i) /\ Y by A35;
A55: R . (i + 1) c= R . i by A15, KURATO_0:def_3;
f9 . (i + 1) = (R . (i + 1)) /\ Y by A35;
hence f9 . (i + 1) c= f9 . i by A54, A55, XBOOLE_1:26; ::_thesis: verum
end;
then f9 is V172() by KURATO_0:def_3;
then meet f9 <> {} by A1, A53, Th10;
then consider z being set such that
A56: z in meet f9 by XBOOLE_0:def_1;
reconsider z = z as Point of M by A56;
A57: x0 = z
proof
assume x0 <> z ; ::_thesis: contradiction
then dist (x0,z) <> 0 by METRIC_1:2;
then dist (x0,z) > 0 by METRIC_1:5;
then consider i being Element of NAT such that
A58: for j being Element of NAT st i <= j holds
abs (((diameter R9) . j) - 0) < dist (x0,z) by A31, A28, SEQ_2:def_7;
A59: f9 . i = (R . i) /\ Y by A35;
z in f9 . i by A56, KURATO_0:3;
then A60: z in R . i by A59, XBOOLE_0:def_4;
A61: R9 . i is bounded by Def1;
then A62: 0 <= diameter (R9 . i) by TBSP_1:21;
x0 in R . i by A30, KURATO_0:3;
then dist (x0,z) <= diameter (R9 . i) by A60, A61, TBSP_1:def_8;
then A63: abs (diameter (R9 . i)) >= dist (x0,z) by A62, ABSVALUE:def_1;
abs (((diameter R9) . i) - 0) < dist (x0,z) by A58;
hence contradiction by A63, Def2; ::_thesis: verum
end;
z in f9 . 0 by A56, KURATO_0:3;
hence x0 in y by A57, A48, XBOOLE_0:def_4; ::_thesis: verum
end;
F <> {} by A4, FINSET_1:def_3;
hence not meet F is empty by A32, SETFAM_1:def_1; ::_thesis: verum
end;
assume A64: for F being Subset-Family of (TopSpaceMetr M) st F is closed & F is centered & ( for r being Real st r > 0 holds
ex A being Subset of M st
( A in F & A is bounded & diameter A < r ) ) holds
not meet F is empty ; ::_thesis: M is complete
now__::_thesis:_for_S_being_non-empty_pointwise_bounded_closed_SetSequence_of_M_st_S_is_V172()_&_lim_(diameter_S)_=_0_holds_
not_meet_S_is_empty
let S be non-empty pointwise_bounded closed SetSequence of M; ::_thesis: ( S is V172() & lim (diameter S) = 0 implies not meet S is empty )
assume that
A65: S is V172() and
A66: lim (diameter S) = 0 ; ::_thesis: not meet S is empty
reconsider RS = rng S as Subset-Family of (TopSpaceMetr M) ;
A67: now__::_thesis:_for_r_being_Real_st_r_>_0_holds_
ex_Sn_being_Element_of_bool_the_carrier_of_M_st_
(_Sn_in_RS_&_Sn_is_bounded_&_diameter_Sn_<_r_)
set d = diameter S;
A68: dom S = NAT by FUNCT_2:def_1;
A69: diameter S is bounded_below by Th1;
A70: diameter S is V103() by A65, Th2;
let r be Real; ::_thesis: ( r > 0 implies ex Sn being Element of bool the carrier of M st
( Sn in RS & Sn is bounded & diameter Sn < r ) )
assume r > 0 ; ::_thesis: ex Sn being Element of bool the carrier of M st
( Sn in RS & Sn is bounded & diameter Sn < r )
then consider n being Element of NAT such that
A71: for m being Element of NAT st n <= m holds
abs (((diameter S) . m) - 0) < r by A66, A69, A70, SEQ_2:def_7;
take Sn = S . n; ::_thesis: ( Sn in RS & Sn is bounded & diameter Sn < r )
A72: (diameter S) . n = diameter Sn by Def2;
Sn is bounded by Def1;
then A73: diameter Sn >= 0 by TBSP_1:21;
abs (((diameter S) . n) - 0) < r by A71;
hence ( Sn in RS & Sn is bounded & diameter Sn < r ) by A72, A73, A68, Def1, ABSVALUE:def_1, FUNCT_1:def_3; ::_thesis: verum
end;
RS is closed by Th12;
then not meet RS is empty by A64, A65, A67, Th11;
then consider x being set such that
A74: x in meet RS by XBOOLE_0:def_1;
now__::_thesis:_for_i_being_Element_of_NAT_holds_x_in_S_._i
let i be Element of NAT ; ::_thesis: x in S . i
dom S = NAT by FUNCT_2:def_1;
then S . i in RS by FUNCT_1:def_3;
hence x in S . i by A74, SETFAM_1:def_1; ::_thesis: verum
end;
hence not meet S is empty by KURATO_0:3; ::_thesis: verum
end;
hence M is complete by Th10; ::_thesis: verum
end;
theorem Th15: :: COMPL_SP:15
for M being non empty MetrSpace
for A being non empty Subset of M
for B being Subset of M
for B9 being Subset of (M | A) st B = B9 holds
( B9 is bounded iff B is bounded )
proof
let M be non empty MetrSpace; ::_thesis: for A being non empty Subset of M
for B being Subset of M
for B9 being Subset of (M | A) st B = B9 holds
( B9 is bounded iff B is bounded )
let A be non empty Subset of M; ::_thesis: for B being Subset of M
for B9 being Subset of (M | A) st B = B9 holds
( B9 is bounded iff B is bounded )
let B be Subset of M; ::_thesis: for B9 being Subset of (M | A) st B = B9 holds
( B9 is bounded iff B is bounded )
let B9 be Subset of (M | A); ::_thesis: ( B = B9 implies ( B9 is bounded iff B is bounded ) )
assume A1: B = B9 ; ::_thesis: ( B9 is bounded iff B is bounded )
thus ( B9 is bounded implies B is bounded ) by A1, HAUSDORF:17; ::_thesis: ( B is bounded implies B9 is bounded )
assume A2: B is bounded ; ::_thesis: B9 is bounded
percases ( B9 = {} (M | A) or B9 <> {} (M | A) ) ;
suppose B9 = {} (M | A) ; ::_thesis: B9 is bounded
hence B9 is bounded ; ::_thesis: verum
end;
suppose B9 <> {} (M | A) ; ::_thesis: B9 is bounded
then consider p being set such that
A3: p in B9 by XBOOLE_0:def_1;
reconsider p = p as Point of (M | A) by A3;
A4: now__::_thesis:_for_q_being_Point_of_(M_|_A)_st_q_in_B9_holds_
dist_(p,q)_<=_(diameter_B)_+_1
let q be Point of (M | A); ::_thesis: ( q in B9 implies dist (p,q) <= (diameter B) + 1 )
assume A5: q in B9 ; ::_thesis: dist (p,q) <= (diameter B) + 1
reconsider p9 = p, q9 = q as Point of M by TOPMETR:8;
A6: dist (p,q) = dist (p9,q9) by TOPMETR:def_1;
A7: (diameter B) + 0 <= (diameter B) + 1 by XREAL_1:8;
dist (p9,q9) <= diameter B by A1, A2, A3, A5, TBSP_1:def_8;
hence dist (p,q) <= (diameter B) + 1 by A6, A7, XXREAL_0:2; ::_thesis: verum
end;
0 + 0 < (diameter B) + 1 by A2, TBSP_1:21, XREAL_1:8;
hence B9 is bounded by A4, TBSP_1:10; ::_thesis: verum
end;
end;
end;
theorem Th16: :: COMPL_SP:16
for M being non empty MetrSpace
for A being non empty Subset of M
for B being Subset of M
for B9 being Subset of (M | A) st B = B9 & B is bounded holds
diameter B9 <= diameter B
proof
let M be non empty MetrSpace; ::_thesis: for A being non empty Subset of M
for B being Subset of M
for B9 being Subset of (M | A) st B = B9 & B is bounded holds
diameter B9 <= diameter B
let A be non empty Subset of M; ::_thesis: for B being Subset of M
for B9 being Subset of (M | A) st B = B9 & B is bounded holds
diameter B9 <= diameter B
let B be Subset of M; ::_thesis: for B9 being Subset of (M | A) st B = B9 & B is bounded holds
diameter B9 <= diameter B
let B9 be Subset of (M | A); ::_thesis: ( B = B9 & B is bounded implies diameter B9 <= diameter B )
assume that
A1: B = B9 and
A2: B is bounded ; ::_thesis: diameter B9 <= diameter B
A3: B9 is bounded by A1, A2, Th15;
percases ( B9 = {} (M | A) or B9 <> {} (M | A) ) ;
supposeA4: B9 = {} (M | A) ; ::_thesis: diameter B9 <= diameter B
then diameter B9 = 0 by TBSP_1:def_8;
hence diameter B9 <= diameter B by A1, A4, TBSP_1:def_8; ::_thesis: verum
end;
supposeA5: B9 <> {} (M | A) ; ::_thesis: diameter B9 <= diameter B
now__::_thesis:_for_x,_y_being_Point_of_(M_|_A)_st_x_in_B9_&_y_in_B9_holds_
dist_(x,y)_<=_diameter_B
let x, y be Point of (M | A); ::_thesis: ( x in B9 & y in B9 implies dist (x,y) <= diameter B )
assume that
A6: x in B9 and
A7: y in B9 ; ::_thesis: dist (x,y) <= diameter B
reconsider x9 = x, y9 = y as Point of M by TOPMETR:8;
dist (x,y) = dist (x9,y9) by TOPMETR:def_1;
hence dist (x,y) <= diameter B by A1, A2, A6, A7, TBSP_1:def_8; ::_thesis: verum
end;
hence diameter B9 <= diameter B by A3, A5, TBSP_1:def_8; ::_thesis: verum
end;
end;
end;
theorem Th17: :: COMPL_SP:17
for M being non empty MetrSpace
for A being non empty Subset of M
for S being sequence of (M | A) holds S is sequence of M
proof
let M be non empty MetrSpace; ::_thesis: for A being non empty Subset of M
for S being sequence of (M | A) holds S is sequence of M
let A be non empty Subset of M; ::_thesis: for S being sequence of (M | A) holds S is sequence of M
let S be sequence of (M | A); ::_thesis: S is sequence of M
A c= the carrier of M ;
then the carrier of (M | A) c= the carrier of M by TOPMETR:def_2;
hence S is sequence of M by FUNCT_2:7; ::_thesis: verum
end;
theorem Th18: :: COMPL_SP:18
for M being non empty MetrSpace
for A being non empty Subset of M
for S being sequence of (M | A)
for S9 being sequence of M st S = S9 holds
( S9 is Cauchy iff S is Cauchy )
proof
let M be non empty MetrSpace; ::_thesis: for A being non empty Subset of M
for S being sequence of (M | A)
for S9 being sequence of M st S = S9 holds
( S9 is Cauchy iff S is Cauchy )
let A be non empty Subset of M; ::_thesis: for S being sequence of (M | A)
for S9 being sequence of M st S = S9 holds
( S9 is Cauchy iff S is Cauchy )
let S be sequence of (M | A); ::_thesis: for S9 being sequence of M st S = S9 holds
( S9 is Cauchy iff S is Cauchy )
let S9 be sequence of M; ::_thesis: ( S = S9 implies ( S9 is Cauchy iff S is Cauchy ) )
assume A1: S = S9 ; ::_thesis: ( S9 is Cauchy iff S is Cauchy )
thus ( S9 is Cauchy implies S is Cauchy ) ::_thesis: ( S is Cauchy implies S9 is Cauchy )
proof
assume A2: S9 is Cauchy ; ::_thesis: S is Cauchy
let r be Real; :: according to TBSP_1:def_4 ::_thesis: ( r <= 0 or ex b1 being Element of NAT st
for b2, b3 being Element of NAT holds
( not b1 <= b2 or not b1 <= b3 or not r <= dist ((S . b2),(S . b3)) ) )
assume r > 0 ; ::_thesis: ex b1 being Element of NAT st
for b2, b3 being Element of NAT holds
( not b1 <= b2 or not b1 <= b3 or not r <= dist ((S . b2),(S . b3)) )
then consider p being Element of NAT such that
A3: for n, m being Element of NAT st p <= n & p <= m holds
dist ((S9 . n),(S9 . m)) < r by A2, TBSP_1:def_4;
take p ; ::_thesis: for b1, b2 being Element of NAT holds
( not p <= b1 or not p <= b2 or not r <= dist ((S . b1),(S . b2)) )
let n, m be Element of NAT ; ::_thesis: ( not p <= n or not p <= m or not r <= dist ((S . n),(S . m)) )
assume that
A4: p <= n and
A5: p <= m ; ::_thesis: not r <= dist ((S . n),(S . m))
dist ((S . n),(S . m)) = dist ((S9 . n),(S9 . m)) by A1, TOPMETR:def_1;
hence not r <= dist ((S . n),(S . m)) by A3, A4, A5; ::_thesis: verum
end;
assume A6: S is Cauchy ; ::_thesis: S9 is Cauchy
let r be Real; :: according to TBSP_1:def_4 ::_thesis: ( r <= 0 or ex b1 being Element of NAT st
for b2, b3 being Element of NAT holds
( not b1 <= b2 or not b1 <= b3 or not r <= dist ((S9 . b2),(S9 . b3)) ) )
assume r > 0 ; ::_thesis: ex b1 being Element of NAT st
for b2, b3 being Element of NAT holds
( not b1 <= b2 or not b1 <= b3 or not r <= dist ((S9 . b2),(S9 . b3)) )
then consider p being Element of NAT such that
A7: for n, m being Element of NAT st p <= n & p <= m holds
dist ((S . n),(S . m)) < r by A6, TBSP_1:def_4;
take p ; ::_thesis: for b1, b2 being Element of NAT holds
( not p <= b1 or not p <= b2 or not r <= dist ((S9 . b1),(S9 . b2)) )
let n, m be Element of NAT ; ::_thesis: ( not p <= n or not p <= m or not r <= dist ((S9 . n),(S9 . m)) )
assume that
A8: p <= n and
A9: p <= m ; ::_thesis: not r <= dist ((S9 . n),(S9 . m))
dist ((S . n),(S . m)) = dist ((S9 . n),(S9 . m)) by A1, TOPMETR:def_1;
hence not r <= dist ((S9 . n),(S9 . m)) by A7, A8, A9; ::_thesis: verum
end;
theorem :: COMPL_SP:19
for M being non empty MetrSpace st M is complete holds
for A being non empty Subset of M
for A9 being Subset of (TopSpaceMetr M) st A = A9 holds
( M | A is complete iff A9 is closed )
proof
let M be non empty MetrSpace; ::_thesis: ( M is complete implies for A being non empty Subset of M
for A9 being Subset of (TopSpaceMetr M) st A = A9 holds
( M | A is complete iff A9 is closed ) )
assume A1: M is complete ; ::_thesis: for A being non empty Subset of M
for A9 being Subset of (TopSpaceMetr M) st A = A9 holds
( M | A is complete iff A9 is closed )
set T = TopSpaceMetr M;
let A be non empty Subset of M; ::_thesis: for A9 being Subset of (TopSpaceMetr M) st A = A9 holds
( M | A is complete iff A9 is closed )
let A9 be Subset of (TopSpaceMetr M); ::_thesis: ( A = A9 implies ( M | A is complete iff A9 is closed ) )
assume A2: A = A9 ; ::_thesis: ( M | A is complete iff A9 is closed )
set MA = M | A;
set TA = TopSpaceMetr (M | A);
thus ( M | A is complete implies A9 is closed ) ::_thesis: ( A9 is closed implies M | A is complete )
proof
assume A3: M | A is complete ; ::_thesis: A9 is closed
A4: Cl A9 c= A9
proof
let p be set ; :: according to TARSKI:def_3 ::_thesis: ( not p in Cl A9 or p in A9 )
assume A5: p in Cl A9 ; ::_thesis: p in A9
reconsider p = p as Point of M by A5;
defpred S1[ set , set ] means for i being Nat st i = $1 holds
$2 = A /\ (cl_Ball (p,(1 / (i + 1))));
A6: for x being set st x in NAT holds
ex y being set st
( y in bool the carrier of (M | A) & S1[x,y] )
proof
let x be set ; ::_thesis: ( x in NAT implies ex y being set st
( y in bool the carrier of (M | A) & S1[x,y] ) )
assume x in NAT ; ::_thesis: ex y being set st
( y in bool the carrier of (M | A) & S1[x,y] )
then reconsider i = x as Element of NAT ;
take A /\ (cl_Ball (p,(1 / (i + 1)))) ; ::_thesis: ( A /\ (cl_Ball (p,(1 / (i + 1)))) in bool the carrier of (M | A) & S1[x,A /\ (cl_Ball (p,(1 / (i + 1))))] )
A /\ (cl_Ball (p,(1 / (i + 1)))) c= A by XBOOLE_1:17;
then A /\ (cl_Ball (p,(1 / (i + 1)))) c= the carrier of (M | A) by TOPMETR:def_2;
hence ( A /\ (cl_Ball (p,(1 / (i + 1)))) in bool the carrier of (M | A) & S1[x,A /\ (cl_Ball (p,(1 / (i + 1))))] ) ; ::_thesis: verum
end;
consider f being SetSequence of (M | A) such that
A7: for x being set st x in NAT holds
S1[x,f . x] from FUNCT_2:sch_1(A6);
A8: now__::_thesis:_for_x_being_set_st_x_in_dom_f_holds_
not_f_._x_is_empty
let x be set ; ::_thesis: ( x in dom f implies not f . x is empty )
assume x in dom f ; ::_thesis: not f . x is empty
then reconsider i = x as Element of NAT ;
reconsider B = Ball (p,(1 / (i + 1))) as Subset of (TopSpaceMetr M) ;
Ball (p,(1 / (i + 1))) in Family_open_set M by PCOMPS_1:29;
then A9: B is open by PRE_TOPC:def_2;
p in B by TBSP_1:11, XREAL_1:139;
then B meets A9 by A5, A9, PRE_TOPC:24;
then consider y being set such that
A10: y in B and
A11: y in A9 by XBOOLE_0:3;
reconsider y = y as Point of M by A10;
dist (p,y) < 1 / (i + 1) by A10, METRIC_1:11;
then y in cl_Ball (p,(1 / (i + 1))) by METRIC_1:12;
then y in A /\ (cl_Ball (p,(1 / (i + 1)))) by A2, A11, XBOOLE_0:def_4;
hence not f . x is empty by A7; ::_thesis: verum
end;
A12: now__::_thesis:_for_i_being_Nat_holds_f_._i_is_closed
let i be Nat; ::_thesis: f . i is closed
reconsider cB = cl_Ball (p,(1 / (i + 1))) as Subset of (TopSpaceMetr M) ;
reconsider fi = f . i as Subset of (TopSpaceMetr (M | A)) ;
A13: i in NAT by ORDINAL1:def_12;
([#] M) \ cB in Family_open_set M by NAGATA_1:14;
then cB ` is open by PRE_TOPC:def_2;
then A14: cB is closed by TOPS_1:3;
A15: TopSpaceMetr (M | A) = (TopSpaceMetr M) | A9 by A2, HAUSDORF:16;
then [#] ((TopSpaceMetr M) | A9) = A by TOPMETR:def_2;
then fi = cB /\ ([#] ((TopSpaceMetr M) | A9)) by A7, A13;
then fi is closed by A14, A15, PRE_TOPC:13;
hence f . i is closed by Th6; ::_thesis: verum
end;
now__::_thesis:_for_i_being_Nat_holds_f_._i_is_bounded
let i be Nat; ::_thesis: f . i is bounded
set ACL = A /\ (cl_Ball (p,(1 / (i + 1))));
cl_Ball (p,(1 / (i + 1))) is bounded by TOPREAL6:59;
then A16: A /\ (cl_Ball (p,(1 / (i + 1)))) is bounded by TBSP_1:14, XBOOLE_1:17;
i in NAT by ORDINAL1:def_12;
then f . i = A /\ (cl_Ball (p,(1 / (i + 1)))) by A7;
hence f . i is bounded by A16, Th15; ::_thesis: verum
end;
then reconsider f = f as non-empty pointwise_bounded closed SetSequence of (M | A) by A8, A12, Def1, Def8, FUNCT_1:def_9;
set df = diameter f;
reconsider NULL = 0 , TWO = 2 as Real ;
deffunc H1( Element of NAT ) -> Element of REAL = 1 / (1 + $1);
consider seq being Real_Sequence such that
A17: for n being Element of NAT holds seq . n = H1(n) from SEQ_1:sch_1();
now__::_thesis:_for_i_being_Element_of_NAT_holds_f_._(i_+_1)_c=_f_._i
let i be Element of NAT ; ::_thesis: f . (i + 1) c= f . i
set i1 = i + 1;
cl_Ball (p,(1 / ((i + 1) + 1))) c= cl_Ball (p,(1 / (i + 1)))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in cl_Ball (p,(1 / ((i + 1) + 1))) or x in cl_Ball (p,(1 / (i + 1))) )
assume A18: x in cl_Ball (p,(1 / ((i + 1) + 1))) ; ::_thesis: x in cl_Ball (p,(1 / (i + 1)))
reconsider q = x as Point of M by A18;
i + 1 < (i + 1) + 1 by NAT_1:13;
then A19: 1 / ((i + 1) + 1) < 1 / (i + 1) by XREAL_1:76;
dist (p,q) <= 1 / ((i + 1) + 1) by A18, METRIC_1:12;
then dist (p,q) <= 1 / (i + 1) by A19, XXREAL_0:2;
hence x in cl_Ball (p,(1 / (i + 1))) by METRIC_1:12; ::_thesis: verum
end;
then A20: A /\ (cl_Ball (p,(1 / ((i + 1) + 1)))) c= A /\ (cl_Ball (p,(1 / (i + 1)))) by XBOOLE_1:26;
f . i = A /\ (cl_Ball (p,(1 / (i + 1)))) by A7;
hence f . (i + 1) c= f . i by A7, A20; ::_thesis: verum
end;
then A21: f is V172() by KURATO_0:def_3;
set Ts = TWO (#) seq;
set Ns = NULL (#) seq;
A22: for n being Element of NAT holds seq . n = 1 / (n + 1) by A17;
then A23: NULL (#) seq is convergent by SEQ_2:7, SEQ_4:30;
A24: now__::_thesis:_for_n_being_Element_of_NAT_holds_
(_(NULL_(#)_seq)_._n_<=_(diameter_f)_._n_&_(diameter_f)_._n_<=_(TWO_(#)_seq)_._n_)
let n be Element of NAT ; ::_thesis: ( (NULL (#) seq) . n <= (diameter f) . n & (diameter f) . n <= (TWO (#) seq) . n )
set cB = cl_Ball (p,(1 / (n + 1)));
A25: (NULL (#) seq) . n = NULL * (seq . n) by SEQ_1:9;
A26: (TWO (#) seq) . n = TWO * (seq . n) by SEQ_1:9;
A27: cl_Ball (p,(1 / (n + 1))) is bounded by TOPREAL6:59;
then A28: A /\ (cl_Ball (p,(1 / (n + 1)))) is bounded by TBSP_1:14, XBOOLE_1:17;
A29: diameter (A /\ (cl_Ball (p,(1 / (n + 1))))) <= diameter (cl_Ball (p,(1 / (n + 1)))) by A27, TBSP_1:24, XBOOLE_1:17;
diameter (cl_Ball (p,(1 / (n + 1)))) <= 2 * H1(n) by Th5;
then A30: diameter (A /\ (cl_Ball (p,(1 / (n + 1))))) <= 2 * H1(n) by A29, XXREAL_0:2;
A31: f . n = A /\ (cl_Ball (p,(1 / (n + 1)))) by A7;
then f . n is bounded by A28, Th15;
then A32: 0 <= diameter (f . n) by TBSP_1:21;
diameter (f . n) <= diameter (A /\ (cl_Ball (p,(1 / (n + 1))))) by A28, A31, Th16;
then A33: diameter (f . n) <= 2 * H1(n) by A30, XXREAL_0:2;
H1(n) = seq . n by A17;
hence ( (NULL (#) seq) . n <= (diameter f) . n & (diameter f) . n <= (TWO (#) seq) . n ) by A32, A33, A25, A26, Def2; ::_thesis: verum
end;
A34: TWO (#) seq is convergent by A22, SEQ_2:7, SEQ_4:30;
A35: lim seq = 0 by A22, SEQ_4:30;
then A36: lim (TWO (#) seq) = TWO * 0 by A22, SEQ_2:8, SEQ_4:30;
lim (NULL (#) seq) = NULL * 0 by A22, A35, SEQ_2:8, SEQ_4:30;
then lim (diameter f) = 0 by A23, A34, A36, A24, SEQ_2:20;
then not meet f is empty by A3, A21, Th10;
then consider q being set such that
A37: q in meet f by XBOOLE_0:def_1;
reconsider q = q as Point of M by A37, TOPMETR:8;
A38: seq is convergent by A22, SEQ_4:30;
p = q
proof
assume p <> q ; ::_thesis: contradiction
then dist (p,q) <> 0 by METRIC_1:2;
then dist (p,q) > 0 by METRIC_1:5;
then consider n being Element of NAT such that
A39: for m being Element of NAT st n <= m holds
abs ((seq . m) - 0) < dist (p,q) by A38, A35, SEQ_2:def_7;
set cB = cl_Ball (p,(1 / (n + 1)));
A40: q in f . n by A37, KURATO_0:3;
f . n = A /\ (cl_Ball (p,(1 / (n + 1)))) by A7;
then q in cl_Ball (p,(1 / (n + 1))) by A40, XBOOLE_0:def_4;
then A41: dist (p,q) <= H1(n) by METRIC_1:12;
seq . n = H1(n) by A17;
then abs ((seq . n) - 0) = H1(n) by ABSVALUE:def_1;
hence contradiction by A39, A41; ::_thesis: verum
end;
then A42: p in f . 0 by A37, KURATO_0:3;
f . 0 = A /\ (cl_Ball (p,(1 / (0 + 1)))) by A7;
hence p in A9 by A2, A42, XBOOLE_0:def_4; ::_thesis: verum
end;
A9 c= Cl A9 by PRE_TOPC:18;
hence A9 is closed by A4, XBOOLE_0:def_10; ::_thesis: verum
end;
assume A43: A9 is closed ; ::_thesis: M | A is complete
let S be sequence of (M | A); :: according to TBSP_1:def_5 ::_thesis: ( not S is Cauchy or S is convergent )
assume A44: S is Cauchy ; ::_thesis: S is convergent
reconsider S9 = S as sequence of M by Th17;
S9 is Cauchy by A44, Th18;
then A45: S9 is convergent by A1, TBSP_1:def_5;
A46: now__::_thesis:_for_n_being_Element_of_NAT_holds_S9_._n_in_A9
let n be Element of NAT ; ::_thesis: S9 . n in A9
S . n in the carrier of (M | A) ;
hence S9 . n in A9 by A2, TOPMETR:def_2; ::_thesis: verum
end;
the carrier of (M | A) = A9 by A2, TOPMETR:def_2;
then reconsider limS = lim S9 as Point of (M | A) by A43, A45, A46, TOPMETR3:1;
take limS ; :: according to TBSP_1:def_2 ::_thesis: for b1 being Element of REAL holds
( b1 <= 0 or ex b2 being Element of NAT st
for b3 being Element of NAT holds
( not b2 <= b3 or not b1 <= dist ((S . b3),limS) ) )
let r be Real; ::_thesis: ( r <= 0 or ex b1 being Element of NAT st
for b2 being Element of NAT holds
( not b1 <= b2 or not r <= dist ((S . b2),limS) ) )
assume r > 0 ; ::_thesis: ex b1 being Element of NAT st
for b2 being Element of NAT holds
( not b1 <= b2 or not r <= dist ((S . b2),limS) )
then consider n being Element of NAT such that
A47: for m being Element of NAT st m >= n holds
dist ((S9 . m),(lim S9)) < r by A45, TBSP_1:def_3;
take n ; ::_thesis: for b1 being Element of NAT holds
( not n <= b1 or not r <= dist ((S . b1),limS) )
let m be Element of NAT ; ::_thesis: ( not n <= m or not r <= dist ((S . m),limS) )
assume A48: m >= n ; ::_thesis: not r <= dist ((S . m),limS)
dist ((S . m),limS) = dist ((S9 . m),(lim S9)) by TOPMETR:def_1;
hence not r <= dist ((S . m),limS) by A47, A48; ::_thesis: verum
end;
begin
definition
let T be TopStruct ;
attrT is countably_compact means :Def9: :: COMPL_SP:def 9
for F being Subset-Family of T st F is Cover of T & F is open & F is countable holds
ex G being Subset-Family of T st
( G c= F & G is Cover of T & G is finite );
end;
:: deftheorem Def9 defines countably_compact COMPL_SP:def_9_:_
for T being TopStruct holds
( T is countably_compact iff for F being Subset-Family of T st F is Cover of T & F is open & F is countable holds
ex G being Subset-Family of T st
( G c= F & G is Cover of T & G is finite ) );
theorem Th20: :: COMPL_SP:20
for T being TopStruct st T is compact holds
T is countably_compact
proof
let T be TopStruct ; ::_thesis: ( T is compact implies T is countably_compact )
assume A1: T is compact ; ::_thesis: T is countably_compact
let F be Subset-Family of T; :: according to COMPL_SP:def_9 ::_thesis: ( F is Cover of T & F is open & F is countable implies ex G being Subset-Family of T st
( G c= F & G is Cover of T & G is finite ) )
assume that
A2: F is Cover of T and
A3: F is open and
F is countable ; ::_thesis: ex G being Subset-Family of T st
( G c= F & G is Cover of T & G is finite )
thus ex G being Subset-Family of T st
( G c= F & G is Cover of T & G is finite ) by A1, A2, A3, COMPTS_1:def_1; ::_thesis: verum
end;
theorem Th21: :: COMPL_SP:21
for T being non empty TopSpace holds
( T is countably_compact iff for F being Subset-Family of T st F is centered & F is closed & F is countable holds
meet F <> {} )
proof
let T be non empty TopSpace; ::_thesis: ( T is countably_compact iff for F being Subset-Family of T st F is centered & F is closed & F is countable holds
meet F <> {} )
thus ( T is countably_compact implies for F being Subset-Family of T st F is centered & F is closed & F is countable holds
meet F <> {} ) ::_thesis: ( ( for F being Subset-Family of T st F is centered & F is closed & F is countable holds
meet F <> {} ) implies T is countably_compact )
proof
assume A1: T is countably_compact ; ::_thesis: for F being Subset-Family of T st F is centered & F is closed & F is countable holds
meet F <> {}
let F be Subset-Family of T; ::_thesis: ( F is centered & F is closed & F is countable implies meet F <> {} )
assume that
A2: F is centered and
A3: F is closed and
A4: F is countable ; ::_thesis: meet F <> {}
assume A5: meet F = {} ; ::_thesis: contradiction
F <> {} by A2, FINSET_1:def_3;
then union (COMPLEMENT F) = (meet F) ` by TOPS_2:7
.= [#] T by A5 ;
then A6: COMPLEMENT F is Cover of T by SETFAM_1:45;
A7: COMPLEMENT F is countable by A4, TOPGEN_4:1;
COMPLEMENT F is open by A3, TOPS_2:9;
then consider G9 being Subset-Family of T such that
A8: G9 c= COMPLEMENT F and
A9: G9 is Cover of T and
A10: G9 is finite by A1, A6, A7, Def9;
A11: COMPLEMENT G9 is finite by A10, TOPS_2:8;
A12: COMPLEMENT G9 c= F
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in COMPLEMENT G9 or x in F )
assume A13: x in COMPLEMENT G9 ; ::_thesis: x in F
reconsider x9 = x as Subset of T by A13;
x9 ` in G9 by A13, SETFAM_1:def_7;
then (x9 `) ` in F by A8, SETFAM_1:def_7;
hence x in F ; ::_thesis: verum
end;
G9 <> {} by A9, TOPS_2:3;
then A14: COMPLEMENT G9 <> {} by TOPS_2:5;
meet (COMPLEMENT G9) = (union G9) ` by A9, TOPS_2:3, TOPS_2:6
.= ([#] T) \ ([#] T) by A9, SETFAM_1:45
.= {} by XBOOLE_1:37 ;
hence contradiction by A2, A12, A14, A11, FINSET_1:def_3; ::_thesis: verum
end;
assume A15: for F being Subset-Family of T st F is centered & F is closed & F is countable holds
meet F <> {} ; ::_thesis: T is countably_compact
let F be Subset-Family of T; :: according to COMPL_SP:def_9 ::_thesis: ( F is Cover of T & F is open & F is countable implies ex G being Subset-Family of T st
( G c= F & G is Cover of T & G is finite ) )
assume that
A16: F is Cover of T and
A17: F is open and
A18: F is countable ; ::_thesis: ex G being Subset-Family of T st
( G c= F & G is Cover of T & G is finite )
A19: COMPLEMENT F is countable by A18, TOPGEN_4:1;
F <> {} by A16, TOPS_2:3;
then A20: COMPLEMENT F <> {} by TOPS_2:5;
A21: COMPLEMENT F is closed by A17, TOPS_2:10;
meet (COMPLEMENT F) = (union F) ` by A16, TOPS_2:3, TOPS_2:6
.= ([#] T) \ ([#] T) by A16, SETFAM_1:45
.= {} by XBOOLE_1:37 ;
then not COMPLEMENT F is centered by A15, A19, A21;
then consider G9 being set such that
A22: G9 <> {} and
A23: G9 c= COMPLEMENT F and
A24: G9 is finite and
A25: meet G9 = {} by A20, FINSET_1:def_3;
reconsider G9 = G9 as Subset-Family of T by A23, XBOOLE_1:1;
take F9 = COMPLEMENT G9; ::_thesis: ( F9 c= F & F9 is Cover of T & F9 is finite )
thus F9 c= F ::_thesis: ( F9 is Cover of T & F9 is finite )
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in F9 or x in F )
assume A26: x in F9 ; ::_thesis: x in F
reconsider x9 = x as Subset of T by A26;
x9 ` in G9 by A26, SETFAM_1:def_7;
then (x9 `) ` in F by A23, SETFAM_1:def_7;
hence x in F ; ::_thesis: verum
end;
union F9 = (meet G9) ` by A22, TOPS_2:7
.= [#] T by A25 ;
hence ( F9 is Cover of T & F9 is finite ) by A24, SETFAM_1:45, TOPS_2:8; ::_thesis: verum
end;
theorem Th22: :: COMPL_SP:22
for T being non empty TopSpace holds
( T is countably_compact iff for S being non-empty closed SetSequence of T st S is V172() holds
meet S <> {} )
proof
let T be non empty TopSpace; ::_thesis: ( T is countably_compact iff for S being non-empty closed SetSequence of T st S is V172() holds
meet S <> {} )
thus ( T is countably_compact implies for S being non-empty closed SetSequence of T st S is V172() holds
meet S <> {} ) ::_thesis: ( ( for S being non-empty closed SetSequence of T st S is V172() holds
meet S <> {} ) implies T is countably_compact )
proof
assume A1: T is countably_compact ; ::_thesis: for S being non-empty closed SetSequence of T st S is V172() holds
meet S <> {}
let S be non-empty closed SetSequence of T; ::_thesis: ( S is V172() implies meet S <> {} )
assume A2: S is V172() ; ::_thesis: meet S <> {}
reconsider rngS = rng S as Subset-Family of T ;
dom S = NAT by FUNCT_2:def_1;
then A3: rngS is countable by CARD_3:93;
now__::_thesis:_for_D_being_Subset_of_T_st_D_in_rngS_holds_
D_is_closed
let D be Subset of T; ::_thesis: ( D in rngS implies D is closed )
assume D in rngS ; ::_thesis: D is closed
then ex x being set st
( x in dom S & S . x = D ) by FUNCT_1:def_3;
hence D is closed by Def6; ::_thesis: verum
end;
then A4: rngS is closed by TOPS_2:def_2;
rngS is centered by A2, Th11;
then meet rngS <> {} by A1, A3, A4, Th21;
then consider x being set such that
A5: x in meet rngS by XBOOLE_0:def_1;
now__::_thesis:_for_n_being_Element_of_NAT_holds_x_in_S_._n
let n be Element of NAT ; ::_thesis: x in S . n
dom S = NAT by FUNCT_2:def_1;
then S . n in rngS by FUNCT_1:def_3;
hence x in S . n by A5, SETFAM_1:def_1; ::_thesis: verum
end;
hence meet S <> {} by KURATO_0:3; ::_thesis: verum
end;
assume A6: for S being non-empty closed SetSequence of T st S is V172() holds
meet S <> {} ; ::_thesis: T is countably_compact
now__::_thesis:_for_F_being_Subset-Family_of_T_st_F_is_centered_&_F_is_closed_&_F_is_countable_holds_
meet_F_<>_{}
let F be Subset-Family of T; ::_thesis: ( F is centered & F is closed & F is countable implies meet F <> {} )
assume that
A7: F is centered and
A8: F is closed and
A9: F is countable ; ::_thesis: meet F <> {}
A10: card F c= omega by A9, CARD_3:def_14;
now__::_thesis:_not_meet_F_is_empty
percases ( card F = omega or card F in omega ) by A10, CARD_1:3;
suppose card F = omega ; ::_thesis: not meet F is empty
then NAT ,F are_equipotent by CARD_1:5, CARD_1:47;
then consider s being Function such that
s is one-to-one and
A11: dom s = NAT and
A12: rng s = F by WELLORD2:def_4;
reconsider s = s as SetSequence of T by A11, A12, FUNCT_2:2;
consider R being SetSequence of T such that
A13: R is V172() and
A14: ( F is centered implies R is non-empty ) and
( F is open implies R is open ) and
A15: ( F is closed implies R is closed ) and
A16: for i being Nat holds R . i = meet { (s . j) where j is Element of NAT : j <= i } by A12, Th13;
meet R <> {} by A6, A7, A8, A13, A14, A15;
then consider x being set such that
A17: x in meet R by XBOOLE_0:def_1;
A18: now__::_thesis:_for_y_being_set_st_y_in_F_holds_
x_in_y
let y be set ; ::_thesis: ( y in F implies x in y )
assume y in F ; ::_thesis: x in y
then consider z being set such that
A19: z in dom s and
A20: s . z = y by A12, FUNCT_1:def_3;
reconsider z = z as Element of NAT by A19;
A21: s . z in { (s . j) where j is Element of NAT : j <= z } ;
A22: x in R . z by A17, KURATO_0:3;
R . z = meet { (s . j) where j is Element of NAT : j <= z } by A16;
then R . z c= s . z by A21, SETFAM_1:3;
hence x in y by A20, A22; ::_thesis: verum
end;
not F is empty by A11, A12, RELAT_1:42;
hence not meet F is empty by A18, SETFAM_1:def_1; ::_thesis: verum
end;
supposeA23: card F in omega ; ::_thesis: not meet F is empty
F is finite by A23;
hence not meet F is empty by A7, FINSET_1:def_3; ::_thesis: verum
end;
end;
end;
hence meet F <> {} ; ::_thesis: verum
end;
hence T is countably_compact by Th21; ::_thesis: verum
end;
theorem Th23: :: COMPL_SP:23
for T being non empty TopSpace
for F being Subset-Family of T
for S being SetSequence of T st rng S c= F & S is non-empty holds
ex R being non-empty closed SetSequence of T st
( R is V172() & ( F is locally_finite & S is one-to-one implies meet R = {} ) & ( for i being Nat ex Si being Subset-Family of T st
( R . i = Cl (union Si) & Si = { (S . j) where j is Element of NAT : j >= i } ) ) )
proof
let T be non empty TopSpace; ::_thesis: for F being Subset-Family of T
for S being SetSequence of T st rng S c= F & S is non-empty holds
ex R being non-empty closed SetSequence of T st
( R is V172() & ( F is locally_finite & S is one-to-one implies meet R = {} ) & ( for i being Nat ex Si being Subset-Family of T st
( R . i = Cl (union Si) & Si = { (S . j) where j is Element of NAT : j >= i } ) ) )
let F be Subset-Family of T; ::_thesis: for S being SetSequence of T st rng S c= F & S is non-empty holds
ex R being non-empty closed SetSequence of T st
( R is V172() & ( F is locally_finite & S is one-to-one implies meet R = {} ) & ( for i being Nat ex Si being Subset-Family of T st
( R . i = Cl (union Si) & Si = { (S . j) where j is Element of NAT : j >= i } ) ) )
let S be SetSequence of T; ::_thesis: ( rng S c= F & S is non-empty implies ex R being non-empty closed SetSequence of T st
( R is V172() & ( F is locally_finite & S is one-to-one implies meet R = {} ) & ( for i being Nat ex Si being Subset-Family of T st
( R . i = Cl (union Si) & Si = { (S . j) where j is Element of NAT : j >= i } ) ) ) )
assume that
A1: rng S c= F and
A2: S is non-empty ; ::_thesis: ex R being non-empty closed SetSequence of T st
( R is V172() & ( F is locally_finite & S is one-to-one implies meet R = {} ) & ( for i being Nat ex Si being Subset-Family of T st
( R . i = Cl (union Si) & Si = { (S . j) where j is Element of NAT : j >= i } ) ) )
defpred S1[ set , set ] means for i being Nat st i = $1 holds
ex SS being Subset-Family of T st
( SS c= F & SS = { (S . j) where j is Element of NAT : j >= i } & $2 = Cl (union SS) );
A3: for x being set st x in NAT holds
ex y being set st
( y in bool the carrier of T & S1[x,y] )
proof
let x be set ; ::_thesis: ( x in NAT implies ex y being set st
( y in bool the carrier of T & S1[x,y] ) )
assume x in NAT ; ::_thesis: ex y being set st
( y in bool the carrier of T & S1[x,y] )
then reconsider x9 = x as Element of NAT ;
set SS = { (S . j) where j is Element of NAT : j >= x9 } ;
{ (S . j) where j is Element of NAT : j >= x9 } c= bool the carrier of T
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in { (S . j) where j is Element of NAT : j >= x9 } or y in bool the carrier of T )
assume y in { (S . j) where j is Element of NAT : j >= x9 } ; ::_thesis: y in bool the carrier of T
then ex j being Element of NAT st
( S . j = y & j >= x9 ) ;
hence y in bool the carrier of T ; ::_thesis: verum
end;
then reconsider SS = { (S . j) where j is Element of NAT : j >= x9 } as Subset-Family of T ;
take Cl (union SS) ; ::_thesis: ( Cl (union SS) in bool the carrier of T & S1[x, Cl (union SS)] )
SS c= F
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in SS or y in F )
assume y in SS ; ::_thesis: y in F
then A4: ex j being Element of NAT st
( S . j = y & j >= x9 ) ;
dom S = NAT by FUNCT_2:def_1;
then y in rng S by A4, FUNCT_1:def_3;
hence y in F by A1; ::_thesis: verum
end;
hence ( Cl (union SS) in bool the carrier of T & S1[x, Cl (union SS)] ) ; ::_thesis: verum
end;
consider R being SetSequence of T such that
A5: for x being set st x in NAT holds
S1[x,R . x] from FUNCT_2:sch_1(A3);
A6: now__::_thesis:_for_n_being_set_st_n_in_dom_R_holds_
not_R_._n_is_empty
let n be set ; ::_thesis: ( n in dom R implies not R . n is empty )
assume n in dom R ; ::_thesis: not R . n is empty
then reconsider n9 = n as Element of NAT ;
A7: S . n9 c= Cl (S . n9) by PRE_TOPC:18;
consider SS being Subset-Family of T such that
SS c= F and
A8: SS = { (S . j) where j is Element of NAT : j >= n9 } and
A9: R . n9 = Cl (union SS) by A5;
S . n9 in SS by A8;
then A10: Cl (S . n9) c= Cl (union SS) by PRE_TOPC:19, ZFMISC_1:74;
dom S = NAT by FUNCT_2:def_1;
hence not R . n is empty by A2, A9, A7, A10, FUNCT_1:def_9; ::_thesis: verum
end;
now__::_thesis:_for_n_being_Nat_holds_R_._n_is_closed
let n be Nat; ::_thesis: R . n is closed
reconsider n9 = n as Element of NAT by ORDINAL1:def_12;
ex SS being Subset-Family of T st
( SS c= F & SS = { (S . j) where j is Element of NAT : j >= n9 } & R . n9 = Cl (union SS) ) by A5;
hence R . n is closed ; ::_thesis: verum
end;
then reconsider R = R as non-empty closed SetSequence of T by A6, Def6, FUNCT_1:def_9;
take R ; ::_thesis: ( R is V172() & ( F is locally_finite & S is one-to-one implies meet R = {} ) & ( for i being Nat ex Si being Subset-Family of T st
( R . i = Cl (union Si) & Si = { (S . j) where j is Element of NAT : j >= i } ) ) )
now__::_thesis:_for_n_being_Element_of_NAT_holds_R_._(n_+_1)_c=_R_._n
let n be Element of NAT ; ::_thesis: R . (n + 1) c= R . n
consider Sn being Subset-Family of T such that
Sn c= F and
A11: Sn = { (S . j) where j is Element of NAT : j >= n } and
A12: R . n = Cl (union Sn) by A5;
consider Sn1 being Subset-Family of T such that
Sn1 c= F and
A13: Sn1 = { (S . j) where j is Element of NAT : j >= n + 1 } and
A14: R . (n + 1) = Cl (union Sn1) by A5;
Sn1 c= Sn
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in Sn1 or y in Sn )
assume y in Sn1 ; ::_thesis: y in Sn
then consider j being Element of NAT such that
A15: y = S . j and
A16: j >= n + 1 by A13;
j >= n by A16, NAT_1:13;
hence y in Sn by A11, A15; ::_thesis: verum
end;
then union Sn1 c= union Sn by ZFMISC_1:77;
hence R . (n + 1) c= R . n by A12, A14, PRE_TOPC:19; ::_thesis: verum
end;
hence R is V172() by KURATO_0:def_3; ::_thesis: ( ( F is locally_finite & S is one-to-one implies meet R = {} ) & ( for i being Nat ex Si being Subset-Family of T st
( R . i = Cl (union Si) & Si = { (S . j) where j is Element of NAT : j >= i } ) ) )
thus ( F is locally_finite & S is one-to-one implies meet R = {} ) ::_thesis: for i being Nat ex Si being Subset-Family of T st
( R . i = Cl (union Si) & Si = { (S . j) where j is Element of NAT : j >= i } )
proof
A17: dom S = NAT by FUNCT_2:def_1;
then reconsider rngS = rng S as non empty Subset-Family of T by RELAT_1:42;
reconsider Sp = S as Function of NAT,rngS by A17, FUNCT_2:1;
assume that
A18: F is locally_finite and
A19: S is one-to-one ; ::_thesis: meet R = {}
reconsider S9 = Sp " as Function ;
reconsider S9 = S9 as Function of rngS,NAT by A19, FUNCT_2:25;
deffunc H1( Element of rngS) -> Element of NAT = S9 . $1;
assume meet R <> {} ; ::_thesis: contradiction
then consider x being set such that
A20: x in meet R by XBOOLE_0:def_1;
reconsider x = x as Point of T by A20;
rng S is locally_finite by A1, A18, PCOMPS_1:9;
then consider W being Subset of T such that
A21: x in W and
A22: W is open and
A23: { V where V is Subset of T : ( V in rngS & V meets W ) } is finite by PCOMPS_1:def_1;
set X = { V where V is Subset of T : ( V in rngS & V meets W ) } ;
set Y = { H1(z) where z is Element of rngS : z in { V where V is Subset of T : ( V in rngS & V meets W ) } } ;
A24: { H1(z) where z is Element of rngS : z in { V where V is Subset of T : ( V in rngS & V meets W ) } } is finite from FRAENKEL:sch_21(A23);
{ H1(z) where z is Element of rngS : z in { V where V is Subset of T : ( V in rngS & V meets W ) } } c= NAT
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in { H1(z) where z is Element of rngS : z in { V where V is Subset of T : ( V in rngS & V meets W ) } } or y in NAT )
assume y in { H1(z) where z is Element of rngS : z in { V where V is Subset of T : ( V in rngS & V meets W ) } } ; ::_thesis: y in NAT
then ex z being Element of rngS st
( y = H1(z) & z in { V where V is Subset of T : ( V in rngS & V meets W ) } ) ;
hence y in NAT ; ::_thesis: verum
end;
then reconsider Y = { H1(z) where z is Element of rngS : z in { V where V is Subset of T : ( V in rngS & V meets W ) } } as Subset of NAT ;
consider n being Nat such that
A25: for k being Nat st k in Y holds
k <= n by A24, STIRL2_1:56;
set n1 = n + 1;
A26: x in R . (n + 1) by A20, KURATO_0:3;
consider Sn being Subset-Family of T such that
A27: Sn c= F and
A28: Sn = { (S . j) where j is Element of NAT : j >= n + 1 } and
A29: R . (n + 1) = Cl (union Sn) by A5;
Cl (union Sn) = union (clf Sn) by A18, A27, PCOMPS_1:9, PCOMPS_1:20;
then consider CLF being set such that
A30: x in CLF and
A31: CLF in clf Sn by A29, A26, TARSKI:def_4;
reconsider CLF = CLF as Subset of T by A31;
consider U being Subset of T such that
A32: CLF = Cl U and
A33: U in Sn by A31, PCOMPS_1:def_2;
consider j being Element of NAT such that
A34: U = S . j and
A35: j >= n + 1 by A28, A33;
A36: Sp . j in rngS ;
Sp . j meets W by A21, A22, A30, A32, A34, TOPS_1:12;
then A37: Sp . j in { V where V is Subset of T : ( V in rngS & V meets W ) } by A36;
(S ") . (S . j) = j by A19, FUNCT_2:26;
then j in Y by A37;
then j <= n by A25;
hence contradiction by A35, NAT_1:13; ::_thesis: verum
end;
let i be Nat; ::_thesis: ex Si being Subset-Family of T st
( R . i = Cl (union Si) & Si = { (S . j) where j is Element of NAT : j >= i } )
i in NAT by ORDINAL1:def_12;
then ex SS being Subset-Family of T st
( SS c= F & SS = { (S . j) where j is Element of NAT : j >= i } & R . i = Cl (union SS) ) by A5;
hence ex Si being Subset-Family of T st
( R . i = Cl (union Si) & Si = { (S . j) where j is Element of NAT : j >= i } ) ; ::_thesis: verum
end;
Lm2: for T being non empty TopSpace st T is countably_compact holds
for F being Subset-Family of T st F is locally_finite & F is with_non-empty_elements holds
F is finite
proof
let T be non empty TopSpace; ::_thesis: ( T is countably_compact implies for F being Subset-Family of T st F is locally_finite & F is with_non-empty_elements holds
F is finite )
assume A1: T is countably_compact ; ::_thesis: for F being Subset-Family of T st F is locally_finite & F is with_non-empty_elements holds
F is finite
given F being Subset-Family of T such that A2: F is locally_finite and
A3: F is with_non-empty_elements and
A4: F is infinite ; ::_thesis: contradiction
consider f being Function of NAT,F such that
A5: f is one-to-one by A4, DICKSON:3;
A6: rng f c= F ;
reconsider f = f as SetSequence of T by A4, FUNCT_2:7;
now__::_thesis:_for_x_being_set_st_x_in_dom_f_holds_
not_f_._x_is_empty
let x be set ; ::_thesis: ( x in dom f implies not f . x is empty )
assume x in dom f ; ::_thesis: not f . x is empty
then f . x in rng f by FUNCT_1:def_3;
hence not f . x is empty by A3, A6, SETFAM_1:def_8; ::_thesis: verum
end;
then f is non-empty by FUNCT_1:def_9;
then ex R being non-empty closed SetSequence of T st
( R is V172() & ( F is locally_finite & f is one-to-one implies meet R = {} ) & ( for i being Nat ex fi being Subset-Family of T st
( R . i = Cl (union fi) & fi = { (f . j) where j is Element of NAT : j >= i } ) ) ) by A6, Th23;
hence contradiction by A1, A2, A5, Th22; ::_thesis: verum
end;
Lm3: for T being non empty TopSpace st ( for F being Subset-Family of T st F is locally_finite & F is with_non-empty_elements holds
F is finite ) holds
for F being Subset-Family of T st F is locally_finite & ( for A being Subset of T st A in F holds
card A = 1 ) holds
F is finite
proof
let T be non empty TopSpace; ::_thesis: ( ( for F being Subset-Family of T st F is locally_finite & F is with_non-empty_elements holds
F is finite ) implies for F being Subset-Family of T st F is locally_finite & ( for A being Subset of T st A in F holds
card A = 1 ) holds
F is finite )
assume A1: for F being Subset-Family of T st F is locally_finite & F is with_non-empty_elements holds
F is finite ; ::_thesis: for F being Subset-Family of T st F is locally_finite & ( for A being Subset of T st A in F holds
card A = 1 ) holds
F is finite
let F be Subset-Family of T; ::_thesis: ( F is locally_finite & ( for A being Subset of T st A in F holds
card A = 1 ) implies F is finite )
assume that
A2: F is locally_finite and
A3: for A being Subset of T st A in F holds
card A = 1 ; ::_thesis: F is finite
not {} T in F by A3, CARD_1:27;
then F is with_non-empty_elements by SETFAM_1:def_8;
hence F is finite by A1, A2; ::_thesis: verum
end;
Lm4: for T being non empty TopSpace st ( for F being Subset-Family of T st F is locally_finite & ( for A being Subset of T st A in F holds
card A = 1 ) holds
F is finite ) holds
for A being Subset of T st A is infinite holds
not Der A is empty
proof
deffunc H1( set ) -> set = meet $1;
let T be non empty TopSpace; ::_thesis: ( ( for F being Subset-Family of T st F is locally_finite & ( for A being Subset of T st A in F holds
card A = 1 ) holds
F is finite ) implies for A being Subset of T st A is infinite holds
not Der A is empty )
assume A1: for F being Subset-Family of T st F is locally_finite & ( for A being Subset of T st A in F holds
card A = 1 ) holds
F is finite ; ::_thesis: for A being Subset of T st A is infinite holds
not Der A is empty
let A be Subset of T; ::_thesis: ( A is infinite implies not Der A is empty )
assume A2: A is infinite ; ::_thesis: not Der A is empty
set F = { {x} where x is Element of T : x in A } ;
reconsider F = { {x} where x is Element of T : x in A } as Subset-Family of T by RELSET_2:16;
set PP = { H1(y) where y is Element of bool the carrier of T : y in F } ;
A3: A c= { H1(y) where y is Element of bool the carrier of T : y in F }
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in A or y in { H1(y) where y is Element of bool the carrier of T : y in F } )
assume A4: y in A ; ::_thesis: y in { H1(y) where y is Element of bool the carrier of T : y in F }
reconsider y9 = y as Point of T by A4;
{y9} in F by A4;
then H1({y9}) in { H1(y) where y is Element of bool the carrier of T : y in F } ;
hence y in { H1(y) where y is Element of bool the carrier of T : y in F } by SETFAM_1:10; ::_thesis: verum
end;
assume A5: Der A is empty ; ::_thesis: contradiction
A6: F is locally_finite
proof
let x be Point of T; :: according to PCOMPS_1:def_1 ::_thesis: ex b1 being Element of bool the carrier of T st
( x in b1 & b1 is open & { b2 where b2 is Element of bool the carrier of T : ( b2 in F & not b2 misses b1 ) } is finite )
consider U being open Subset of T such that
A7: x in U and
A8: for y being Point of T st y in A /\ U holds
x = y by A5, TOPGEN_1:17;
set M = { V where V is Subset of T : ( V in F & V meets U ) } ;
take U ; ::_thesis: ( x in U & U is open & { b1 where b1 is Element of bool the carrier of T : ( b1 in F & not b1 misses U ) } is finite )
{ V where V is Subset of T : ( V in F & V meets U ) } c= {{x}}
proof
A9: {x} in {{x}} by TARSKI:def_1;
let v be set ; :: according to TARSKI:def_3 ::_thesis: ( not v in { V where V is Subset of T : ( V in F & V meets U ) } or v in {{x}} )
assume v in { V where V is Subset of T : ( V in F & V meets U ) } ; ::_thesis: v in {{x}}
then consider V being Subset of T such that
A10: v = V and
A11: V in F and
A12: V meets U ;
consider y being Point of T such that
A13: V = {y} and
A14: y in A by A11;
y in U by A12, A13, ZFMISC_1:50;
then y in A /\ U by A14, XBOOLE_0:def_4;
hence v in {{x}} by A8, A10, A13, A9; ::_thesis: verum
end;
hence ( x in U & U is open & { b1 where b1 is Element of bool the carrier of T : ( b1 in F & not b1 misses U ) } is finite ) by A7; ::_thesis: verum
end;
now__::_thesis:_for_a_being_Subset_of_T_st_a_in_F_holds_
card_a_=_1
let a be Subset of T; ::_thesis: ( a in F implies card a = 1 )
assume a in F ; ::_thesis: card a = 1
then ex y being Point of T st
( a = {y} & y in A ) ;
hence card a = 1 by CARD_1:30; ::_thesis: verum
end;
then A15: F is finite by A1, A6;
{ H1(y) where y is Element of bool the carrier of T : y in F } is finite from FRAENKEL:sch_21(A15);
hence contradiction by A2, A3; ::_thesis: verum
end;
theorem :: COMPL_SP:24
canceled;
theorem Th25: :: COMPL_SP:25
for X being non empty set
for F being SetSequence of X st F is V172() holds
for S being Function of NAT,X st ( for n being Nat holds S . n in F . n ) & rng S is finite holds
not meet F is empty
proof
let X be non empty set ; ::_thesis: for F being SetSequence of X st F is V172() holds
for S being Function of NAT,X st ( for n being Nat holds S . n in F . n ) & rng S is finite holds
not meet F is empty
let F be SetSequence of X; ::_thesis: ( F is V172() implies for S being Function of NAT,X st ( for n being Nat holds S . n in F . n ) & rng S is finite holds
not meet F is empty )
assume A1: F is V172() ; ::_thesis: for S being Function of NAT,X st ( for n being Nat holds S . n in F . n ) & rng S is finite holds
not meet F is empty
let S be Function of NAT,X; ::_thesis: ( ( for n being Nat holds S . n in F . n ) & rng S is finite implies not meet F is empty )
assume A2: for n being Nat holds S . n in F . n ; ::_thesis: ( not rng S is finite or not meet F is empty )
A3: dom S = NAT by FUNCT_2:def_1;
assume rng S is finite ; ::_thesis: not meet F is empty
then consider x being set such that
x in rng S and
A4: S " {x} is infinite by A3, CARD_2:101;
now__::_thesis:_for_n_being_Element_of_NAT_holds_x_in_F_._n
let n be Element of NAT ; ::_thesis: x in F . n
ex i being Nat st
( x in F . i & i >= n )
proof
assume A5: for i being Nat st x in F . i holds
i < n ; ::_thesis: contradiction
S " {x} c= n
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in S " {x} or y in n )
assume A6: y in S " {x} ; ::_thesis: y in n
reconsider i = y as Element of NAT by A6;
S . i in {x} by A6, FUNCT_1:def_7;
then A7: S . i = x by TARSKI:def_1;
S . i in F . i by A2;
then i < n by A5, A7;
hence y in n by NAT_1:44; ::_thesis: verum
end;
hence contradiction by A4; ::_thesis: verum
end;
then consider i being Nat such that
A8: x in F . i and
A9: i >= n ;
i in NAT by ORDINAL1:def_12;
then F . i c= F . n by A1, A9, PROB_1:def_4;
hence x in F . n by A8; ::_thesis: verum
end;
hence not meet F is empty by KURATO_0:3; ::_thesis: verum
end;
Lm5: for T being non empty T_1 TopSpace st ( for A being Subset of T st A is infinite & A is countable holds
not Der A is empty ) holds
T is countably_compact
proof
let T be non empty T_1 TopSpace; ::_thesis: ( ( for A being Subset of T st A is infinite & A is countable holds
not Der A is empty ) implies T is countably_compact )
assume A1: for A being Subset of T st A is infinite & A is countable holds
not Der A is empty ; ::_thesis: T is countably_compact
assume not T is countably_compact ; ::_thesis: contradiction
then consider S being non-empty closed SetSequence of T such that
A2: S is V172() and
A3: meet S = {} by Th22;
defpred S1[ set , set ] means $2 in S . $1;
A4: for x being set st x in NAT holds
ex y being set st
( y in the carrier of T & S1[x,y] )
proof
let x be set ; ::_thesis: ( x in NAT implies ex y being set st
( y in the carrier of T & S1[x,y] ) )
assume x in NAT ; ::_thesis: ex y being set st
( y in the carrier of T & S1[x,y] )
then reconsider x9 = x as Element of NAT ;
dom S = NAT by FUNCT_2:def_1;
then not S . x9 is empty by FUNCT_1:def_9;
then consider y being set such that
A5: y in S . x9 by XBOOLE_0:def_1;
take y ; ::_thesis: ( y in the carrier of T & S1[x,y] )
thus ( y in the carrier of T & S1[x,y] ) by A5; ::_thesis: verum
end;
consider F being sequence of T such that
A6: for x being set st x in NAT holds
S1[x,F . x] from FUNCT_2:sch_1(A4);
reconsider rngF = rng F as Subset of T ;
A7: now__::_thesis:_for_n_being_Nat_holds_F_._n_in_S_._n
let n be Nat; ::_thesis: F . n in S . n
n in NAT by ORDINAL1:def_12;
hence F . n in S . n by A6; ::_thesis: verum
end;
dom F = NAT by FUNCT_2:def_1;
then rng F is countable by CARD_3:93;
then not Der rngF is empty by A1, A2, A3, A7, Th25;
then consider p being set such that
A8: p in Der rngF by XBOOLE_0:def_1;
reconsider p = p as Point of T by A8;
consider n being Element of NAT such that
A9: not p in S . n by A3, KURATO_0:3;
A10: p in (S . n) ` by A9, XBOOLE_0:def_5;
deffunc H1( set ) -> set = F . $1;
set F1n = { H1(i) where i is Element of NAT : i in n + 1 } ;
A11: { H1(i) where i is Element of NAT : i in n + 1 } c= the carrier of T
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { H1(i) where i is Element of NAT : i in n + 1 } or x in the carrier of T )
assume x in { H1(i) where i is Element of NAT : i in n + 1 } ; ::_thesis: x in the carrier of T
then ex i being Element of NAT st
( x = F . i & i in n + 1 ) ;
hence x in the carrier of T ; ::_thesis: verum
end;
A12: n + 1 is finite ;
A13: { H1(i) where i is Element of NAT : i in n + 1 } is finite from FRAENKEL:sch_21(A12);
reconsider F1n = { H1(i) where i is Element of NAT : i in n + 1 } as Subset of T by A11;
set U = ((S . n) `) \ (F1n \ {p});
reconsider U = ((S . n) `) \ (F1n \ {p}) as Subset of T ;
p in {p} by TARSKI:def_1;
then not p in F1n \ {p} by XBOOLE_0:def_5;
then A14: p in U by A10, XBOOLE_0:def_5;
S . n is closed by Def6;
then U is open by A13, FRECHET:4;
then consider q being Point of T such that
A15: q in rngF /\ U and
A16: q <> p by A8, A14, TOPGEN_1:17;
q in rngF by A15, XBOOLE_0:def_4;
then consider i being set such that
A17: i in dom F and
A18: F . i = q by FUNCT_1:def_3;
reconsider i = i as Element of NAT by A17;
percases ( i <= n or i > n ) ;
suppose i <= n ; ::_thesis: contradiction
then i < n + 1 by NAT_1:13;
then i in n + 1 by NAT_1:44;
then q in F1n by A18;
then q in F1n \ {p} by A16, ZFMISC_1:56;
then not q in U by XBOOLE_0:def_5;
hence contradiction by A15, XBOOLE_0:def_4; ::_thesis: verum
end;
suppose i > n ; ::_thesis: contradiction
then A19: S . i c= S . n by A2, PROB_1:def_4;
q in S . i by A6, A18;
then not q in (S . n) ` by A19, XBOOLE_0:def_5;
then not q in U by XBOOLE_0:def_5;
hence contradiction by A15, XBOOLE_0:def_4; ::_thesis: verum
end;
end;
end;
Lm6: for T being non empty TopSpace st ( for F being Subset-Family of T st F is locally_finite & ( for A being Subset of T st A in F holds
card A = 1 ) holds
F is finite ) holds
T is countably_compact
proof
deffunc H1( set ) -> set = meet $1;
let T be non empty TopSpace; ::_thesis: ( ( for F being Subset-Family of T st F is locally_finite & ( for A being Subset of T st A in F holds
card A = 1 ) holds
F is finite ) implies T is countably_compact )
assume A1: for F being Subset-Family of T st F is locally_finite & ( for A being Subset of T st A in F holds
card A = 1 ) holds
F is finite ; ::_thesis: T is countably_compact
assume not T is countably_compact ; ::_thesis: contradiction
then consider S being non-empty closed SetSequence of T such that
A2: S is V172() and
A3: meet S = {} by Th22;
defpred S1[ set , set ] means $2 in S . $1;
A4: for x being set st x in NAT holds
ex y being set st
( y in the carrier of T & S1[x,y] )
proof
let x be set ; ::_thesis: ( x in NAT implies ex y being set st
( y in the carrier of T & S1[x,y] ) )
assume x in NAT ; ::_thesis: ex y being set st
( y in the carrier of T & S1[x,y] )
then reconsider x9 = x as Element of NAT ;
dom S = NAT by FUNCT_2:def_1;
then not S . x9 is empty by FUNCT_1:def_9;
then consider y being set such that
A5: y in S . x9 by XBOOLE_0:def_1;
take y ; ::_thesis: ( y in the carrier of T & S1[x,y] )
thus ( y in the carrier of T & S1[x,y] ) by A5; ::_thesis: verum
end;
consider F being sequence of T such that
A6: for x being set st x in NAT holds
S1[x,F . x] from FUNCT_2:sch_1(A4);
reconsider rngF = rng F as Subset of T ;
set A = { {x} where x is Element of T : x in rngF } ;
reconsider A = { {x} where x is Element of T : x in rngF } as Subset-Family of T by RELSET_2:16;
A7: A is locally_finite
proof
deffunc H2( set ) -> set = {(F . $1)};
let x be Point of T; :: according to PCOMPS_1:def_1 ::_thesis: ex b1 being Element of bool the carrier of T st
( x in b1 & b1 is open & { b2 where b2 is Element of bool the carrier of T : ( b2 in A & not b2 misses b1 ) } is finite )
consider i being Element of NAT such that
A8: not x in S . i by A3, KURATO_0:3;
take Si9 = (S . i) ` ; ::_thesis: ( x in Si9 & Si9 is open & { b1 where b1 is Element of bool the carrier of T : ( b1 in A & not b1 misses Si9 ) } is finite )
S . i is closed by Def6;
hence ( x in Si9 & Si9 is open ) by A8, SUBSET_1:29; ::_thesis: { b1 where b1 is Element of bool the carrier of T : ( b1 in A & not b1 misses Si9 ) } is finite
set meetS = { V where V is Subset of T : ( V in A & V meets Si9 ) } ;
set SI = { H2(j) where j is Element of NAT : j in i } ;
A9: { V where V is Subset of T : ( V in A & V meets Si9 ) } c= { H2(j) where j is Element of NAT : j in i }
proof
let v be set ; :: according to TARSKI:def_3 ::_thesis: ( not v in { V where V is Subset of T : ( V in A & V meets Si9 ) } or v in { H2(j) where j is Element of NAT : j in i } )
assume v in { V where V is Subset of T : ( V in A & V meets Si9 ) } ; ::_thesis: v in { H2(j) where j is Element of NAT : j in i }
then consider V being Subset of T such that
A10: V = v and
A11: V in A and
A12: V meets Si9 ;
consider y being Point of T such that
A13: V = {y} and
A14: y in rng F by A11;
consider z being set such that
A15: z in dom F and
A16: y = F . z by A14, FUNCT_1:def_3;
reconsider z = z as Element of NAT by A15;
z in i
proof
assume not z in i ; ::_thesis: contradiction
then z >= i by NAT_1:44;
then A17: S . z c= S . i by A2, PROB_1:def_4;
A18: y in Si9 by A12, A13, ZFMISC_1:50;
y in S . z by A6, A16;
hence contradiction by A17, A18, XBOOLE_0:def_5; ::_thesis: verum
end;
hence v in { H2(j) where j is Element of NAT : j in i } by A10, A13, A16; ::_thesis: verum
end;
A19: i is finite ;
{ H2(j) where j is Element of NAT : j in i } is finite from FRAENKEL:sch_21(A19);
hence { b1 where b1 is Element of bool the carrier of T : ( b1 in A & not b1 misses Si9 ) } is finite by A9; ::_thesis: verum
end;
set PP = { H1(y) where y is Element of bool the carrier of T : y in A } ;
A20: rngF c= { H1(y) where y is Element of bool the carrier of T : y in A }
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rngF or y in { H1(y) where y is Element of bool the carrier of T : y in A } )
assume A21: y in rngF ; ::_thesis: y in { H1(y) where y is Element of bool the carrier of T : y in A }
reconsider y9 = y as Point of T by A21;
{y9} in A by A21;
then H1({y9}) in { H1(y) where y is Element of bool the carrier of T : y in A } ;
hence y in { H1(y) where y is Element of bool the carrier of T : y in A } by SETFAM_1:10; ::_thesis: verum
end;
A22: now__::_thesis:_for_n_being_Nat_holds_F_._n_in_S_._n
let n be Nat; ::_thesis: F . n in S . n
n in NAT by ORDINAL1:def_12;
hence F . n in S . n by A6; ::_thesis: verum
end;
now__::_thesis:_for_a_being_Subset_of_T_st_a_in_A_holds_
card_a_=_1
let a be Subset of T; ::_thesis: ( a in A implies card a = 1 )
assume a in A ; ::_thesis: card a = 1
then ex y being Point of T st
( a = {y} & y in rngF ) ;
hence card a = 1 by CARD_1:30; ::_thesis: verum
end;
then A23: A is finite by A1, A7;
{ H1(y) where y is Element of bool the carrier of T : y in A } is finite from FRAENKEL:sch_21(A23);
hence contradiction by A2, A3, A22, A20, Th25; ::_thesis: verum
end;
theorem Th26: :: COMPL_SP:26
for T being non empty TopSpace holds
( T is countably_compact iff for F being Subset-Family of T st F is locally_finite & F is with_non-empty_elements holds
F is finite )
proof
let T be non empty TopSpace; ::_thesis: ( T is countably_compact iff for F being Subset-Family of T st F is locally_finite & F is with_non-empty_elements holds
F is finite )
thus ( T is countably_compact implies for F being Subset-Family of T st F is locally_finite & F is with_non-empty_elements holds
F is finite ) by Lm2; ::_thesis: ( ( for F being Subset-Family of T st F is locally_finite & F is with_non-empty_elements holds
F is finite ) implies T is countably_compact )
assume for F being Subset-Family of T st F is locally_finite & F is with_non-empty_elements holds
F is finite ; ::_thesis: T is countably_compact
then for F being Subset-Family of T st F is locally_finite & ( for A being Subset of T st A in F holds
card A = 1 ) holds
F is finite by Lm3;
hence T is countably_compact by Lm6; ::_thesis: verum
end;
theorem Th27: :: COMPL_SP:27
for T being non empty TopSpace holds
( T is countably_compact iff for F being Subset-Family of T st F is locally_finite & ( for A being Subset of T st A in F holds
card A = 1 ) holds
F is finite )
proof
let T be non empty TopSpace; ::_thesis: ( T is countably_compact iff for F being Subset-Family of T st F is locally_finite & ( for A being Subset of T st A in F holds
card A = 1 ) holds
F is finite )
thus ( T is countably_compact implies for F being Subset-Family of T st F is locally_finite & ( for A being Subset of T st A in F holds
card A = 1 ) holds
F is finite ) ::_thesis: ( ( for F being Subset-Family of T st F is locally_finite & ( for A being Subset of T st A in F holds
card A = 1 ) holds
F is finite ) implies T is countably_compact )
proof
assume T is countably_compact ; ::_thesis: for F being Subset-Family of T st F is locally_finite & ( for A being Subset of T st A in F holds
card A = 1 ) holds
F is finite
then for F being Subset-Family of T st F is locally_finite & F is with_non-empty_elements holds
F is finite by Th26;
hence for F being Subset-Family of T st F is locally_finite & ( for A being Subset of T st A in F holds
card A = 1 ) holds
F is finite by Lm3; ::_thesis: verum
end;
thus ( ( for F being Subset-Family of T st F is locally_finite & ( for A being Subset of T st A in F holds
card A = 1 ) holds
F is finite ) implies T is countably_compact ) by Lm6; ::_thesis: verum
end;
theorem Th28: :: COMPL_SP:28
for T being non empty T_1 TopSpace holds
( T is countably_compact iff for A being Subset of T st A is infinite holds
not Der A is empty )
proof
let T be non empty T_1 TopSpace; ::_thesis: ( T is countably_compact iff for A being Subset of T st A is infinite holds
not Der A is empty )
thus ( T is countably_compact implies for A being Subset of T st A is infinite holds
not Der A is empty ) ::_thesis: ( ( for A being Subset of T st A is infinite holds
not Der A is empty ) implies T is countably_compact )
proof
assume T is countably_compact ; ::_thesis: for A being Subset of T st A is infinite holds
not Der A is empty
then for F being Subset-Family of T st F is locally_finite & ( for A being Subset of T st A in F holds
card A = 1 ) holds
F is finite by Th27;
hence for A being Subset of T st A is infinite holds
not Der A is empty by Lm4; ::_thesis: verum
end;
assume for A being Subset of T st A is infinite holds
not Der A is empty ; ::_thesis: T is countably_compact
then for A being Subset of T st A is infinite & A is countable holds
not Der A is empty ;
hence T is countably_compact by Lm5; ::_thesis: verum
end;
theorem :: COMPL_SP:29
for T being non empty T_1 TopSpace holds
( T is countably_compact iff for A being Subset of T st A is infinite & A is countable holds
not Der A is empty ) by Lm5, Th28;
scheme :: COMPL_SP:sch 1
Th39{ F1() -> non empty set , P1[ set , set ] } :
ex A being Subset of F1() st
( ( for x, y being Element of F1() st x in A & y in A & x <> y holds
P1[x,y] ) & ( for x being Element of F1() ex y being Element of F1() st
( y in A & P1[x,y] ) ) )
provided
A1: for x, y being Element of F1() holds
( P1[x,y] iff P1[y,x] ) and
A2: for x being Element of F1() holds P1[x,x]
proof
set bX = bool F1();
consider R being Relation such that
A3: R well_orders F1() by WELLORD2:17;
R /\ [:F1(),F1():] c= [:F1(),F1():] by XBOOLE_1:17;
then reconsider R2 = R |_2 F1() as Relation of F1() by WELLORD1:def_6;
reconsider RS = RelStr(# F1(),R2 #) as non empty RelStr ;
set cRS = the carrier of RS;
defpred S1[ set , set , set ] means for p being Element of F1()
for f being PartFunc of the carrier of RS,(bool F1()) st $1 = p & $2 = f holds
( ( ( for q being Element of F1() st q in union (rng f) holds
P1[p,q] ) implies $3 = (union (rng f)) \/ {p} ) & ( ex q being Element of F1() st
( q in union (rng f) & P1[p,q] ) implies $3 = union (rng f) ) );
A4: for x, y being set st x in the carrier of RS & y in PFuncs ( the carrier of RS,(bool F1())) holds
ex z being set st
( z in bool F1() & S1[x,y,z] )
proof
let x, y be set ; ::_thesis: ( x in the carrier of RS & y in PFuncs ( the carrier of RS,(bool F1())) implies ex z being set st
( z in bool F1() & S1[x,y,z] ) )
assume that
A5: x in the carrier of RS and
A6: y in PFuncs ( the carrier of RS,(bool F1())) ; ::_thesis: ex z being set st
( z in bool F1() & S1[x,y,z] )
reconsider f = y as PartFunc of the carrier of RS,(bool F1()) by A6, PARTFUN1:46;
reconsider p = x as Element of F1() by A5;
percases ( for q being Element of F1() st q in union (rng f) holds
P1[p,q] or ex q being Element of F1() st
( q in union (rng f) & P1[p,q] ) ) ;
supposeA7: for q being Element of F1() st q in union (rng f) holds
P1[p,q] ; ::_thesis: ex z being set st
( z in bool F1() & S1[x,y,z] )
take (union (rng f)) \/ {p} ; ::_thesis: ( (union (rng f)) \/ {p} in bool F1() & S1[x,y,(union (rng f)) \/ {p}] )
thus ( (union (rng f)) \/ {p} in bool F1() & S1[x,y,(union (rng f)) \/ {p}] ) by A7; ::_thesis: verum
end;
supposeA8: ex q being Element of F1() st
( q in union (rng f) & P1[p,q] ) ; ::_thesis: ex z being set st
( z in bool F1() & S1[x,y,z] )
take union (rng f) ; ::_thesis: ( union (rng f) in bool F1() & S1[x,y, union (rng f)] )
thus ( union (rng f) in bool F1() & S1[x,y, union (rng f)] ) by A8; ::_thesis: verum
end;
end;
end;
consider h being Function of [: the carrier of RS,(PFuncs ( the carrier of RS,(bool F1()))):],(bool F1()) such that
A9: for x, y being set st x in the carrier of RS & y in PFuncs ( the carrier of RS,(bool F1())) holds
S1[x,y,h . (x,y)] from BINOP_1:sch_1(A4);
set IRS = the InternalRel of RS;
A10: R2 well_orders F1() by A3, PCOMPS_2:1;
then R2 is_well_founded_in F1() by WELLORD1:def_5;
then A11: RS is well_founded by WELLFND1:def_2;
then consider f being Function of the carrier of RS,(bool F1()) such that
A12: f is_recursively_expressed_by h by WELLFND1:11;
defpred S2[ set ] means ( f . $1 c= ( the InternalRel of RS -Seg $1) \/ {$1} & ( $1 in f . $1 implies for r being Element of F1() st r in union (rng (f | ( the InternalRel of RS -Seg $1))) holds
P1[$1,r] ) & ( not $1 in f . $1 implies ex r being Element of F1() st
( r in union (rng (f | ( the InternalRel of RS -Seg $1))) & P1[$1,r] ) ) );
reconsider rngf = rng f as Subset of (bool F1()) ;
take A = union rngf; ::_thesis: ( ( for x, y being Element of F1() st x in A & y in A & x <> y holds
P1[x,y] ) & ( for x being Element of F1() ex y being Element of F1() st
( y in A & P1[x,y] ) ) )
A13: field R2 = F1() by A3, PCOMPS_2:1;
then A14: R2 is well-ordering by A10, WELLORD1:4;
A15: for x being Element of RS st ( for y being Element of RS st y <> x & [y,x] in the InternalRel of RS holds
S2[y] ) holds
S2[x]
proof
let x be Element of RS; ::_thesis: ( ( for y being Element of RS st y <> x & [y,x] in the InternalRel of RS holds
S2[y] ) implies S2[x] )
assume A16: for y being Element of RS st y <> x & [y,x] in the InternalRel of RS holds
S2[y] ; ::_thesis: S2[x]
set fIx = f | ( the InternalRel of RS -Seg x);
A17: union (rng (f | ( the InternalRel of RS -Seg x))) c= the InternalRel of RS -Seg x
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in union (rng (f | ( the InternalRel of RS -Seg x))) or y in the InternalRel of RS -Seg x )
assume y in union (rng (f | ( the InternalRel of RS -Seg x))) ; ::_thesis: y in the InternalRel of RS -Seg x
then consider z being set such that
A18: y in z and
A19: z in rng (f | ( the InternalRel of RS -Seg x)) by TARSKI:def_4;
consider t being set such that
A20: t in dom (f | ( the InternalRel of RS -Seg x)) and
A21: (f | ( the InternalRel of RS -Seg x)) . t = z by A19, FUNCT_1:def_3;
A22: t in the InternalRel of RS -Seg x by A20, RELAT_1:57;
reconsider t = t as Element of RS by A20;
A23: {t} c= the InternalRel of RS -Seg x by A22, ZFMISC_1:31;
A24: [t,x] in the InternalRel of RS by A22, WELLORD1:1;
then the InternalRel of RS -Seg t c= the InternalRel of RS -Seg x by A13, A14, WELLORD1:29;
then A25: ( the InternalRel of RS -Seg t) \/ {t} c= the InternalRel of RS -Seg x by A23, XBOOLE_1:8;
t <> x by A22, WELLORD1:1;
then A26: f . t c= ( the InternalRel of RS -Seg t) \/ {t} by A16, A24;
(f | ( the InternalRel of RS -Seg x)) . t = f . t by A20, FUNCT_1:47;
then y in ( the InternalRel of RS -Seg t) \/ {t} by A18, A21, A26;
hence y in the InternalRel of RS -Seg x by A25; ::_thesis: verum
end;
A27: f | ( the InternalRel of RS -Seg x) in PFuncs ( the carrier of RS,(bool F1())) by PARTFUN1:45;
A28: f . x = h . (x,(f | ( the InternalRel of RS -Seg x))) by A12, WELLFND1:def_5;
percases ( for q being Element of F1() st q in union (rng (f | ( the InternalRel of RS -Seg x))) holds
P1[x,q] or ex q being Element of F1() st
( q in union (rng (f | ( the InternalRel of RS -Seg x))) & P1[x,q] ) ) ;
supposeA29: for q being Element of F1() st q in union (rng (f | ( the InternalRel of RS -Seg x))) holds
P1[x,q] ; ::_thesis: S2[x]
then A30: f . x = (union (rng (f | ( the InternalRel of RS -Seg x)))) \/ {x} by A9, A28, A27;
hence f . x c= ( the InternalRel of RS -Seg x) \/ {x} by A17, XBOOLE_1:9; ::_thesis: ( ( x in f . x implies for r being Element of F1() st r in union (rng (f | ( the InternalRel of RS -Seg x))) holds
P1[x,r] ) & ( not x in f . x implies ex r being Element of F1() st
( r in union (rng (f | ( the InternalRel of RS -Seg x))) & P1[x,r] ) ) )
thus ( x in f . x implies for r being Element of F1() st r in union (rng (f | ( the InternalRel of RS -Seg x))) holds
P1[x,r] ) by A29; ::_thesis: ( not x in f . x implies ex r being Element of F1() st
( r in union (rng (f | ( the InternalRel of RS -Seg x))) & P1[x,r] ) )
A31: x in {x} by TARSKI:def_1;
assume not x in f . x ; ::_thesis: ex r being Element of F1() st
( r in union (rng (f | ( the InternalRel of RS -Seg x))) & P1[x,r] )
hence ex r being Element of F1() st
( r in union (rng (f | ( the InternalRel of RS -Seg x))) & P1[x,r] ) by A30, A31, XBOOLE_0:def_3; ::_thesis: verum
end;
supposeA32: ex q being Element of F1() st
( q in union (rng (f | ( the InternalRel of RS -Seg x))) & P1[x,q] ) ; ::_thesis: S2[x]
then A33: f . x c= the InternalRel of RS -Seg x by A9, A17, A28, A27;
the InternalRel of RS -Seg x c= ( the InternalRel of RS -Seg x) \/ {x} by XBOOLE_1:7;
hence f . x c= ( the InternalRel of RS -Seg x) \/ {x} by A33, XBOOLE_1:1; ::_thesis: ( ( x in f . x implies for r being Element of F1() st r in union (rng (f | ( the InternalRel of RS -Seg x))) holds
P1[x,r] ) & ( not x in f . x implies ex r being Element of F1() st
( r in union (rng (f | ( the InternalRel of RS -Seg x))) & P1[x,r] ) ) )
thus ( ( x in f . x implies for r being Element of F1() st r in union (rng (f | ( the InternalRel of RS -Seg x))) holds
P1[x,r] ) & ( not x in f . x implies ex r being Element of F1() st
( r in union (rng (f | ( the InternalRel of RS -Seg x))) & P1[x,r] ) ) ) by A32, A33, WELLORD1:1; ::_thesis: verum
end;
end;
end;
A34: for x being Element of RS holds S2[x] from WELLFND1:sch_3(A15, A11);
thus for x, y being Element of F1() st x in A & y in A & x <> y holds
P1[x,y] ::_thesis: for x being Element of F1() ex y being Element of F1() st
( y in A & P1[x,y] )
proof
A35: now__::_thesis:_for_x_being_Element_of_F1()_st_x_in_A_holds_
x_in_f_._x
let x be Element of F1(); ::_thesis: ( x in A implies x in f . x )
assume x in A ; ::_thesis: x in f . x
then consider y being set such that
A36: x in y and
A37: y in rng f by TARSKI:def_4;
defpred S3[ set ] means x in f . $1;
consider z being set such that
A38: z in dom f and
A39: f . z = y by A37, FUNCT_1:def_3;
reconsider z = z as Element of RS by A38;
A40: S3[z] by A36, A39;
consider p being Element of RS such that
A41: S3[p] and
A42: for q being Element of RS holds
( not p <> q or not S3[q] or not [q,p] in the InternalRel of RS ) from WELLFND1:sch_2(A40, A11);
p = x
proof
set fIp = f | ( the InternalRel of RS -Seg p);
A43: f | ( the InternalRel of RS -Seg p) in PFuncs ( the carrier of RS,(bool F1())) by PARTFUN1:45;
A44: f . p = h . (p,(f | ( the InternalRel of RS -Seg p))) by A12, WELLFND1:def_5;
assume A45: p <> x ; ::_thesis: contradiction
now__::_thesis:_x_in_union_(rng_(f_|_(_the_InternalRel_of_RS_-Seg_p)))
percases ( for q being Element of F1() st q in union (rng (f | ( the InternalRel of RS -Seg p))) holds
P1[p,q] or ex q being Element of F1() st
( q in union (rng (f | ( the InternalRel of RS -Seg p))) & P1[p,q] ) ) ;
supposeA46: for q being Element of F1() st q in union (rng (f | ( the InternalRel of RS -Seg p))) holds
P1[p,q] ; ::_thesis: x in union (rng (f | ( the InternalRel of RS -Seg p)))
A47: not x in {p} by A45, TARSKI:def_1;
f . p = (union (rng (f | ( the InternalRel of RS -Seg p)))) \/ {p} by A9, A44, A43, A46;
hence x in union (rng (f | ( the InternalRel of RS -Seg p))) by A41, A47, XBOOLE_0:def_3; ::_thesis: verum
end;
suppose ex q being Element of F1() st
( q in union (rng (f | ( the InternalRel of RS -Seg p))) & P1[p,q] ) ; ::_thesis: x in union (rng (f | ( the InternalRel of RS -Seg p)))
hence x in union (rng (f | ( the InternalRel of RS -Seg p))) by A9, A41, A44, A43; ::_thesis: verum
end;
end;
end;
then consider y9 being set such that
A48: x in y9 and
A49: y9 in rng (f | ( the InternalRel of RS -Seg p)) by TARSKI:def_4;
consider z9 being set such that
A50: z9 in dom (f | ( the InternalRel of RS -Seg p)) and
A51: (f | ( the InternalRel of RS -Seg p)) . z9 = y9 by A49, FUNCT_1:def_3;
reconsider z9 = z9 as Point of RS by A50;
A52: z9 in the InternalRel of RS -Seg p by A50, RELAT_1:57;
then A53: z9 <> p by WELLORD1:1;
A54: [z9,p] in the InternalRel of RS by A52, WELLORD1:1;
S3[z9] by A48, A50, A51, FUNCT_1:47;
hence contradiction by A42, A53, A54; ::_thesis: verum
end;
hence x in f . x by A41; ::_thesis: verum
end;
A55: now__::_thesis:_for_x,_y_being_Element_of_F1()_st_x_in_A_&_y_in_A_&_x_<>_y_&_[x,y]_in_the_InternalRel_of_RS_holds_
P1[x,y]
A56: dom f = the carrier of RS by FUNCT_2:def_1;
let x, y be Element of F1(); ::_thesis: ( x in A & y in A & x <> y & [x,y] in the InternalRel of RS implies P1[x,y] )
assume that
A57: x in A and
A58: y in A and
A59: x <> y and
A60: [x,y] in the InternalRel of RS ; ::_thesis: P1[x,y]
A61: y in f . y by A35, A58;
set fIy = f | ( the InternalRel of RS -Seg y);
x in the InternalRel of RS -Seg y by A59, A60, WELLORD1:1;
then A62: x in dom (f | ( the InternalRel of RS -Seg y)) by A56, RELAT_1:57;
then A63: (f | ( the InternalRel of RS -Seg y)) . x in rng (f | ( the InternalRel of RS -Seg y)) by FUNCT_1:def_3;
A64: (f | ( the InternalRel of RS -Seg y)) . x = f . x by A62, FUNCT_1:47;
x in f . x by A35, A57;
then x in union (rng (f | ( the InternalRel of RS -Seg y))) by A63, A64, TARSKI:def_4;
then P1[y,x] by A34, A61;
hence P1[x,y] by A1; ::_thesis: verum
end;
let x, y be Element of F1(); ::_thesis: ( x in A & y in A & x <> y implies P1[x,y] )
assume that
A65: x in A and
A66: y in A and
A67: x <> y ; ::_thesis: P1[x,y]
R2 well_orders F1() by A3, PCOMPS_2:1;
then R2 is_connected_in F1() by WELLORD1:def_5;
then ( [x,y] in the InternalRel of RS or [y,x] in the InternalRel of RS ) by A67, RELAT_2:def_6;
then ( P1[x,y] or P1[y,x] ) by A55, A65, A66, A67;
hence P1[x,y] by A1; ::_thesis: verum
end;
let x be Element of F1(); ::_thesis: ex y being Element of F1() st
( y in A & P1[x,y] )
percases ( x in A or not x in A ) ;
supposeA68: x in A ; ::_thesis: ex y being Element of F1() st
( y in A & P1[x,y] )
take x ; ::_thesis: ( x in A & P1[x,x] )
thus ( x in A & P1[x,x] ) by A2, A68; ::_thesis: verum
end;
supposeA69: not x in A ; ::_thesis: ex y being Element of F1() st
( y in A & P1[x,y] )
not x in f . x
proof
dom f = the carrier of RS by FUNCT_2:def_1;
then A70: f . x in rng f by FUNCT_1:def_3;
assume x in f . x ; ::_thesis: contradiction
hence contradiction by A69, A70, TARSKI:def_4; ::_thesis: verum
end;
then consider r being Element of F1() such that
A71: r in union (rng (f | ( the InternalRel of RS -Seg x))) and
A72: P1[x,r] by A34;
take r ; ::_thesis: ( r in A & P1[x,r] )
union (rng (f | ( the InternalRel of RS -Seg x))) c= A by RELAT_1:70, ZFMISC_1:77;
hence ( r in A & P1[x,r] ) by A71, A72; ::_thesis: verum
end;
end;
end;
theorem Th30: :: COMPL_SP:30
for M being non empty Reflexive symmetric MetrStruct
for r being Real st r > 0 holds
ex A being Subset of M st
( ( for p, q being Point of M st p <> q & p in A & q in A holds
dist (p,q) >= r ) & ( for p being Point of M ex q being Point of M st
( q in A & p in Ball (q,r) ) ) )
proof
let M be non empty Reflexive symmetric MetrStruct ; ::_thesis: for r being Real st r > 0 holds
ex A being Subset of M st
( ( for p, q being Point of M st p <> q & p in A & q in A holds
dist (p,q) >= r ) & ( for p being Point of M ex q being Point of M st
( q in A & p in Ball (q,r) ) ) )
let r be Real; ::_thesis: ( r > 0 implies ex A being Subset of M st
( ( for p, q being Point of M st p <> q & p in A & q in A holds
dist (p,q) >= r ) & ( for p being Point of M ex q being Point of M st
( q in A & p in Ball (q,r) ) ) ) )
assume A1: r > 0 ; ::_thesis: ex A being Subset of M st
( ( for p, q being Point of M st p <> q & p in A & q in A holds
dist (p,q) >= r ) & ( for p being Point of M ex q being Point of M st
( q in A & p in Ball (q,r) ) ) )
set cM = the carrier of M;
defpred S1[ set , set ] means for p, q being Point of M st p = $1 & q = $2 holds
dist (p,q) >= r;
A2: for x being Element of the carrier of M holds not S1[x,x]
proof
let x be Element of the carrier of M; ::_thesis: not S1[x,x]
dist (x,x) = 0 by METRIC_1:1;
hence not S1[x,x] by A1; ::_thesis: verum
end;
A3: for x, y being Element of the carrier of M holds
( S1[x,y] iff S1[y,x] ) ;
consider A being Subset of the carrier of M such that
A4: for x, y being Element of the carrier of M st x in A & y in A & x <> y holds
S1[x,y] and
A5: for x being Element of the carrier of M ex y being Element of the carrier of M st
( y in A & not S1[x,y] ) from COMPL_SP:sch_1(A3, A2);
take A ; ::_thesis: ( ( for p, q being Point of M st p <> q & p in A & q in A holds
dist (p,q) >= r ) & ( for p being Point of M ex q being Point of M st
( q in A & p in Ball (q,r) ) ) )
thus for p, q being Point of M st p <> q & p in A & q in A holds
dist (p,q) >= r by A4; ::_thesis: for p being Point of M ex q being Point of M st
( q in A & p in Ball (q,r) )
let p be Point of M; ::_thesis: ex q being Point of M st
( q in A & p in Ball (q,r) )
consider y being Element of the carrier of M such that
A6: y in A and
A7: not S1[p,y] by A5;
take y ; ::_thesis: ( y in A & p in Ball (y,r) )
thus ( y in A & p in Ball (y,r) ) by A6, A7, METRIC_1:11; ::_thesis: verum
end;
theorem Th31: :: COMPL_SP:31
for M being non empty Reflexive symmetric triangle MetrStruct holds
( M is totally_bounded iff for r being Real
for A being Subset of M st r > 0 & ( for p, q being Point of M st p <> q & p in A & q in A holds
dist (p,q) >= r ) holds
A is finite )
proof
let M be non empty Reflexive symmetric triangle MetrStruct ; ::_thesis: ( M is totally_bounded iff for r being Real
for A being Subset of M st r > 0 & ( for p, q being Point of M st p <> q & p in A & q in A holds
dist (p,q) >= r ) holds
A is finite )
thus ( M is totally_bounded implies for r being Real
for A being Subset of M st r > 0 & ( for p, q being Point of M st p <> q & p in A & q in A holds
dist (p,q) >= r ) holds
A is finite ) ::_thesis: ( ( for r being Real
for A being Subset of M st r > 0 & ( for p, q being Point of M st p <> q & p in A & q in A holds
dist (p,q) >= r ) holds
A is finite ) implies M is totally_bounded )
proof
assume A1: M is totally_bounded ; ::_thesis: for r being Real
for A being Subset of M st r > 0 & ( for p, q being Point of M st p <> q & p in A & q in A holds
dist (p,q) >= r ) holds
A is finite
let r be Real; ::_thesis: for A being Subset of M st r > 0 & ( for p, q being Point of M st p <> q & p in A & q in A holds
dist (p,q) >= r ) holds
A is finite
let A be Subset of M; ::_thesis: ( r > 0 & ( for p, q being Point of M st p <> q & p in A & q in A holds
dist (p,q) >= r ) implies A is finite )
assume that
A2: r > 0 and
A3: for p, q being Point of M st p <> q & p in A & q in A holds
dist (p,q) >= r ; ::_thesis: A is finite
r / 2 > 0 by A2, XREAL_1:215;
then consider G being Subset-Family of M such that
A4: G is finite and
A5: the carrier of M = union G and
A6: for C being Subset of M st C in G holds
ex w being Point of M st C = Ball (w,(r / 2)) by A1, TBSP_1:def_1;
defpred S1[ set , set ] means ( $1 in $2 & $2 in G );
A7: for x being set st x in A holds
ex y being set st
( y in bool the carrier of M & S1[x,y] )
proof
let x be set ; ::_thesis: ( x in A implies ex y being set st
( y in bool the carrier of M & S1[x,y] ) )
assume x in A ; ::_thesis: ex y being set st
( y in bool the carrier of M & S1[x,y] )
then consider y being set such that
A8: x in y and
A9: y in G by A5, TARSKI:def_4;
reconsider y = y as Subset of M by A9;
take y ; ::_thesis: ( y in bool the carrier of M & S1[x,y] )
thus ( y in bool the carrier of M & S1[x,y] ) by A8, A9; ::_thesis: verum
end;
consider F being Function of A,(bool the carrier of M) such that
A10: for x being set st x in A holds
S1[x,F . x] from FUNCT_2:sch_1(A7);
now__::_thesis:_for_x1,_x2_being_set_st_x1_in_A_&_x2_in_A_&_F_._x1_=_F_._x2_holds_
x1_=_x2
let x1, x2 be set ; ::_thesis: ( x1 in A & x2 in A & F . x1 = F . x2 implies x1 = x2 )
assume that
A11: x1 in A and
A12: x2 in A and
A13: F . x1 = F . x2 ; ::_thesis: x1 = x2
reconsider p1 = x1, p2 = x2 as Point of M by A11, A12;
F . x1 in G by A10, A11;
then consider w being Point of M such that
A14: F . x1 = Ball (w,(r / 2)) by A6;
p1 in Ball (w,(r / 2)) by A10, A11, A14;
then A15: dist (p1,w) < r / 2 by METRIC_1:11;
A16: dist (p1,p2) <= (dist (p1,w)) + (dist (w,p2)) by METRIC_1:4;
p2 in Ball (w,(r / 2)) by A10, A12, A13, A14;
then dist (w,p2) < r / 2 by METRIC_1:11;
then (dist (p1,w)) + (dist (w,p2)) < (r / 2) + (r / 2) by A15, XREAL_1:8;
then dist (p1,p2) < (r / 2) + (r / 2) by A16, XXREAL_0:2;
hence x1 = x2 by A3, A11, A12; ::_thesis: verum
end;
then A17: F is one-to-one by FUNCT_2:19;
A18: rng F c= G
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng F or x in G )
assume x in rng F ; ::_thesis: x in G
then ex y being set st
( y in dom F & x = F . y ) by FUNCT_1:def_3;
hence x in G by A10; ::_thesis: verum
end;
dom F = A by FUNCT_2:def_1;
then A, rng F are_equipotent by A17, WELLORD2:def_4;
hence A is finite by A4, A18, CARD_1:38; ::_thesis: verum
end;
assume A19: for r being Real
for A being Subset of M st r > 0 & ( for p, q being Point of M st p <> q & p in A & q in A holds
dist (p,q) >= r ) holds
A is finite ; ::_thesis: M is totally_bounded
let r be Real; :: according to TBSP_1:def_1 ::_thesis: ( r <= 0 or ex b1 being Element of bool (bool the carrier of M) st
( b1 is finite & the carrier of M = union b1 & ( for b2 being Element of bool the carrier of M holds
( not b2 in b1 or ex b3 being Element of the carrier of M st b2 = Ball (b3,r) ) ) ) )
assume A20: r > 0 ; ::_thesis: ex b1 being Element of bool (bool the carrier of M) st
( b1 is finite & the carrier of M = union b1 & ( for b2 being Element of bool the carrier of M holds
( not b2 in b1 or ex b3 being Element of the carrier of M st b2 = Ball (b3,r) ) ) )
consider A being Subset of M such that
A21: for p, q being Point of M st p <> q & p in A & q in A holds
dist (p,q) >= r and
A22: for p being Point of M ex q being Point of M st
( q in A & p in Ball (q,r) ) by A20, Th30;
deffunc H1( Point of M) -> Element of bool the carrier of M = Ball ($1,r);
set BA = { H1(p) where p is Point of M : p in A } ;
A23: A is finite by A19, A20, A21;
A24: { H1(p) where p is Point of M : p in A } is finite from FRAENKEL:sch_21(A23);
{ H1(p) where p is Point of M : p in A } c= bool the carrier of M
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { H1(p) where p is Point of M : p in A } or x in bool the carrier of M )
assume x in { H1(p) where p is Point of M : p in A } ; ::_thesis: x in bool the carrier of M
then ex p being Point of M st
( x = H1(p) & p in A ) ;
hence x in bool the carrier of M ; ::_thesis: verum
end;
then reconsider BA = { H1(p) where p is Point of M : p in A } as Subset-Family of M ;
take BA ; ::_thesis: ( BA is finite & the carrier of M = union BA & ( for b1 being Element of bool the carrier of M holds
( not b1 in BA or ex b2 being Element of the carrier of M st b1 = Ball (b2,r) ) ) )
the carrier of M c= union BA
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the carrier of M or x in union BA )
assume x in the carrier of M ; ::_thesis: x in union BA
then reconsider p = x as Point of M ;
consider q being Point of M such that
A25: q in A and
A26: p in H1(q) by A22;
H1(q) in BA by A25;
hence x in union BA by A26, TARSKI:def_4; ::_thesis: verum
end;
hence ( BA is finite & union BA = the carrier of M ) by A24, XBOOLE_0:def_10; ::_thesis: for b1 being Element of bool the carrier of M holds
( not b1 in BA or ex b2 being Element of the carrier of M st b1 = Ball (b2,r) )
let C be Subset of M; ::_thesis: ( not C in BA or ex b1 being Element of the carrier of M st C = Ball (b1,r) )
assume C in BA ; ::_thesis: ex b1 being Element of the carrier of M st C = Ball (b1,r)
then ex p being Point of M st
( C = H1(p) & p in A ) ;
hence ex b1 being Element of the carrier of M st C = Ball (b1,r) ; ::_thesis: verum
end;
Lm7: for M being non empty Reflexive symmetric triangle MetrStruct
for r being Real
for A being Subset of M st r > 0 & ( for p, q being Point of M st p <> q & p in A & q in A holds
dist (p,q) >= r ) holds
for F being Subset-Family of (TopSpaceMetr M) st F = { {x} where x is Element of (TopSpaceMetr M) : x in A } holds
F is locally_finite
proof
let M be non empty Reflexive symmetric triangle MetrStruct ; ::_thesis: for r being Real
for A being Subset of M st r > 0 & ( for p, q being Point of M st p <> q & p in A & q in A holds
dist (p,q) >= r ) holds
for F being Subset-Family of (TopSpaceMetr M) st F = { {x} where x is Element of (TopSpaceMetr M) : x in A } holds
F is locally_finite
set T = TopSpaceMetr M;
let r be Real; ::_thesis: for A being Subset of M st r > 0 & ( for p, q being Point of M st p <> q & p in A & q in A holds
dist (p,q) >= r ) holds
for F being Subset-Family of (TopSpaceMetr M) st F = { {x} where x is Element of (TopSpaceMetr M) : x in A } holds
F is locally_finite
let A be Subset of M; ::_thesis: ( r > 0 & ( for p, q being Point of M st p <> q & p in A & q in A holds
dist (p,q) >= r ) implies for F being Subset-Family of (TopSpaceMetr M) st F = { {x} where x is Element of (TopSpaceMetr M) : x in A } holds
F is locally_finite )
assume that
A1: r > 0 and
A2: for p, q being Point of M st p <> q & p in A & q in A holds
dist (p,q) >= r ; ::_thesis: for F being Subset-Family of (TopSpaceMetr M) st F = { {x} where x is Element of (TopSpaceMetr M) : x in A } holds
F is locally_finite
A3: r / 2 > 0 by A1, XREAL_1:215;
let F be Subset-Family of (TopSpaceMetr M); ::_thesis: ( F = { {x} where x is Element of (TopSpaceMetr M) : x in A } implies F is locally_finite )
assume A4: F = { {x} where x is Element of (TopSpaceMetr M) : x in A } ; ::_thesis: F is locally_finite
let x be Point of (TopSpaceMetr M); :: according to PCOMPS_1:def_1 ::_thesis: ex b1 being Element of bool the carrier of (TopSpaceMetr M) st
( x in b1 & b1 is open & { b2 where b2 is Element of bool the carrier of (TopSpaceMetr M) : ( b2 in F & not b2 misses b1 ) } is finite )
reconsider x9 = x as Point of M ;
reconsider B = Ball (x9,(r / 2)) as Subset of (TopSpaceMetr M) ;
take B ; ::_thesis: ( x in B & B is open & { b1 where b1 is Element of bool the carrier of (TopSpaceMetr M) : ( b1 in F & not b1 misses B ) } is finite )
A5: dist (x9,x9) = 0 by METRIC_1:1;
B in Family_open_set M by PCOMPS_1:29;
hence ( x in B & B is open ) by A5, A3, METRIC_1:11, PRE_TOPC:def_2; ::_thesis: { b1 where b1 is Element of bool the carrier of (TopSpaceMetr M) : ( b1 in F & not b1 misses B ) } is finite
set VV = { V where V is Subset of (TopSpaceMetr M) : ( V in F & V meets B ) } ;
percases ( for p being Point of M st p in A holds
dist (p,x9) >= r / 2 or ex p being Point of M st
( p in A & dist (p,x9) < r / 2 ) ) ;
supposeA6: for p being Point of M st p in A holds
dist (p,x9) >= r / 2 ; ::_thesis: { b1 where b1 is Element of bool the carrier of (TopSpaceMetr M) : ( b1 in F & not b1 misses B ) } is finite
{ V where V is Subset of (TopSpaceMetr M) : ( V in F & V meets B ) } is empty
proof
assume not { V where V is Subset of (TopSpaceMetr M) : ( V in F & V meets B ) } is empty ; ::_thesis: contradiction
then consider v being set such that
A7: v in { V where V is Subset of (TopSpaceMetr M) : ( V in F & V meets B ) } by XBOOLE_0:def_1;
consider V being Subset of (TopSpaceMetr M) such that
v = V and
A8: V in F and
A9: V meets B by A7;
consider y being Point of (TopSpaceMetr M) such that
A10: V = {y} and
A11: y in A by A4, A8;
reconsider y = y as Point of M ;
y in B by A9, A10, ZFMISC_1:50;
then dist (x9,y) < r / 2 by METRIC_1:11;
hence contradiction by A6, A11; ::_thesis: verum
end;
hence { b1 where b1 is Element of bool the carrier of (TopSpaceMetr M) : ( b1 in F & not b1 misses B ) } is finite ; ::_thesis: verum
end;
suppose ex p being Point of M st
( p in A & dist (p,x9) < r / 2 ) ; ::_thesis: { b1 where b1 is Element of bool the carrier of (TopSpaceMetr M) : ( b1 in F & not b1 misses B ) } is finite
then consider p being Point of M such that
A12: p in A and
A13: dist (p,x9) < r / 2 ;
{ V where V is Subset of (TopSpaceMetr M) : ( V in F & V meets B ) } c= {{p}}
proof
let v be set ; :: according to TARSKI:def_3 ::_thesis: ( not v in { V where V is Subset of (TopSpaceMetr M) : ( V in F & V meets B ) } or v in {{p}} )
assume v in { V where V is Subset of (TopSpaceMetr M) : ( V in F & V meets B ) } ; ::_thesis: v in {{p}}
then consider V being Subset of (TopSpaceMetr M) such that
A14: v = V and
A15: V in F and
A16: V meets B ;
consider y being Point of (TopSpaceMetr M) such that
A17: V = {y} and
A18: y in A by A4, A15;
reconsider y = y as Point of M ;
y in B by A16, A17, ZFMISC_1:50;
then dist (x9,y) < r / 2 by METRIC_1:11;
then A19: (dist (p,x9)) + (dist (x9,y)) < (r / 2) + (r / 2) by A13, XREAL_1:8;
dist (p,y) <= (dist (p,x9)) + (dist (x9,y)) by METRIC_1:4;
then dist (p,y) < (r / 2) + (r / 2) by A19, XXREAL_0:2;
then p = y by A2, A12, A18;
hence v in {{p}} by A14, A17, TARSKI:def_1; ::_thesis: verum
end;
hence { b1 where b1 is Element of bool the carrier of (TopSpaceMetr M) : ( b1 in F & not b1 misses B ) } is finite ; ::_thesis: verum
end;
end;
end;
theorem Th32: :: COMPL_SP:32
for M being non empty Reflexive symmetric triangle MetrStruct st TopSpaceMetr M is countably_compact holds
M is totally_bounded
proof
deffunc H1( set ) -> set = meet $1;
let M be non empty Reflexive symmetric triangle MetrStruct ; ::_thesis: ( TopSpaceMetr M is countably_compact implies M is totally_bounded )
assume A1: TopSpaceMetr M is countably_compact ; ::_thesis: M is totally_bounded
set T = TopSpaceMetr M;
assume not M is totally_bounded ; ::_thesis: contradiction
then consider r being Real, A being Subset of M such that
A2: r > 0 and
A3: for p, q being Point of M st p <> q & p in A & q in A holds
dist (p,q) >= r and
A4: A is infinite by Th31;
reconsider A = A as Subset of (TopSpaceMetr M) ;
set F = { {x} where x is Element of (TopSpaceMetr M) : x in A } ;
reconsider F = { {x} where x is Element of (TopSpaceMetr M) : x in A } as Subset-Family of (TopSpaceMetr M) by RELSET_2:16;
A5: now__::_thesis:_for_a_being_Subset_of_(TopSpaceMetr_M)_st_a_in_F_holds_
card_a_=_1
let a be Subset of (TopSpaceMetr M); ::_thesis: ( a in F implies card a = 1 )
assume a in F ; ::_thesis: card a = 1
then ex y being Point of (TopSpaceMetr M) st
( a = {y} & y in A ) ;
hence card a = 1 by CARD_1:30; ::_thesis: verum
end;
set PP = { H1(y) where y is Subset of (TopSpaceMetr M) : y in F } ;
A6: A c= { H1(y) where y is Subset of (TopSpaceMetr M) : y in F }
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in A or y in { H1(y) where y is Subset of (TopSpaceMetr M) : y in F } )
assume A7: y in A ; ::_thesis: y in { H1(y) where y is Subset of (TopSpaceMetr M) : y in F }
reconsider y9 = y as Point of (TopSpaceMetr M) by A7;
{y9} in F by A7;
then H1({y9}) in { H1(y) where y is Subset of (TopSpaceMetr M) : y in F } ;
hence y in { H1(y) where y is Subset of (TopSpaceMetr M) : y in F } by SETFAM_1:10; ::_thesis: verum
end;
F is locally_finite by A2, A3, Lm7;
then A8: F is finite by A1, A5, Th27;
{ H1(y) where y is Subset of (TopSpaceMetr M) : y in F } is finite from FRAENKEL:sch_21(A8);
hence contradiction by A4, A6; ::_thesis: verum
end;
theorem Th33: :: COMPL_SP:33
for M being non empty MetrSpace st M is totally_bounded holds
TopSpaceMetr M is second-countable
proof
let M be non empty MetrSpace; ::_thesis: ( M is totally_bounded implies TopSpaceMetr M is second-countable )
assume A1: M is totally_bounded ; ::_thesis: TopSpaceMetr M is second-countable
set T = TopSpaceMetr M;
defpred S1[ set , set ] means for i being Nat st i = $1 holds
for G being Subset-Family of (TopSpaceMetr M) st $2 = G holds
( G is finite & the carrier of M = union G & ( for C being Subset of M st C in G holds
ex w being Point of M st C = Ball (w,(1 / (i + 1))) ) );
A2: for x being set st x in NAT holds
ex y being set st
( y in bool (bool the carrier of (TopSpaceMetr M)) & S1[x,y] )
proof
let x be set ; ::_thesis: ( x in NAT implies ex y being set st
( y in bool (bool the carrier of (TopSpaceMetr M)) & S1[x,y] ) )
assume x in NAT ; ::_thesis: ex y being set st
( y in bool (bool the carrier of (TopSpaceMetr M)) & S1[x,y] )
then reconsider i = x as Element of NAT ;
1 / (i + 1) > 0 by XREAL_1:139;
then consider G being Subset-Family of (TopSpaceMetr M) such that
A3: G is finite and
A4: the carrier of M = union G and
A5: for C being Subset of M st C in G holds
ex w being Point of M st C = Ball (w,(1 / (i + 1))) by A1, TBSP_1:def_1;
take G ; ::_thesis: ( G in bool (bool the carrier of (TopSpaceMetr M)) & S1[x,G] )
thus ( G in bool (bool the carrier of (TopSpaceMetr M)) & S1[x,G] ) by A3, A4, A5; ::_thesis: verum
end;
consider f being Function of NAT,(bool (bool the carrier of (TopSpaceMetr M))) such that
A6: for x being set st x in NAT holds
S1[x,f . x] from FUNCT_2:sch_1(A2);
set U = Union f;
A7: dom f = NAT by FUNCT_2:def_1;
A8: for A being Subset of (TopSpaceMetr M) st A is open holds
for p being Point of (TopSpaceMetr M) st p in A holds
ex a being Subset of (TopSpaceMetr M) st
( a in Union f & p in a & a c= A )
proof
let A be Subset of (TopSpaceMetr M); ::_thesis: ( A is open implies for p being Point of (TopSpaceMetr M) st p in A holds
ex a being Subset of (TopSpaceMetr M) st
( a in Union f & p in a & a c= A ) )
assume A9: A is open ; ::_thesis: for p being Point of (TopSpaceMetr M) st p in A holds
ex a being Subset of (TopSpaceMetr M) st
( a in Union f & p in a & a c= A )
let p be Point of (TopSpaceMetr M); ::_thesis: ( p in A implies ex a being Subset of (TopSpaceMetr M) st
( a in Union f & p in a & a c= A ) )
assume A10: p in A ; ::_thesis: ex a being Subset of (TopSpaceMetr M) st
( a in Union f & p in a & a c= A )
reconsider p9 = p as Point of M ;
consider r being real number such that
A11: r > 0 and
A12: Ball (p9,r) c= A by A9, A10, TOPMETR:15;
reconsider r = r as Real by XREAL_0:def_1;
consider n being Element of NAT such that
A13: n > 0 and
A14: 1 / n < r / 2 by A11, UNIFORM1:1, XREAL_1:215;
A15: (1 / n) + (1 / n) < (r / 2) + (r / 2) by A14, XREAL_1:8;
reconsider n1 = n - 1 as Element of NAT by A13, NAT_1:20;
reconsider fn = f . n1 as Subset-Family of (TopSpaceMetr M) ;
the carrier of M = union fn by A6;
then consider x being set such that
A16: p in x and
A17: x in fn by TARSKI:def_4;
reconsider x = x as Subset of M by A17;
consider w being Point of M such that
A18: x = Ball (w,(1 / (n1 + 1))) by A6, A17;
reconsider B = Ball (w,(1 / n)) as Subset of (TopSpaceMetr M) ;
take B ; ::_thesis: ( B in Union f & p in B & B c= A )
f . n1 in rng f by A7, FUNCT_1:def_3;
then B in union (rng f) by A17, A18, TARSKI:def_4;
hence ( B in Union f & p in B ) by A16, A18, CARD_3:def_4; ::_thesis: B c= A
let q be set ; :: according to TARSKI:def_3 ::_thesis: ( not q in B or q in A )
assume A19: q in B ; ::_thesis: q in A
reconsider q9 = q as Point of M by A19;
A20: dist (q9,w) < 1 / n by A19, METRIC_1:11;
dist (w,p9) < 1 / (n1 + 1) by A16, A18, METRIC_1:11;
then A21: (dist (q9,w)) + (dist (w,p9)) < (1 / n) + (1 / n) by A20, XREAL_1:8;
dist (q9,p9) <= (dist (q9,w)) + (dist (w,p9)) by METRIC_1:4;
then dist (q9,p9) < (1 / n) + (1 / n) by A21, XXREAL_0:2;
then dist (q9,p9) < (r / 2) + (r / 2) by A15, XXREAL_0:2;
then q in Ball (p9,r) by METRIC_1:11;
hence q in A by A12; ::_thesis: verum
end;
set CB = { (card B) where B is Basis of (TopSpaceMetr M) : verum } ;
Union f c= the topology of (TopSpaceMetr M)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Union f or x in the topology of (TopSpaceMetr M) )
assume x in Union f ; ::_thesis: x in the topology of (TopSpaceMetr M)
then x in union (rng f) by CARD_3:def_4;
then consider y being set such that
A22: x in y and
A23: y in rng f by TARSKI:def_4;
reconsider X = x as Subset of (TopSpaceMetr M) by A22, A23;
consider z being set such that
A24: z in dom f and
A25: f . z = y by A23, FUNCT_1:def_3;
reconsider z = z as Element of NAT by A24;
ex w being Point of M st X = Ball (w,(1 / (z + 1))) by A6, A22, A25;
hence x in the topology of (TopSpaceMetr M) by PCOMPS_1:29; ::_thesis: verum
end;
then Union f is Basis of (TopSpaceMetr M) by A8, YELLOW_9:32;
then A26: card (Union f) in { (card B) where B is Basis of (TopSpaceMetr M) : verum } ;
now__::_thesis:_for_x_being_set_st_x_in_dom_f_holds_
f_._x_is_countable
let x be set ; ::_thesis: ( x in dom f implies f . x is countable )
assume x in dom f ; ::_thesis: f . x is countable
then reconsider i = x as Element of NAT ;
reconsider fx = f . i as Subset-Family of (TopSpaceMetr M) ;
fx is finite by A6;
hence f . x is countable by CARD_4:1; ::_thesis: verum
end;
then Union f is countable by A7, CARD_4:2, CARD_4:11;
then A27: card (Union f) c= omega by CARD_3:def_14;
weight (TopSpaceMetr M) = meet { (card B) where B is Basis of (TopSpaceMetr M) : verum } by WAYBEL23:def_5;
then weight (TopSpaceMetr M) c= card (Union f) by A26, SETFAM_1:3;
then weight (TopSpaceMetr M) c= omega by A27, XBOOLE_1:1;
hence TopSpaceMetr M is second-countable by WAYBEL23:def_6; ::_thesis: verum
end;
theorem Th34: :: COMPL_SP:34
for T being non empty TopSpace st T is second-countable holds
for F being Subset-Family of T st F is Cover of T & F is open holds
ex G being Subset-Family of T st
( G c= F & G is Cover of T & G is countable )
proof
let T be non empty TopSpace; ::_thesis: ( T is second-countable implies for F being Subset-Family of T st F is Cover of T & F is open holds
ex G being Subset-Family of T st
( G c= F & G is Cover of T & G is countable ) )
assume T is second-countable ; ::_thesis: for F being Subset-Family of T st F is Cover of T & F is open holds
ex G being Subset-Family of T st
( G c= F & G is Cover of T & G is countable )
then consider B being Basis of T such that
A1: B is countable by TOPGEN_4:57;
A2: card B c= omega by A1, CARD_3:def_14;
let F be Subset-Family of T; ::_thesis: ( F is Cover of T & F is open implies ex G being Subset-Family of T st
( G c= F & G is Cover of T & G is countable ) )
assume that
A3: F is Cover of T and
A4: F is open ; ::_thesis: ex G being Subset-Family of T st
( G c= F & G is Cover of T & G is countable )
defpred S1[ set , set ] means for b being Subset of T st b = $1 holds
( ( ex y being set st
( y in F & b c= y ) implies ( $2 in F & b c= $2 ) ) & ( ( for y being set st y in F holds
not b c= y ) implies $2 = {} ) );
A5: for x being set st x in B holds
ex y being set st
( y in bool the carrier of T & S1[x,y] )
proof
let x be set ; ::_thesis: ( x in B implies ex y being set st
( y in bool the carrier of T & S1[x,y] ) )
assume x in B ; ::_thesis: ex y being set st
( y in bool the carrier of T & S1[x,y] )
then reconsider b = x as Subset of T ;
percases ( ex y being set st
( y in F & b c= y ) or for y being set st y in F holds
not b c= y ) ;
suppose ex y being set st
( y in F & b c= y ) ; ::_thesis: ex y being set st
( y in bool the carrier of T & S1[x,y] )
then consider y being set such that
A6: y in F and
A7: b c= y ;
reconsider y = y as Subset of T by A6;
take y ; ::_thesis: ( y in bool the carrier of T & S1[x,y] )
thus ( y in bool the carrier of T & S1[x,y] ) by A6, A7; ::_thesis: verum
end;
supposeA8: for y being set st y in F holds
not b c= y ; ::_thesis: ex y being set st
( y in bool the carrier of T & S1[x,y] )
take {} T ; ::_thesis: ( {} T in bool the carrier of T & S1[x, {} T] )
thus ( {} T in bool the carrier of T & S1[x, {} T] ) by A8; ::_thesis: verum
end;
end;
end;
consider p being Function of B,(bool the carrier of T) such that
A9: for x being set st x in B holds
S1[x,p . x] from FUNCT_2:sch_1(A5);
take RNG = (rng p) \ {{}}; ::_thesis: ( RNG c= F & RNG is Cover of T & RNG is countable )
A10: dom p = B by FUNCT_2:def_1;
thus RNG c= F ::_thesis: ( RNG is Cover of T & RNG is countable )
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in RNG or y in F )
assume A11: y in RNG ; ::_thesis: y in F
y in rng p by A11, XBOOLE_0:def_5;
then consider z being set such that
A12: z in dom p and
A13: p . z = y by FUNCT_1:def_3;
reconsider z = z as Subset of T by A10, A12;
( ex y being set st
( y in F & z c= y ) or for y being set st y in F holds
not z c= y ) ;
then ( ( p . z in F & z c= p . z ) or p . z = {} ) by A9, A12;
hence y in F by A11, A13, ZFMISC_1:56; ::_thesis: verum
end;
the carrier of T c= union RNG
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in the carrier of T or y in union RNG )
assume y in the carrier of T ; ::_thesis: y in union RNG
then reconsider q = y as Point of T ;
consider W being Subset of T such that
A14: q in W and
A15: W in F by A3, PCOMPS_1:3;
W is open by A4, A15, TOPS_2:def_1;
then consider U being Subset of T such that
A16: U in B and
A17: q in U and
A18: U c= W by A14, YELLOW_9:31;
A19: p . U in rng p by A10, A16, FUNCT_1:def_3;
then reconsider pU = p . U as Subset of T ;
A20: U c= pU by A9, A15, A16, A18;
then pU in RNG by A17, A19, ZFMISC_1:56;
hence y in union RNG by A17, A20, TARSKI:def_4; ::_thesis: verum
end;
then [#] T = union RNG by XBOOLE_0:def_10;
hence RNG is Cover of T by SETFAM_1:45; ::_thesis: RNG is countable
card (rng p) c= card B by A10, CARD_2:61;
then card (rng p) c= omega by A2, XBOOLE_1:1;
then rng p is countable by CARD_3:def_14;
hence RNG is countable by CARD_3:95; ::_thesis: verum
end;
begin
theorem Th35: :: COMPL_SP:35
for M being non empty MetrSpace holds
( TopSpaceMetr M is compact iff TopSpaceMetr M is countably_compact )
proof
let M be non empty MetrSpace; ::_thesis: ( TopSpaceMetr M is compact iff TopSpaceMetr M is countably_compact )
set T = TopSpaceMetr M;
thus ( TopSpaceMetr M is compact implies TopSpaceMetr M is countably_compact ) by Th20; ::_thesis: ( TopSpaceMetr M is countably_compact implies TopSpaceMetr M is compact )
assume A1: TopSpaceMetr M is countably_compact ; ::_thesis: TopSpaceMetr M is compact
let F be Subset-Family of (TopSpaceMetr M); :: according to COMPTS_1:def_1 ::_thesis: ( not F is Cover of the carrier of (TopSpaceMetr M) or not F is open or ex b1 being Element of bool (bool the carrier of (TopSpaceMetr M)) st
( b1 c= F & b1 is Cover of the carrier of (TopSpaceMetr M) & b1 is finite ) )
assume that
A2: F is Cover of (TopSpaceMetr M) and
A3: F is open ; ::_thesis: ex b1 being Element of bool (bool the carrier of (TopSpaceMetr M)) st
( b1 c= F & b1 is Cover of the carrier of (TopSpaceMetr M) & b1 is finite )
M is totally_bounded by A1, Th32;
then TopSpaceMetr M is second-countable by Th33;
then consider G being Subset-Family of (TopSpaceMetr M) such that
A4: G c= F and
A5: G is Cover of (TopSpaceMetr M) and
A6: G is countable by A2, A3, Th34;
G is open by A3, A4, TOPS_2:11;
then ex H being Subset-Family of (TopSpaceMetr M) st
( H c= G & H is Cover of (TopSpaceMetr M) & H is finite ) by A1, A5, A6, Def9;
hence ex b1 being Element of bool (bool the carrier of (TopSpaceMetr M)) st
( b1 c= F & b1 is Cover of the carrier of (TopSpaceMetr M) & b1 is finite ) by A4, XBOOLE_1:1; ::_thesis: verum
end;
theorem Th36: :: COMPL_SP:36
for X being set
for F being Subset-Family of X st F is finite holds
for A being Subset of X st A is infinite & A c= union F holds
ex Y being Subset of X st
( Y in F & Y /\ A is infinite )
proof
defpred S1[ set , set ] means $1 in $2;
let X be set ; ::_thesis: for F being Subset-Family of X st F is finite holds
for A being Subset of X st A is infinite & A c= union F holds
ex Y being Subset of X st
( Y in F & Y /\ A is infinite )
let F be Subset-Family of X; ::_thesis: ( F is finite implies for A being Subset of X st A is infinite & A c= union F holds
ex Y being Subset of X st
( Y in F & Y /\ A is infinite ) )
assume A1: F is finite ; ::_thesis: for A being Subset of X st A is infinite & A c= union F holds
ex Y being Subset of X st
( Y in F & Y /\ A is infinite )
let A be Subset of X; ::_thesis: ( A is infinite & A c= union F implies ex Y being Subset of X st
( Y in F & Y /\ A is infinite ) )
assume that
A2: A is infinite and
A3: A c= union F ; ::_thesis: ex Y being Subset of X st
( Y in F & Y /\ A is infinite )
set I = INTERSECTION (F,{A});
card [:F,{A}:] = card F by CARD_1:69;
then card (INTERSECTION (F,{A})) c= card F by TOPGEN_4:25;
then A4: INTERSECTION (F,{A}) is finite by A1;
A5: for x being set st x in A holds
ex y being set st
( y in INTERSECTION (F,{A}) & S1[x,y] )
proof
let x be set ; ::_thesis: ( x in A implies ex y being set st
( y in INTERSECTION (F,{A}) & S1[x,y] ) )
assume A6: x in A ; ::_thesis: ex y being set st
( y in INTERSECTION (F,{A}) & S1[x,y] )
consider y being set such that
A7: x in y and
A8: y in F by A3, A6, TARSKI:def_4;
take y /\ A ; ::_thesis: ( y /\ A in INTERSECTION (F,{A}) & S1[x,y /\ A] )
A in {A} by TARSKI:def_1;
hence ( y /\ A in INTERSECTION (F,{A}) & S1[x,y /\ A] ) by A6, A7, A8, SETFAM_1:def_5, XBOOLE_0:def_4; ::_thesis: verum
end;
consider p being Function of A,(INTERSECTION (F,{A})) such that
A9: for x being set st x in A holds
S1[x,p . x] from FUNCT_2:sch_1(A5);
consider x being set such that
A10: x in A by A2, XBOOLE_0:def_1;
ex y being set st
( y in INTERSECTION (F,{A}) & S1[x,y] ) by A5, A10;
then dom p = A by FUNCT_2:def_1;
then consider t being set such that
A11: t in rng p and
A12: p " {t} is infinite by A2, A4, CARD_2:101;
consider Y, Z being set such that
A13: Y in F and
A14: Z in {A} and
A15: t = Y /\ Z by A11, SETFAM_1:def_5;
reconsider Y = Y as Subset of X by A13;
take Y ; ::_thesis: ( Y in F & Y /\ A is infinite )
A16: Z = A by A14, TARSKI:def_1;
p " {t} c= Y /\ A
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in p " {t} or z in Y /\ A )
assume A17: z in p " {t} ; ::_thesis: z in Y /\ A
p . z in {t} by A17, FUNCT_1:def_7;
then p . z = t by TARSKI:def_1;
hence z in Y /\ A by A9, A15, A16, A17; ::_thesis: verum
end;
hence ( Y in F & Y /\ A is infinite ) by A12, A13; ::_thesis: verum
end;
theorem :: COMPL_SP:37
for M being non empty MetrSpace holds
( TopSpaceMetr M is compact iff ( M is totally_bounded & M is complete ) )
proof
let M be non empty MetrSpace; ::_thesis: ( TopSpaceMetr M is compact iff ( M is totally_bounded & M is complete ) )
set T = TopSpaceMetr M;
thus ( TopSpaceMetr M is compact implies ( M is totally_bounded & M is complete ) ) by TBSP_1:8, TBSP_1:9; ::_thesis: ( M is totally_bounded & M is complete implies TopSpaceMetr M is compact )
assume that
A1: M is totally_bounded and
A2: M is complete ; ::_thesis: TopSpaceMetr M is compact
now__::_thesis:_for_A_being_Subset_of_(TopSpaceMetr_M)_st_A_is_infinite_holds_
not_Der_A_is_empty
reconsider NULL = 0 as Real ;
deffunc H1( Element of NAT ) -> Element of REAL = 1 / (1 + $1);
set cM = the carrier of M;
defpred S1[ set , set ] means for a, b being Subset of M st $1 = a & $2 = b holds
( b c= a & diameter b <= (diameter a) / 2 );
defpred S2[ set ] means for a being Subset of M st a = $1 holds
( a is bounded & a is infinite & a is closed );
consider seq being Real_Sequence such that
A3: for n being Element of NAT holds seq . n = H1(n) from SEQ_1:sch_1();
set Ns = NULL (#) seq;
A4: for x being set st x in bool the carrier of M & S2[x] holds
ex y being set st
( y in bool the carrier of M & S1[x,y] & S2[y] )
proof
let x be set ; ::_thesis: ( x in bool the carrier of M & S2[x] implies ex y being set st
( y in bool the carrier of M & S1[x,y] & S2[y] ) )
assume that
A5: x in bool the carrier of M and
A6: S2[x] ; ::_thesis: ex y being set st
( y in bool the carrier of M & S1[x,y] & S2[y] )
reconsider X = x as Subset of M by A5;
reconsider X9 = X as Subset of (TopSpaceMetr M) ;
set d = diameter X;
percases ( diameter X = 0 or diameter X > 0 ) by A6, TBSP_1:21;
supposeA7: diameter X = 0 ; ::_thesis: ex y being set st
( y in bool the carrier of M & S1[x,y] & S2[y] )
take Y = X; ::_thesis: ( Y in bool the carrier of M & S1[x,Y] & S2[Y] )
thus ( Y in bool the carrier of M & S1[x,Y] & S2[Y] ) by A6, A7; ::_thesis: verum
end;
supposeA8: diameter X > 0 ; ::_thesis: ex y being set st
( y in bool the carrier of M & S1[x,y] & S2[y] )
then (diameter X) / 4 > 0 by XREAL_1:224;
then consider F being Subset-Family of M such that
A9: F is finite and
A10: the carrier of M = union F and
A11: for C being Subset of M st C in F holds
ex w being Point of M st C = Ball (w,((diameter X) / 4)) by A1, TBSP_1:def_1;
X is infinite by A6;
then consider Y being Subset of M such that
A12: Y in F and
A13: Y /\ X is infinite by A9, A10, Th36;
set YX = Y /\ X;
A14: ex w being Point of M st Y = Ball (w,((diameter X) / 4)) by A11, A12;
then A15: Y is bounded ;
then A16: diameter (Y /\ X) <= diameter Y by TBSP_1:24, XBOOLE_1:17;
diameter Y <= 2 * ((diameter X) / 4) by A8, A14, TBSP_1:23, XREAL_1:224;
then A17: diameter (Y /\ X) <= (diameter X) / 2 by A16, XXREAL_0:2;
reconsider yx = Y /\ X as Subset of (TopSpaceMetr M) ;
reconsider CYX = Cl yx as Subset of M ;
take CYX ; ::_thesis: ( CYX in bool the carrier of M & S1[x,CYX] & S2[CYX] )
A18: yx c= Cl yx by PRE_TOPC:18;
A19: yx c= X9 by XBOOLE_1:17;
X is closed by A6;
then A20: X9 is closed by Th6;
Y /\ X is bounded by A15, TBSP_1:14, XBOOLE_1:17;
hence ( CYX in bool the carrier of M & S1[x,CYX] & S2[CYX] ) by A13, A17, A18, A20, A19, Th6, Th8, TOPS_1:5; ::_thesis: verum
end;
end;
end;
consider G being Subset-Family of M such that
A21: G is finite and
A22: the carrier of M = union G and
A23: for C being Subset of M st C in G holds
ex w being Point of M st C = Ball (w,(1 / 2)) by A1, TBSP_1:def_1;
let A be Subset of (TopSpaceMetr M); ::_thesis: ( A is infinite implies not Der A is empty )
assume A is infinite ; ::_thesis: not Der A is empty
then consider X being Subset of M such that
A24: X in G and
A25: X /\ A is infinite by A21, A22, Th36;
reconsider XA = X /\ A as Subset of M ;
reconsider xa = XA as Subset of (TopSpaceMetr M) ;
reconsider CXA = Cl xa as Subset of M ;
A26: ( XA is bounded & diameter XA <= 1 )
proof
A27: ex w being Point of M st X = Ball (w,(1 / 2)) by A23, A24;
then A28: diameter X <= 2 * (1 / 2) by TBSP_1:23;
A29: X is bounded by A27;
then diameter XA <= diameter X by TBSP_1:24, XBOOLE_1:17;
hence ( XA is bounded & diameter XA <= 1 ) by A29, A28, TBSP_1:14, XBOOLE_1:17, XXREAL_0:2; ::_thesis: verum
end;
then CXA is bounded by Th8;
then A30: 0 <= diameter CXA by TBSP_1:21;
xa c= Cl xa by PRE_TOPC:18;
then A31: ( CXA in bool the carrier of M & S2[CXA] ) by A25, A26, Th6, Th8;
consider f being Function such that
A32: ( dom f = NAT & rng f c= bool the carrier of M ) and
A33: f . 0 = CXA and
A34: for k being Element of NAT holds
( S1[f . k,f . (k + 1)] & S2[f . k] ) from TREES_2:sch_5(A31, A4);
reconsider f = f as SetSequence of M by A32, FUNCT_2:2;
A35: now__::_thesis:_for_n_being_Nat_holds_f_._n_is_bounded
let n be Nat; ::_thesis: f . n is bounded
n in NAT by ORDINAL1:def_12;
hence f . n is bounded by A34; ::_thesis: verum
end;
A36: now__::_thesis:_for_x_being_set_st_x_in_dom_f_holds_
not_f_._x_is_empty
let x be set ; ::_thesis: ( x in dom f implies not f . x is empty )
assume x in dom f ; ::_thesis: not f . x is empty
then reconsider i = x as Element of NAT ;
f . i is infinite by A34;
hence not f . x is empty ; ::_thesis: verum
end;
now__::_thesis:_for_n_being_Nat_holds_f_._n_is_closed
let n be Nat; ::_thesis: f . n is closed
n in NAT by ORDINAL1:def_12;
hence f . n is closed by A34; ::_thesis: verum
end;
then reconsider f = f as non-empty pointwise_bounded closed SetSequence of M by A36, A35, Def1, Def8, FUNCT_1:def_9;
A37: (NULL (#) seq) . 0 = NULL * (seq . 0) by SEQ_1:9;
for n being Element of NAT holds f . (n + 1) c= f . n by A34;
then A38: f is V172() by KURATO_0:def_3;
set df = diameter f;
defpred S3[ Element of NAT ] means ( (NULL (#) seq) . $1 <= (diameter f) . $1 & (diameter f) . $1 <= seq . $1 );
A39: for n being Element of NAT st S3[n] holds
S3[n + 1]
proof
let n be Element of NAT ; ::_thesis: ( S3[n] implies S3[n + 1] )
assume S3[n] ; ::_thesis: S3[n + 1]
then (diameter f) . n <= H1(n) by A3;
then A40: ((diameter f) . n) / 2 <= H1(n) / 2 by XREAL_1:72;
set n1 = n + 1;
A41: diameter (f . n) = (diameter f) . n by Def2;
diameter (f . (n + 1)) <= (diameter (f . n)) / 2 by A34;
then (diameter f) . (n + 1) <= ((diameter f) . n) / 2 by A41, Def2;
then A42: (diameter f) . (n + 1) <= H1(n) / 2 by A40, XXREAL_0:2;
A43: (NULL (#) seq) . (n + 1) = NULL * (seq . (n + 1)) by SEQ_1:9;
f . (n + 1) is bounded by Def1;
then A44: 0 <= diameter (f . (n + 1)) by TBSP_1:21;
(n + 1) + 1 <= ((n + 1) + 1) + n by NAT_1:11;
then A45: H1(n + 1) >= 1 / (2 * (n + 1)) by XREAL_1:118;
1 / (2 * (n + 1)) = H1(n) / 2 by XCMPLX_1:78;
then H1(n + 1) >= (diameter f) . (n + 1) by A42, A45, XXREAL_0:2;
hence S3[n + 1] by A3, A44, A43, Def2; ::_thesis: verum
end;
A46: seq . 0 = 1 / (1 + 0) by A3;
A47: for n being Element of NAT holds seq . n = 1 / (n + 1) by A3;
then A48: seq is convergent by SEQ_4:30;
diameter CXA <= 1 by A26, Th8;
then A49: S3[ 0 ] by A33, A30, A46, A37, Def2;
A50: for n being Element of NAT holds S3[n] from NAT_1:sch_1(A49, A39);
A51: NULL (#) seq is convergent by A47, SEQ_2:7, SEQ_4:30;
A52: lim seq = 0 by A47, SEQ_4:30;
then A53: lim (NULL (#) seq) = NULL * 0 by A47, SEQ_2:8, SEQ_4:30;
then A54: lim (diameter f) = 0 by A48, A52, A51, A50, SEQ_2:20;
then not meet f is empty by A2, A38, Th10;
then consider p being set such that
A55: p in meet f by XBOOLE_0:def_1;
reconsider p = p as Point of (TopSpaceMetr M) by A55;
reconsider p9 = p as Point of M ;
A56: diameter f is convergent by A48, A52, A51, A53, A50, SEQ_2:19;
now__::_thesis:_for_U_being_open_Subset_of_(TopSpaceMetr_M)_st_p_in_U_holds_
ex_s9_being_Point_of_(TopSpaceMetr_M)_st_
(_s9_in_A_/\_U_&_s9_<>_p_)
let U be open Subset of (TopSpaceMetr M); ::_thesis: ( p in U implies ex s9 being Point of (TopSpaceMetr M) st
( s9 in A /\ U & s9 <> p ) )
assume p in U ; ::_thesis: ex s9 being Point of (TopSpaceMetr M) st
( s9 in A /\ U & s9 <> p )
then consider r being real number such that
A57: r > 0 and
A58: Ball (p9,r) c= U by TOPMETR:15;
r / 2 > 0 by A57, XREAL_1:215;
then consider n being Element of NAT such that
A59: for m being Element of NAT st n <= m holds
abs (((diameter f) . m) - 0) < r / 2 by A54, A56, SEQ_2:def_7;
p in f . n by A55, KURATO_0:3;
then A60: {p} c= f . n by ZFMISC_1:31;
f . n is infinite by A34;
then {p} c< f . n by A60, XBOOLE_0:def_8;
then (f . n) \ {p} <> {} by XBOOLE_1:105;
then consider q being set such that
A61: q in (f . n) \ {p} by XBOOLE_0:def_1;
reconsider q = q as Point of (TopSpaceMetr M) by A61;
A62: q in f . n by A61, ZFMISC_1:56;
A63: q in f . n by A61, ZFMISC_1:56;
reconsider q9 = q as Point of M ;
q <> p by A61, ZFMISC_1:56;
then A64: dist (p9,q9) <> 0 by METRIC_1:2;
reconsider B = Ball (q9,(dist (p9,q9))) as Subset of (TopSpaceMetr M) ;
A65: dist (p9,q9) >= 0 by METRIC_1:5;
dist (q9,q9) = 0 by METRIC_1:1;
then A66: q in B by A64, A65, METRIC_1:11;
Ball (q9,(dist (p9,q9))) in Family_open_set M by PCOMPS_1:29;
then A67: B is open by PRE_TOPC:def_2;
f . n c= Cl xa by A33, A38, PROB_1:def_4;
then B meets xa by A67, A66, A62, PRE_TOPC:24;
then consider s being set such that
A68: s in B and
A69: s in xa by XBOOLE_0:3;
reconsider s = s as Point of M by A68;
reconsider s9 = s as Point of (TopSpaceMetr M) ;
take s9 = s9; ::_thesis: ( s9 in A /\ U & s9 <> p )
A70: (NULL (#) seq) . n = NULL * (seq . n) by SEQ_1:9;
A71: abs (((diameter f) . n) - 0) < r / 2 by A59;
A72: f . n is bounded by A34;
(diameter f) . n >= (NULL (#) seq) . n by A50;
then (diameter f) . n < r / 2 by A70, A71, ABSVALUE:def_1;
then A73: diameter (f . n) < r / 2 by Def2;
p in f . n by A55, KURATO_0:3;
then dist (p9,q9) <= diameter (f . n) by A63, A72, TBSP_1:def_8;
then A74: dist (p9,q9) < r / 2 by A73, XXREAL_0:2;
dist (q9,s) < dist (p9,q9) by A68, METRIC_1:11;
then dist (q9,s) < r / 2 by A74, XXREAL_0:2;
then A75: (dist (p9,q9)) + (dist (q9,s)) < (r / 2) + (r / 2) by A74, XREAL_1:8;
dist (p9,s) <= (dist (p9,q9)) + (dist (q9,s)) by METRIC_1:4;
then dist (p9,s) < r by A75, XXREAL_0:2;
then A76: s in Ball (p9,r) by METRIC_1:11;
s in A by A69, XBOOLE_0:def_4;
hence ( s9 in A /\ U & s9 <> p ) by A58, A68, A76, METRIC_1:11, XBOOLE_0:def_4; ::_thesis: verum
end;
hence not Der A is empty by TOPGEN_1:17; ::_thesis: verum
end;
then TopSpaceMetr M is countably_compact by Th28;
hence TopSpaceMetr M is compact by Th35; ::_thesis: verum
end;
begin
theorem Th38: :: COMPL_SP:38
for X being set
for M being MetrStruct
for a being Point of M
for x being set holds
( x in [:X,( the carrier of M \ {a}):] \/ {[X,a]} iff ex y being set ex b being Point of M st
( x = [y,b] & ( ( y in X & b <> a ) or ( y = X & b = a ) ) ) )
proof
let X be set ; ::_thesis: for M being MetrStruct
for a being Point of M
for x being set holds
( x in [:X,( the carrier of M \ {a}):] \/ {[X,a]} iff ex y being set ex b being Point of M st
( x = [y,b] & ( ( y in X & b <> a ) or ( y = X & b = a ) ) ) )
let M be MetrStruct ; ::_thesis: for a being Point of M
for x being set holds
( x in [:X,( the carrier of M \ {a}):] \/ {[X,a]} iff ex y being set ex b being Point of M st
( x = [y,b] & ( ( y in X & b <> a ) or ( y = X & b = a ) ) ) )
let a be Point of M; ::_thesis: for x being set holds
( x in [:X,( the carrier of M \ {a}):] \/ {[X,a]} iff ex y being set ex b being Point of M st
( x = [y,b] & ( ( y in X & b <> a ) or ( y = X & b = a ) ) ) )
let x be set ; ::_thesis: ( x in [:X,( the carrier of M \ {a}):] \/ {[X,a]} iff ex y being set ex b being Point of M st
( x = [y,b] & ( ( y in X & b <> a ) or ( y = X & b = a ) ) ) )
thus ( x in [:X,( the carrier of M \ {a}):] \/ {[X,a]} implies ex y being set ex b being Point of M st
( x = [y,b] & ( ( y in X & b <> a ) or ( y = X & b = a ) ) ) ) ::_thesis: ( ex y being set ex b being Point of M st
( x = [y,b] & ( ( y in X & b <> a ) or ( y = X & b = a ) ) ) implies x in [:X,( the carrier of M \ {a}):] \/ {[X,a]} )
proof
assume A1: x in [:X,( the carrier of M \ {a}):] \/ {[X,a]} ; ::_thesis: ex y being set ex b being Point of M st
( x = [y,b] & ( ( y in X & b <> a ) or ( y = X & b = a ) ) )
percases ( x in [:X,( the carrier of M \ {a}):] or x in {[X,a]} ) by A1, XBOOLE_0:def_3;
suppose x in [:X,( the carrier of M \ {a}):] ; ::_thesis: ex y being set ex b being Point of M st
( x = [y,b] & ( ( y in X & b <> a ) or ( y = X & b = a ) ) )
then consider x1, x2 being set such that
A2: x1 in X and
A3: x2 in the carrier of M \ {a} and
A4: x = [x1,x2] by ZFMISC_1:def_2;
reconsider x2 = x2 as Point of M by A3;
take x1 ; ::_thesis: ex b being Point of M st
( x = [x1,b] & ( ( x1 in X & b <> a ) or ( x1 = X & b = a ) ) )
take x2 ; ::_thesis: ( x = [x1,x2] & ( ( x1 in X & x2 <> a ) or ( x1 = X & x2 = a ) ) )
thus ( x = [x1,x2] & ( ( x1 in X & x2 <> a ) or ( x1 = X & x2 = a ) ) ) by A2, A3, A4, ZFMISC_1:56; ::_thesis: verum
end;
suppose x in {[X,a]} ; ::_thesis: ex y being set ex b being Point of M st
( x = [y,b] & ( ( y in X & b <> a ) or ( y = X & b = a ) ) )
then x = [X,a] by TARSKI:def_1;
hence ex y being set ex b being Point of M st
( x = [y,b] & ( ( y in X & b <> a ) or ( y = X & b = a ) ) ) ; ::_thesis: verum
end;
end;
end;
given y being set , b being Point of M such that A5: x = [y,b] and
A6: ( ( y in X & b <> a ) or ( y = X & b = a ) ) ; ::_thesis: x in [:X,( the carrier of M \ {a}):] \/ {[X,a]}
percases ( ( y in X & b <> a ) or ( y = X & b = a ) ) by A6;
supposeA7: ( y in X & b <> a ) ; ::_thesis: x in [:X,( the carrier of M \ {a}):] \/ {[X,a]}
not the carrier of M is empty
proof
assume A8: the carrier of M is empty ; ::_thesis: contradiction
then a = {} by SUBSET_1:def_1;
hence contradiction by A7, A8, SUBSET_1:def_1; ::_thesis: verum
end;
then b in the carrier of M \ {a} by A7, ZFMISC_1:56;
then x in [:X,( the carrier of M \ {a}):] by A5, A7, ZFMISC_1:87;
hence x in [:X,( the carrier of M \ {a}):] \/ {[X,a]} by XBOOLE_0:def_3; ::_thesis: verum
end;
suppose ( y = X & b = a ) ; ::_thesis: x in [:X,( the carrier of M \ {a}):] \/ {[X,a]}
then x in {[X,a]} by A5, TARSKI:def_1;
hence x in [:X,( the carrier of M \ {a}):] \/ {[X,a]} by XBOOLE_0:def_3; ::_thesis: verum
end;
end;
end;
definition
let M be MetrStruct ;
let a be Point of M;
let X be set ;
func well_dist (a,X) -> Function of [:([:X,( the carrier of M \ {a}):] \/ {[X,a]}),([:X,( the carrier of M \ {a}):] \/ {[X,a]}):],REAL means :Def10: :: COMPL_SP:def 10
for x, y being Element of [:X,( the carrier of M \ {a}):] \/ {[X,a]}
for x1, y1 being set
for x2, y2 being Point of M st x = [x1,x2] & y = [y1,y2] holds
( ( x1 = y1 implies it . (x,y) = dist (x2,y2) ) & ( x1 <> y1 implies it . (x,y) = (dist (x2,a)) + (dist (a,y2)) ) );
existence
ex b1 being Function of [:([:X,( the carrier of M \ {a}):] \/ {[X,a]}),([:X,( the carrier of M \ {a}):] \/ {[X,a]}):],REAL st
for x, y being Element of [:X,( the carrier of M \ {a}):] \/ {[X,a]}
for x1, y1 being set
for x2, y2 being Point of M st x = [x1,x2] & y = [y1,y2] holds
( ( x1 = y1 implies b1 . (x,y) = dist (x2,y2) ) & ( x1 <> y1 implies b1 . (x,y) = (dist (x2,a)) + (dist (a,y2)) ) )
proof
set XX = [:X,( the carrier of M \ {a}):] \/ {[X,a]};
defpred S1[ set , set , set ] means for x, y being Element of [:X,( the carrier of M \ {a}):] \/ {[X,a]} st x = $1 & y = $2 holds
for x1, y1 being set
for x2, y2 being Point of M st x = [x1,x2] & y = [y1,y2] holds
( ( x1 = y1 implies $3 = dist (x2,y2) ) & ( x1 <> y1 implies $3 = (dist (x2,a)) + (dist (a,y2)) ) );
A1: for x, y being set st x in [:X,( the carrier of M \ {a}):] \/ {[X,a]} & y in [:X,( the carrier of M \ {a}):] \/ {[X,a]} holds
ex z being set st
( z in REAL & S1[x,y,z] )
proof
let x, y be set ; ::_thesis: ( x in [:X,( the carrier of M \ {a}):] \/ {[X,a]} & y in [:X,( the carrier of M \ {a}):] \/ {[X,a]} implies ex z being set st
( z in REAL & S1[x,y,z] ) )
assume that
A2: x in [:X,( the carrier of M \ {a}):] \/ {[X,a]} and
A3: y in [:X,( the carrier of M \ {a}):] \/ {[X,a]} ; ::_thesis: ex z being set st
( z in REAL & S1[x,y,z] )
consider y1 being set , y2 being Point of M such that
A4: y = [y1,y2] and
( ( y1 in X & y2 <> a ) or ( y1 = X & y2 = a ) ) by A3, Th38;
consider x1 being set , x2 being Point of M such that
A5: x = [x1,x2] and
( ( x1 in X & x2 <> a ) or ( x1 = X & x2 = a ) ) by A2, Th38;
now__::_thesis:_ex_d_being_Element_of_REAL_st_
(_d_in_REAL_&_(_for_x9,_y9_being_Element_of_[:X,(_the_carrier_of_M_\_{a}):]_\/_{[X,a]}_st_x9_=_x_&_y9_=_y_holds_
for_x19,_y19_being_set_
for_x29,_y29_being_Point_of_M_st_x9_=_[x19,x29]_&_y9_=_[y19,y29]_holds_
(_(_x19_=_y19_implies_d_=_dist_(x29,y29)_)_&_(_x19_<>_y19_implies_d_=_(dist_(x29,a))_+_(dist_(a,y29))_)_)_)_)
percases ( x1 = y1 or x1 <> y1 ) ;
supposeA6: x1 = y1 ; ::_thesis: ex d being Element of REAL st
( d in REAL & ( for x9, y9 being Element of [:X,( the carrier of M \ {a}):] \/ {[X,a]} st x9 = x & y9 = y holds
for x19, y19 being set
for x29, y29 being Point of M st x9 = [x19,x29] & y9 = [y19,y29] holds
( ( x19 = y19 implies d = dist (x29,y29) ) & ( x19 <> y19 implies d = (dist (x29,a)) + (dist (a,y29)) ) ) ) )
take d = dist (x2,y2); ::_thesis: ( d in REAL & ( for x9, y9 being Element of [:X,( the carrier of M \ {a}):] \/ {[X,a]} st x9 = x & y9 = y holds
for x19, y19 being set
for x29, y29 being Point of M st x9 = [x19,x29] & y9 = [y19,y29] holds
( ( x19 = y19 implies d = dist (x29,y29) ) & ( x19 <> y19 implies d = (dist (x29,a)) + (dist (a,y29)) ) ) ) )
thus d in REAL ; ::_thesis: for x9, y9 being Element of [:X,( the carrier of M \ {a}):] \/ {[X,a]} st x9 = x & y9 = y holds
for x19, y19 being set
for x29, y29 being Point of M st x9 = [x19,x29] & y9 = [y19,y29] holds
( ( x19 = y19 implies d = dist (x29,y29) ) & ( x19 <> y19 implies d = (dist (x29,a)) + (dist (a,y29)) ) )
let x9, y9 be Element of [:X,( the carrier of M \ {a}):] \/ {[X,a]}; ::_thesis: ( x9 = x & y9 = y implies for x19, y19 being set
for x29, y29 being Point of M st x9 = [x19,x29] & y9 = [y19,y29] holds
( ( x19 = y19 implies d = dist (x29,y29) ) & ( x19 <> y19 implies d = (dist (x29,a)) + (dist (a,y29)) ) ) )
assume that
A7: x9 = x and
A8: y9 = y ; ::_thesis: for x19, y19 being set
for x29, y29 being Point of M st x9 = [x19,x29] & y9 = [y19,y29] holds
( ( x19 = y19 implies d = dist (x29,y29) ) & ( x19 <> y19 implies d = (dist (x29,a)) + (dist (a,y29)) ) )
let x19, y19 be set ; ::_thesis: for x29, y29 being Point of M st x9 = [x19,x29] & y9 = [y19,y29] holds
( ( x19 = y19 implies d = dist (x29,y29) ) & ( x19 <> y19 implies d = (dist (x29,a)) + (dist (a,y29)) ) )
let x29, y29 be Point of M; ::_thesis: ( x9 = [x19,x29] & y9 = [y19,y29] implies ( ( x19 = y19 implies d = dist (x29,y29) ) & ( x19 <> y19 implies d = (dist (x29,a)) + (dist (a,y29)) ) ) )
assume that
A9: x9 = [x19,x29] and
A10: y9 = [y19,y29] ; ::_thesis: ( ( x19 = y19 implies d = dist (x29,y29) ) & ( x19 <> y19 implies d = (dist (x29,a)) + (dist (a,y29)) ) )
A11: x29 = x2 by A5, A7, A9, XTUPLE_0:1;
x19 = x1 by A5, A7, A9, XTUPLE_0:1;
hence ( ( x19 = y19 implies d = dist (x29,y29) ) & ( x19 <> y19 implies d = (dist (x29,a)) + (dist (a,y29)) ) ) by A4, A6, A8, A10, A11, XTUPLE_0:1; ::_thesis: verum
end;
supposeA12: x1 <> y1 ; ::_thesis: ex d being Element of REAL st
( d in REAL & ( for x9, y9 being Element of [:X,( the carrier of M \ {a}):] \/ {[X,a]} st x9 = x & y9 = y holds
for x19, y19 being set
for x29, y29 being Point of M st x9 = [x19,x29] & y9 = [y19,y29] holds
( ( x19 = y19 implies d = dist (x29,y29) ) & ( x19 <> y19 implies d = (dist (x29,a)) + (dist (a,y29)) ) ) ) )
take d = (dist (x2,a)) + (dist (a,y2)); ::_thesis: ( d in REAL & ( for x9, y9 being Element of [:X,( the carrier of M \ {a}):] \/ {[X,a]} st x9 = x & y9 = y holds
for x19, y19 being set
for x29, y29 being Point of M st x9 = [x19,x29] & y9 = [y19,y29] holds
( ( x19 = y19 implies d = dist (x29,y29) ) & ( x19 <> y19 implies d = (dist (x29,a)) + (dist (a,y29)) ) ) ) )
thus d in REAL ; ::_thesis: for x9, y9 being Element of [:X,( the carrier of M \ {a}):] \/ {[X,a]} st x9 = x & y9 = y holds
for x19, y19 being set
for x29, y29 being Point of M st x9 = [x19,x29] & y9 = [y19,y29] holds
( ( x19 = y19 implies d = dist (x29,y29) ) & ( x19 <> y19 implies d = (dist (x29,a)) + (dist (a,y29)) ) )
let x9, y9 be Element of [:X,( the carrier of M \ {a}):] \/ {[X,a]}; ::_thesis: ( x9 = x & y9 = y implies for x19, y19 being set
for x29, y29 being Point of M st x9 = [x19,x29] & y9 = [y19,y29] holds
( ( x19 = y19 implies d = dist (x29,y29) ) & ( x19 <> y19 implies d = (dist (x29,a)) + (dist (a,y29)) ) ) )
assume that
A13: x9 = x and
A14: y9 = y ; ::_thesis: for x19, y19 being set
for x29, y29 being Point of M st x9 = [x19,x29] & y9 = [y19,y29] holds
( ( x19 = y19 implies d = dist (x29,y29) ) & ( x19 <> y19 implies d = (dist (x29,a)) + (dist (a,y29)) ) )
let x19, y19 be set ; ::_thesis: for x29, y29 being Point of M st x9 = [x19,x29] & y9 = [y19,y29] holds
( ( x19 = y19 implies d = dist (x29,y29) ) & ( x19 <> y19 implies d = (dist (x29,a)) + (dist (a,y29)) ) )
let x29, y29 be Point of M; ::_thesis: ( x9 = [x19,x29] & y9 = [y19,y29] implies ( ( x19 = y19 implies d = dist (x29,y29) ) & ( x19 <> y19 implies d = (dist (x29,a)) + (dist (a,y29)) ) ) )
assume that
A15: x9 = [x19,x29] and
A16: y9 = [y19,y29] ; ::_thesis: ( ( x19 = y19 implies d = dist (x29,y29) ) & ( x19 <> y19 implies d = (dist (x29,a)) + (dist (a,y29)) ) )
A17: x29 = x2 by A5, A13, A15, XTUPLE_0:1;
x19 = x1 by A5, A13, A15, XTUPLE_0:1;
hence ( ( x19 = y19 implies d = dist (x29,y29) ) & ( x19 <> y19 implies d = (dist (x29,a)) + (dist (a,y29)) ) ) by A4, A12, A14, A16, A17, XTUPLE_0:1; ::_thesis: verum
end;
end;
end;
hence ex z being set st
( z in REAL & S1[x,y,z] ) ; ::_thesis: verum
end;
consider f being Function of [:([:X,( the carrier of M \ {a}):] \/ {[X,a]}),([:X,( the carrier of M \ {a}):] \/ {[X,a]}):],REAL such that
A18: for x, y being set st x in [:X,( the carrier of M \ {a}):] \/ {[X,a]} & y in [:X,( the carrier of M \ {a}):] \/ {[X,a]} holds
S1[x,y,f . (x,y)] from BINOP_1:sch_1(A1);
take f ; ::_thesis: for x, y being Element of [:X,( the carrier of M \ {a}):] \/ {[X,a]}
for x1, y1 being set
for x2, y2 being Point of M st x = [x1,x2] & y = [y1,y2] holds
( ( x1 = y1 implies f . (x,y) = dist (x2,y2) ) & ( x1 <> y1 implies f . (x,y) = (dist (x2,a)) + (dist (a,y2)) ) )
thus for x, y being Element of [:X,( the carrier of M \ {a}):] \/ {[X,a]}
for x1, y1 being set
for x2, y2 being Point of M st x = [x1,x2] & y = [y1,y2] holds
( ( x1 = y1 implies f . (x,y) = dist (x2,y2) ) & ( x1 <> y1 implies f . (x,y) = (dist (x2,a)) + (dist (a,y2)) ) ) by A18; ::_thesis: verum
end;
uniqueness
for b1, b2 being Function of [:([:X,( the carrier of M \ {a}):] \/ {[X,a]}),([:X,( the carrier of M \ {a}):] \/ {[X,a]}):],REAL st ( for x, y being Element of [:X,( the carrier of M \ {a}):] \/ {[X,a]}
for x1, y1 being set
for x2, y2 being Point of M st x = [x1,x2] & y = [y1,y2] holds
( ( x1 = y1 implies b1 . (x,y) = dist (x2,y2) ) & ( x1 <> y1 implies b1 . (x,y) = (dist (x2,a)) + (dist (a,y2)) ) ) ) & ( for x, y being Element of [:X,( the carrier of M \ {a}):] \/ {[X,a]}
for x1, y1 being set
for x2, y2 being Point of M st x = [x1,x2] & y = [y1,y2] holds
( ( x1 = y1 implies b2 . (x,y) = dist (x2,y2) ) & ( x1 <> y1 implies b2 . (x,y) = (dist (x2,a)) + (dist (a,y2)) ) ) ) holds
b1 = b2
proof
set XX = [:X,( the carrier of M \ {a}):] \/ {[X,a]};
let w1, w2 be Function of [:([:X,( the carrier of M \ {a}):] \/ {[X,a]}),([:X,( the carrier of M \ {a}):] \/ {[X,a]}):],REAL; ::_thesis: ( ( for x, y being Element of [:X,( the carrier of M \ {a}):] \/ {[X,a]}
for x1, y1 being set
for x2, y2 being Point of M st x = [x1,x2] & y = [y1,y2] holds
( ( x1 = y1 implies w1 . (x,y) = dist (x2,y2) ) & ( x1 <> y1 implies w1 . (x,y) = (dist (x2,a)) + (dist (a,y2)) ) ) ) & ( for x, y being Element of [:X,( the carrier of M \ {a}):] \/ {[X,a]}
for x1, y1 being set
for x2, y2 being Point of M st x = [x1,x2] & y = [y1,y2] holds
( ( x1 = y1 implies w2 . (x,y) = dist (x2,y2) ) & ( x1 <> y1 implies w2 . (x,y) = (dist (x2,a)) + (dist (a,y2)) ) ) ) implies w1 = w2 )
assume that
A19: for x, y being Element of [:X,( the carrier of M \ {a}):] \/ {[X,a]}
for x1, y1 being set
for x2, y2 being Point of M st x = [x1,x2] & y = [y1,y2] holds
( ( x1 = y1 implies w1 . (x,y) = dist (x2,y2) ) & ( x1 <> y1 implies w1 . (x,y) = (dist (x2,a)) + (dist (a,y2)) ) ) and
A20: for x, y being Element of [:X,( the carrier of M \ {a}):] \/ {[X,a]}
for x1, y1 being set
for x2, y2 being Point of M st x = [x1,x2] & y = [y1,y2] holds
( ( x1 = y1 implies w2 . (x,y) = dist (x2,y2) ) & ( x1 <> y1 implies w2 . (x,y) = (dist (x2,a)) + (dist (a,y2)) ) ) ; ::_thesis: w1 = w2
now__::_thesis:_for_x,_y_being_set_st_x_in_[:X,(_the_carrier_of_M_\_{a}):]_\/_{[X,a]}_&_y_in_[:X,(_the_carrier_of_M_\_{a}):]_\/_{[X,a]}_holds_
w1_._(x,y)_=_w2_._(x,y)
let x, y be set ; ::_thesis: ( x in [:X,( the carrier of M \ {a}):] \/ {[X,a]} & y in [:X,( the carrier of M \ {a}):] \/ {[X,a]} implies w1 . (x,y) = w2 . (x,y) )
assume that
A21: x in [:X,( the carrier of M \ {a}):] \/ {[X,a]} and
A22: y in [:X,( the carrier of M \ {a}):] \/ {[X,a]} ; ::_thesis: w1 . (x,y) = w2 . (x,y)
consider y1 being set , y2 being Point of M such that
A23: y = [y1,y2] and
( ( y1 in X & y2 <> a ) or ( y1 = X & y2 = a ) ) by A22, Th38;
consider x1 being set , x2 being Point of M such that
A24: x = [x1,x2] and
( ( x1 in X & x2 <> a ) or ( x1 = X & x2 = a ) ) by A21, Th38;
reconsider x2 = x2, y2 = y2 as Point of M ;
( x1 = y1 or x1 <> y1 ) ;
then ( ( w1 . (x,y) = dist (x2,y2) & w2 . (x,y) = dist (x2,y2) ) or ( w1 . (x,y) = (dist (x2,a)) + (dist (a,y2)) & w2 . (x,y) = (dist (x2,a)) + (dist (a,y2)) ) ) by A19, A20, A21, A22, A24, A23;
hence w1 . (x,y) = w2 . (x,y) ; ::_thesis: verum
end;
hence w1 = w2 by BINOP_1:1; ::_thesis: verum
end;
end;
:: deftheorem Def10 defines well_dist COMPL_SP:def_10_:_
for M being MetrStruct
for a being Point of M
for X being set
for b4 being Function of [:([:X,( the carrier of M \ {a}):] \/ {[X,a]}),([:X,( the carrier of M \ {a}):] \/ {[X,a]}):],REAL holds
( b4 = well_dist (a,X) iff for x, y being Element of [:X,( the carrier of M \ {a}):] \/ {[X,a]}
for x1, y1 being set
for x2, y2 being Point of M st x = [x1,x2] & y = [y1,y2] holds
( ( x1 = y1 implies b4 . (x,y) = dist (x2,y2) ) & ( x1 <> y1 implies b4 . (x,y) = (dist (x2,a)) + (dist (a,y2)) ) ) );
theorem :: COMPL_SP:39
for M being MetrStruct
for a being Point of M
for X being non empty set holds
( ( well_dist (a,X) is Reflexive implies M is Reflexive ) & ( well_dist (a,X) is symmetric implies M is symmetric ) & ( well_dist (a,X) is triangle & well_dist (a,X) is Reflexive implies M is triangle ) & ( well_dist (a,X) is discerning & well_dist (a,X) is Reflexive implies M is discerning ) )
proof
let M be MetrStruct ; ::_thesis: for a being Point of M
for X being non empty set holds
( ( well_dist (a,X) is Reflexive implies M is Reflexive ) & ( well_dist (a,X) is symmetric implies M is symmetric ) & ( well_dist (a,X) is triangle & well_dist (a,X) is Reflexive implies M is triangle ) & ( well_dist (a,X) is discerning & well_dist (a,X) is Reflexive implies M is discerning ) )
let A be Point of M; ::_thesis: for X being non empty set holds
( ( well_dist (A,X) is Reflexive implies M is Reflexive ) & ( well_dist (A,X) is symmetric implies M is symmetric ) & ( well_dist (A,X) is triangle & well_dist (A,X) is Reflexive implies M is triangle ) & ( well_dist (A,X) is discerning & well_dist (A,X) is Reflexive implies M is discerning ) )
let X be non empty set ; ::_thesis: ( ( well_dist (A,X) is Reflexive implies M is Reflexive ) & ( well_dist (A,X) is symmetric implies M is symmetric ) & ( well_dist (A,X) is triangle & well_dist (A,X) is Reflexive implies M is triangle ) & ( well_dist (A,X) is discerning & well_dist (A,X) is Reflexive implies M is discerning ) )
consider x0 being set such that
A1: x0 in X by XBOOLE_0:def_1;
set w = well_dist (A,X);
set XX = [:X,( the carrier of M \ {A}):] \/ {[X,A]};
thus A2: ( well_dist (A,X) is Reflexive implies M is Reflexive ) ::_thesis: ( ( well_dist (A,X) is symmetric implies M is symmetric ) & ( well_dist (A,X) is triangle & well_dist (A,X) is Reflexive implies M is triangle ) & ( well_dist (A,X) is discerning & well_dist (A,X) is Reflexive implies M is discerning ) )
proof
assume A3: well_dist (A,X) is Reflexive ; ::_thesis: M is Reflexive
now__::_thesis:_for_a_being_Element_of_M_holds_dist_(a,a)_=_0
let a be Element of M; ::_thesis: dist (a,a) = 0
now__::_thesis:_dist_(a,a)_=_0
percases ( a = A or a <> A ) ;
suppose a = A ; ::_thesis: dist (a,a) = 0
then A4: [X,a] in [:X,( the carrier of M \ {A}):] \/ {[X,A]} by Th38;
hence dist (a,a) = (well_dist (A,X)) . ([X,a],[X,a]) by Def10
.= 0 by A3, A4, METRIC_1:def_2 ;
::_thesis: verum
end;
suppose a <> A ; ::_thesis: dist (a,a) = 0
then A5: [x0,a] in [:X,( the carrier of M \ {A}):] \/ {[X,A]} by A1, Th38;
hence dist (a,a) = (well_dist (A,X)) . ([x0,a],[x0,a]) by Def10
.= 0 by A3, A5, METRIC_1:def_2 ;
::_thesis: verum
end;
end;
end;
hence dist (a,a) = 0 ; ::_thesis: verum
end;
hence M is Reflexive by METRIC_1:1; ::_thesis: verum
end;
thus ( well_dist (A,X) is symmetric implies M is symmetric ) ::_thesis: ( ( well_dist (A,X) is triangle & well_dist (A,X) is Reflexive implies M is triangle ) & ( well_dist (A,X) is discerning & well_dist (A,X) is Reflexive implies M is discerning ) )
proof
assume A6: well_dist (A,X) is symmetric ; ::_thesis: M is symmetric
now__::_thesis:_for_a,_b_being_Element_of_M_holds_dist_(a,b)_=_dist_(b,a)
let a, b be Element of M; ::_thesis: dist (a,b) = dist (b,a)
now__::_thesis:_dist_(a,b)_=_dist_(b,a)
percases ( ( a = A & b = A ) or ( a = A & b <> A ) or ( a <> A & b = A ) or ( a <> A & b <> A ) ) ;
suppose ( a = A & b = A ) ; ::_thesis: dist (a,b) = dist (b,a)
hence dist (a,b) = dist (b,a) ; ::_thesis: verum
end;
supposeA7: ( a = A & b <> A ) ; ::_thesis: dist (a,b) = dist (b,a)
then A8: [x0,b] in [:X,( the carrier of M \ {A}):] \/ {[X,A]} by A1, Th38;
A9: [X,A] in [:X,( the carrier of M \ {A}):] \/ {[X,A]} by Th38;
A10: X <> x0 by A1;
then (dist (A,A)) + (dist (A,b)) = (well_dist (A,X)) . ([X,A],[x0,b]) by A9, A8, Def10
.= (well_dist (A,X)) . ([x0,b],[X,A]) by A6, A9, A8, METRIC_1:def_4
.= (dist (b,A)) + (dist (A,A)) by A9, A8, A10, Def10 ;
hence dist (a,b) = dist (b,a) by A7; ::_thesis: verum
end;
supposeA11: ( a <> A & b = A ) ; ::_thesis: dist (a,b) = dist (b,a)
then A12: [x0,a] in [:X,( the carrier of M \ {A}):] \/ {[X,A]} by A1, Th38;
A13: [X,A] in [:X,( the carrier of M \ {A}):] \/ {[X,A]} by Th38;
A14: X <> x0 by A1;
then (dist (A,A)) + (dist (A,a)) = (well_dist (A,X)) . ([X,A],[x0,a]) by A13, A12, Def10
.= (well_dist (A,X)) . ([x0,a],[X,A]) by A6, A13, A12, METRIC_1:def_4
.= (dist (a,A)) + (dist (A,A)) by A13, A12, A14, Def10 ;
hence dist (a,b) = dist (b,a) by A11; ::_thesis: verum
end;
supposeA15: ( a <> A & b <> A ) ; ::_thesis: dist (a,b) = dist (b,a)
then A16: [x0,b] in [:X,( the carrier of M \ {A}):] \/ {[X,A]} by A1, Th38;
A17: [x0,a] in [:X,( the carrier of M \ {A}):] \/ {[X,A]} by A1, A15, Th38;
hence dist (a,b) = (well_dist (A,X)) . ([x0,a],[x0,b]) by A16, Def10
.= (well_dist (A,X)) . ([x0,b],[x0,a]) by A6, A17, A16, METRIC_1:def_4
.= dist (b,a) by A17, A16, Def10 ;
::_thesis: verum
end;
end;
end;
hence dist (a,b) = dist (b,a) ; ::_thesis: verum
end;
hence M is symmetric by METRIC_1:3; ::_thesis: verum
end;
thus ( well_dist (A,X) is triangle & well_dist (A,X) is Reflexive implies M is triangle ) ::_thesis: ( well_dist (A,X) is discerning & well_dist (A,X) is Reflexive implies M is discerning )
proof
assume A18: ( well_dist (A,X) is triangle & well_dist (A,X) is Reflexive ) ; ::_thesis: M is triangle
now__::_thesis:_for_a,_b,_c_being_Point_of_M_holds_dist_(a,c)_<=_(dist_(a,b))_+_(dist_(b,c))
let a, b, c be Point of M; ::_thesis: dist (a,c) <= (dist (a,b)) + (dist (b,c))
now__::_thesis:_dist_(a,c)_<=_(dist_(a,b))_+_(dist_(b,c))
percases ( ( a = A & b = A & c = A ) or ( a = A & b = A & c <> A ) or ( a = A & b <> A & c = A ) or ( a = A & b <> A & c <> A ) or ( a <> A & b = A & c = A ) or ( a <> A & b = A & c <> A ) or ( a <> A & b <> A & c = A ) or ( a <> A & b <> A & c <> A ) ) ;
suppose ( a = A & b = A & c = A ) ; ::_thesis: dist (a,c) <= (dist (a,b)) + (dist (b,c))
then reconsider Xa = [X,a], Xb = [X,b], Xc = [X,c] as Element of [:X,( the carrier of M \ {A}):] \/ {[X,A]} by Th38;
A19: dist (a,c) = (well_dist (A,X)) . (Xa,Xc) by Def10;
A20: dist (a,b) = (well_dist (A,X)) . (Xa,Xb) by Def10;
(well_dist (A,X)) . (Xa,Xc) <= ((well_dist (A,X)) . (Xa,Xb)) + ((well_dist (A,X)) . (Xb,Xc)) by A18, METRIC_1:def_5;
hence dist (a,c) <= (dist (a,b)) + (dist (b,c)) by A19, A20, Def10; ::_thesis: verum
end;
supposeA21: ( a = A & b = A & c <> A ) ; ::_thesis: dist (a,c) <= (dist (a,b)) + (dist (b,c))
dist (a,a) = 0 by A2, A18, METRIC_1:1;
hence dist (a,c) <= (dist (a,b)) + (dist (b,c)) by A21; ::_thesis: verum
end;
supposeA22: ( a = A & b <> A & c = A ) ; ::_thesis: dist (a,c) <= (dist (a,b)) + (dist (b,c))
then reconsider Xa = [X,a], Xb = [x0,b], Xc = [X,c] as Element of [:X,( the carrier of M \ {A}):] \/ {[X,A]} by A1, Th38;
A23: x0 <> X by A1;
then A24: (dist (b,c)) + (dist (a,a)) = (well_dist (A,X)) . (Xb,Xc) by A22, Def10;
A25: dist (a,a) = 0 by A2, A18, METRIC_1:1;
A26: (well_dist (A,X)) . (Xa,Xc) <= ((well_dist (A,X)) . (Xa,Xb)) + ((well_dist (A,X)) . (Xb,Xc)) by A18, METRIC_1:def_5;
(dist (a,a)) + (dist (a,b)) = (well_dist (A,X)) . (Xa,Xb) by A22, A23, Def10;
hence dist (a,c) <= (dist (a,b)) + (dist (b,c)) by A26, A24, A25, Def10; ::_thesis: verum
end;
supposeA27: ( a = A & b <> A & c <> A ) ; ::_thesis: dist (a,c) <= (dist (a,b)) + (dist (b,c))
then reconsider Xa = [X,a], Xb = [x0,b], Xc = [x0,c] as Element of [:X,( the carrier of M \ {A}):] \/ {[X,A]} by A1, Th38;
A28: x0 <> X by A1;
then A29: (dist (a,a)) + (dist (a,b)) = (well_dist (A,X)) . (Xa,Xb) by A27, Def10;
A30: dist (a,a) = 0 by A2, A18, METRIC_1:1;
A31: (well_dist (A,X)) . (Xa,Xc) <= ((well_dist (A,X)) . (Xa,Xb)) + ((well_dist (A,X)) . (Xb,Xc)) by A18, METRIC_1:def_5;
(dist (a,a)) + (dist (a,c)) = (well_dist (A,X)) . (Xa,Xc) by A27, A28, Def10;
hence dist (a,c) <= (dist (a,b)) + (dist (b,c)) by A31, A29, A30, Def10; ::_thesis: verum
end;
supposeA32: ( a <> A & b = A & c = A ) ; ::_thesis: dist (a,c) <= (dist (a,b)) + (dist (b,c))
dist (c,c) = 0 by A2, A18, METRIC_1:1;
hence dist (a,c) <= (dist (a,b)) + (dist (b,c)) by A32; ::_thesis: verum
end;
supposeA33: ( a <> A & b = A & c <> A ) ; ::_thesis: dist (a,c) <= (dist (a,b)) + (dist (b,c))
then reconsider Xa = [x0,a], Xb = [X,b], Xc = [x0,c] as Element of [:X,( the carrier of M \ {A}):] \/ {[X,A]} by A1, Th38;
A34: x0 <> X by A1;
then A35: (dist (b,b)) + (dist (b,c)) = (well_dist (A,X)) . (Xb,Xc) by A33, Def10;
A36: dist (b,b) = 0 by A2, A18, METRIC_1:1;
A37: (well_dist (A,X)) . (Xa,Xc) <= ((well_dist (A,X)) . (Xa,Xb)) + ((well_dist (A,X)) . (Xb,Xc)) by A18, METRIC_1:def_5;
(dist (a,b)) + (dist (b,b)) = (well_dist (A,X)) . (Xa,Xb) by A33, A34, Def10;
hence dist (a,c) <= (dist (a,b)) + (dist (b,c)) by A37, A35, A36, Def10; ::_thesis: verum
end;
supposeA38: ( a <> A & b <> A & c = A ) ; ::_thesis: dist (a,c) <= (dist (a,b)) + (dist (b,c))
then reconsider Xa = [x0,a], Xb = [x0,b], Xc = [X,c] as Element of [:X,( the carrier of M \ {A}):] \/ {[X,A]} by A1, Th38;
A39: x0 <> X by A1;
then A40: (dist (b,c)) + (dist (c,c)) = (well_dist (A,X)) . (Xb,Xc) by A38, Def10;
A41: dist (c,c) = 0 by A2, A18, METRIC_1:1;
A42: (well_dist (A,X)) . (Xa,Xc) <= ((well_dist (A,X)) . (Xa,Xb)) + ((well_dist (A,X)) . (Xb,Xc)) by A18, METRIC_1:def_5;
(dist (a,c)) + (dist (c,c)) = (well_dist (A,X)) . (Xa,Xc) by A38, A39, Def10;
hence dist (a,c) <= (dist (a,b)) + (dist (b,c)) by A42, A40, A41, Def10; ::_thesis: verum
end;
suppose ( a <> A & b <> A & c <> A ) ; ::_thesis: dist (a,c) <= (dist (a,b)) + (dist (b,c))
then reconsider Xa = [x0,a], Xb = [x0,b], Xc = [x0,c] as Element of [:X,( the carrier of M \ {A}):] \/ {[X,A]} by A1, Th38;
A43: dist (a,c) = (well_dist (A,X)) . (Xa,Xc) by Def10;
A44: dist (a,b) = (well_dist (A,X)) . (Xa,Xb) by Def10;
(well_dist (A,X)) . (Xa,Xc) <= ((well_dist (A,X)) . (Xa,Xb)) + ((well_dist (A,X)) . (Xb,Xc)) by A18, METRIC_1:def_5;
hence dist (a,c) <= (dist (a,b)) + (dist (b,c)) by A43, A44, Def10; ::_thesis: verum
end;
end;
end;
hence dist (a,c) <= (dist (a,b)) + (dist (b,c)) ; ::_thesis: verum
end;
hence M is triangle by METRIC_1:4; ::_thesis: verum
end;
assume A45: ( well_dist (A,X) is discerning & well_dist (A,X) is Reflexive ) ; ::_thesis: M is discerning
now__::_thesis:_for_a,_b_being_Point_of_M_st_dist_(a,b)_=_0_holds_
a_=_b
let a, b be Point of M; ::_thesis: ( dist (a,b) = 0 implies a = b )
assume A46: dist (a,b) = 0 ; ::_thesis: a = b
now__::_thesis:_a_=_b
percases ( ( a = A & b = A ) or ( a = A & b <> A ) or ( a <> A & b = A ) or ( a <> A & b <> A ) ) ;
suppose ( a = A & b = A ) ; ::_thesis: a = b
hence a = b ; ::_thesis: verum
end;
supposeA47: ( a = A & b <> A ) ; ::_thesis: a = b
then reconsider Xa = [X,a], Xb = [x0,b] as Element of [:X,( the carrier of M \ {A}):] \/ {[X,A]} by A1, Th38;
x0 <> X by A1;
then A48: (dist (a,a)) + (dist (a,b)) = (well_dist (A,X)) . (Xa,Xb) by A47, Def10;
dist (a,a) = 0 by A2, A45, METRIC_1:1;
then Xa = Xb by A45, A46, A48, METRIC_1:def_3;
hence a = b by XTUPLE_0:1; ::_thesis: verum
end;
supposeA49: ( a <> A & b = A ) ; ::_thesis: a = b
then reconsider Xa = [x0,a], Xb = [X,b] as Element of [:X,( the carrier of M \ {A}):] \/ {[X,A]} by A1, Th38;
x0 <> X by A1;
then A50: (dist (a,b)) + (dist (b,b)) = (well_dist (A,X)) . (Xa,Xb) by A49, Def10;
dist (b,b) = 0 by A2, A45, METRIC_1:1;
then Xa = Xb by A45, A46, A50, METRIC_1:def_3;
hence a = b by XTUPLE_0:1; ::_thesis: verum
end;
suppose ( a <> A & b <> A ) ; ::_thesis: a = b
then reconsider Xa = [x0,a], Xb = [x0,b] as Element of [:X,( the carrier of M \ {A}):] \/ {[X,A]} by A1, Th38;
dist (a,b) = (well_dist (A,X)) . (Xa,Xb) by Def10;
then Xa = Xb by A45, A46, METRIC_1:def_3;
hence a = b by XTUPLE_0:1; ::_thesis: verum
end;
end;
end;
hence a = b ; ::_thesis: verum
end;
hence M is discerning by METRIC_1:2; ::_thesis: verum
end;
definition
let M be MetrStruct ;
let a be Point of M;
let X be set ;
func WellSpace (a,X) -> strict MetrStruct equals :: COMPL_SP:def 11
MetrStruct(# ([:X,( the carrier of M \ {a}):] \/ {[X,a]}),(well_dist (a,X)) #);
coherence
MetrStruct(# ([:X,( the carrier of M \ {a}):] \/ {[X,a]}),(well_dist (a,X)) #) is strict MetrStruct ;
end;
:: deftheorem defines WellSpace COMPL_SP:def_11_:_
for M being MetrStruct
for a being Point of M
for X being set holds WellSpace (a,X) = MetrStruct(# ([:X,( the carrier of M \ {a}):] \/ {[X,a]}),(well_dist (a,X)) #);
registration
let M be MetrStruct ;
let a be Point of M;
let X be set ;
cluster WellSpace (a,X) -> non empty strict ;
coherence
not WellSpace (a,X) is empty ;
end;
Lm8: for M being MetrStruct
for a being Point of M
for X being set holds
( ( M is Reflexive implies WellSpace (a,X) is Reflexive ) & ( M is symmetric implies WellSpace (a,X) is symmetric ) & ( M is triangle & M is symmetric & M is Reflexive implies WellSpace (a,X) is triangle ) & ( M is triangle & M is symmetric & M is Reflexive & M is discerning implies WellSpace (a,X) is discerning ) )
proof
let M be MetrStruct ; ::_thesis: for a being Point of M
for X being set holds
( ( M is Reflexive implies WellSpace (a,X) is Reflexive ) & ( M is symmetric implies WellSpace (a,X) is symmetric ) & ( M is triangle & M is symmetric & M is Reflexive implies WellSpace (a,X) is triangle ) & ( M is triangle & M is symmetric & M is Reflexive & M is discerning implies WellSpace (a,X) is discerning ) )
let a be Point of M; ::_thesis: for X being set holds
( ( M is Reflexive implies WellSpace (a,X) is Reflexive ) & ( M is symmetric implies WellSpace (a,X) is symmetric ) & ( M is triangle & M is symmetric & M is Reflexive implies WellSpace (a,X) is triangle ) & ( M is triangle & M is symmetric & M is Reflexive & M is discerning implies WellSpace (a,X) is discerning ) )
let X be set ; ::_thesis: ( ( M is Reflexive implies WellSpace (a,X) is Reflexive ) & ( M is symmetric implies WellSpace (a,X) is symmetric ) & ( M is triangle & M is symmetric & M is Reflexive implies WellSpace (a,X) is triangle ) & ( M is triangle & M is symmetric & M is Reflexive & M is discerning implies WellSpace (a,X) is discerning ) )
set XX = [:X,( the carrier of M \ {a}):] \/ {[X,a]};
set w = well_dist (a,X);
set W = WellSpace (a,X);
thus ( M is Reflexive implies WellSpace (a,X) is Reflexive ) ::_thesis: ( ( M is symmetric implies WellSpace (a,X) is symmetric ) & ( M is triangle & M is symmetric & M is Reflexive implies WellSpace (a,X) is triangle ) & ( M is triangle & M is symmetric & M is Reflexive & M is discerning implies WellSpace (a,X) is discerning ) )
proof
assume A1: M is Reflexive ; ::_thesis: WellSpace (a,X) is Reflexive
now__::_thesis:_for_A_being_Element_of_[:X,(_the_carrier_of_M_\_{a}):]_\/_{[X,a]}_holds_(well_dist_(a,X))_._(A,A)_=_0
let A be Element of [:X,( the carrier of M \ {a}):] \/ {[X,a]}; ::_thesis: (well_dist (a,X)) . (A,A) = 0
consider y being set , b being Point of M such that
A2: A = [y,b] and
( ( y in X & b <> a ) or ( y = X & b = a ) ) by Th38;
thus (well_dist (a,X)) . (A,A) = dist (b,b) by A2, Def10
.= 0 by A1, METRIC_1:1 ; ::_thesis: verum
end;
then well_dist (a,X) is Reflexive by METRIC_1:def_2;
hence WellSpace (a,X) is Reflexive by METRIC_1:def_6; ::_thesis: verum
end;
thus ( M is symmetric implies WellSpace (a,X) is symmetric ) ::_thesis: ( ( M is triangle & M is symmetric & M is Reflexive implies WellSpace (a,X) is triangle ) & ( M is triangle & M is symmetric & M is Reflexive & M is discerning implies WellSpace (a,X) is discerning ) )
proof
assume M is symmetric ; ::_thesis: WellSpace (a,X) is symmetric
then reconsider M = M as symmetric MetrStruct ;
reconsider a = a as Point of M ;
now__::_thesis:_for_A,_B_being_Element_of_[:X,(_the_carrier_of_M_\_{a}):]_\/_{[X,a]}_holds_(well_dist_(a,X))_._(A,B)_=_(well_dist_(a,X))_._(B,A)
let A, B be Element of [:X,( the carrier of M \ {a}):] \/ {[X,a]}; ::_thesis: (well_dist (a,X)) . (A,B) = (well_dist (a,X)) . (B,A)
consider y1 being set , b1 being Point of M such that
A3: A = [y1,b1] and
A4: ( ( y1 in X & b1 <> a ) or ( y1 = X & b1 = a ) ) by Th38;
consider y2 being set , b2 being Point of M such that
A5: B = [y2,b2] and
A6: ( ( y2 in X & b2 <> a ) or ( y2 = X & b2 = a ) ) by Th38;
now__::_thesis:_(well_dist_(a,X))_._(A,B)_=_(well_dist_(a,X))_._(B,A)
percases ( ( b1 = a & y1 = X & b2 = a & y2 = X ) or y1 <> y2 or ( b1 <> a & b2 <> a & y1 = y2 ) ) by A4, A6;
suppose ( b1 = a & y1 = X & b2 = a & y2 = X ) ; ::_thesis: (well_dist (a,X)) . (A,B) = (well_dist (a,X)) . (B,A)
hence (well_dist (a,X)) . (A,B) = (well_dist (a,X)) . (B,A) by A3, A5; ::_thesis: verum
end;
supposeA7: y1 <> y2 ; ::_thesis: (well_dist (a,X)) . (A,B) = (well_dist (a,X)) . (B,A)
hence (well_dist (a,X)) . (A,B) = (dist (b1,a)) + (dist (a,b2)) by A3, A5, Def10
.= (dist (a,b1)) + (dist (a,b2))
.= (dist (a,b1)) + (dist (b2,a))
.= (well_dist (a,X)) . (B,A) by A3, A5, A7, Def10 ;
::_thesis: verum
end;
supposeA8: ( b1 <> a & b2 <> a & y1 = y2 ) ; ::_thesis: (well_dist (a,X)) . (A,B) = (well_dist (a,X)) . (B,A)
hence (well_dist (a,X)) . (A,B) = dist (b1,b2) by A3, A5, Def10
.= dist (b2,b1)
.= (well_dist (a,X)) . (B,A) by A3, A5, A8, Def10 ;
::_thesis: verum
end;
end;
end;
hence (well_dist (a,X)) . (A,B) = (well_dist (a,X)) . (B,A) ; ::_thesis: verum
end;
then well_dist (a,X) is symmetric by METRIC_1:def_4;
hence WellSpace (a,X) is symmetric by METRIC_1:def_8; ::_thesis: verum
end;
thus ( M is triangle & M is symmetric & M is Reflexive implies WellSpace (a,X) is triangle ) ::_thesis: ( M is triangle & M is symmetric & M is Reflexive & M is discerning implies WellSpace (a,X) is discerning )
proof
assume A9: ( M is triangle & M is symmetric & M is Reflexive ) ; ::_thesis: WellSpace (a,X) is triangle
now__::_thesis:_for_A,_B,_C_being_Element_of_[:X,(_the_carrier_of_M_\_{a}):]_\/_{[X,a]}_holds_(well_dist_(a,X))_._(A,C)_<=_((well_dist_(a,X))_._(A,B))_+_((well_dist_(a,X))_._(B,C))
let A, B, C be Element of [:X,( the carrier of M \ {a}):] \/ {[X,a]}; ::_thesis: (well_dist (a,X)) . (A,C) <= ((well_dist (a,X)) . (A,B)) + ((well_dist (a,X)) . (B,C))
consider y1 being set , b1 being Point of M such that
A10: A = [y1,b1] and
( ( y1 in X & b1 <> a ) or ( y1 = X & b1 = a ) ) by Th38;
consider y2 being set , b2 being Point of M such that
A11: B = [y2,b2] and
( ( y2 in X & b2 <> a ) or ( y2 = X & b2 = a ) ) by Th38;
consider y3 being set , b3 being Point of M such that
A12: C = [y3,b3] and
( ( y3 in X & b3 <> a ) or ( y3 = X & b3 = a ) ) by Th38;
now__::_thesis:_(well_dist_(a,X))_._(A,C)_<=_((well_dist_(a,X))_._(A,B))_+_((well_dist_(a,X))_._(B,C))
percases ( ( y1 = y2 & y1 = y3 ) or ( y1 <> y2 & y1 = y3 ) or ( y1 = y2 & y1 <> y3 ) or ( y1 <> y2 & y1 <> y3 & y2 <> y3 ) or ( y1 <> y2 & y1 <> y3 & y2 = y3 ) ) ;
supposeA13: ( y1 = y2 & y1 = y3 ) ; ::_thesis: (well_dist (a,X)) . (A,C) <= ((well_dist (a,X)) . (A,B)) + ((well_dist (a,X)) . (B,C))
then A14: dist (b2,b3) = (well_dist (a,X)) . (B,C) by A11, A12, Def10;
A15: dist (b1,b2) = (well_dist (a,X)) . (A,B) by A10, A11, A13, Def10;
dist (b1,b3) = (well_dist (a,X)) . (A,C) by A10, A12, A13, Def10;
hence (well_dist (a,X)) . (A,C) <= ((well_dist (a,X)) . (A,B)) + ((well_dist (a,X)) . (B,C)) by A9, A15, A14, METRIC_1:4; ::_thesis: verum
end;
supposeA16: ( y1 <> y2 & y1 = y3 ) ; ::_thesis: (well_dist (a,X)) . (A,C) <= ((well_dist (a,X)) . (A,B)) + ((well_dist (a,X)) . (B,C))
then A17: (dist (b2,a)) + (dist (a,b3)) = (well_dist (a,X)) . (B,C) by A11, A12, Def10;
A18: dist (b1,b2) <= (dist (b1,a)) + (dist (a,b2)) by A9, METRIC_1:4;
A19: dist (b2,b3) <= (dist (b2,a)) + (dist (a,b3)) by A9, METRIC_1:4;
(dist (b1,a)) + (dist (a,b2)) = (well_dist (a,X)) . (A,B) by A10, A11, A16, Def10;
then A20: (dist (b1,b2)) + (dist (b2,b3)) <= ((well_dist (a,X)) . (A,B)) + ((well_dist (a,X)) . (B,C)) by A17, A18, A19, XREAL_1:7;
A21: dist (b1,b3) <= (dist (b1,b2)) + (dist (b2,b3)) by A9, METRIC_1:4;
dist (b1,b3) = (well_dist (a,X)) . (A,C) by A10, A12, A16, Def10;
hence (well_dist (a,X)) . (A,C) <= ((well_dist (a,X)) . (A,B)) + ((well_dist (a,X)) . (B,C)) by A20, A21, XXREAL_0:2; ::_thesis: verum
end;
supposeA22: ( y1 = y2 & y1 <> y3 ) ; ::_thesis: (well_dist (a,X)) . (A,C) <= ((well_dist (a,X)) . (A,B)) + ((well_dist (a,X)) . (B,C))
A23: dist (b1,a) <= (dist (b1,b2)) + (dist (b2,a)) by A9, METRIC_1:4;
(dist (b1,a)) + (dist (a,b3)) = (well_dist (a,X)) . (A,C) by A10, A12, A22, Def10;
then A24: (well_dist (a,X)) . (A,C) <= ((dist (b1,b2)) + (dist (b2,a))) + (dist (a,b3)) by A23, XREAL_1:6;
A25: (dist (b2,a)) + (dist (a,b3)) = (well_dist (a,X)) . (B,C) by A11, A12, A22, Def10;
dist (b1,b2) = (well_dist (a,X)) . (A,B) by A10, A11, A22, Def10;
hence (well_dist (a,X)) . (A,C) <= ((well_dist (a,X)) . (A,B)) + ((well_dist (a,X)) . (B,C)) by A25, A24; ::_thesis: verum
end;
supposeA26: ( y1 <> y2 & y1 <> y3 & y2 <> y3 ) ; ::_thesis: (well_dist (a,X)) . (A,C) <= ((well_dist (a,X)) . (A,B)) + ((well_dist (a,X)) . (B,C))
A27: 0 <= dist (b2,a) by A9, METRIC_1:5;
(dist (b2,a)) + (dist (a,b3)) = (well_dist (a,X)) . (B,C) by A11, A12, A26, Def10;
then A28: 0 + (dist (a,b3)) <= (well_dist (a,X)) . (B,C) by A27, XREAL_1:6;
A29: 0 <= dist (a,b2) by A9, METRIC_1:5;
(dist (b1,a)) + (dist (a,b2)) = (well_dist (a,X)) . (A,B) by A10, A11, A26, Def10;
then A30: (dist (b1,a)) + 0 <= (well_dist (a,X)) . (A,B) by A29, XREAL_1:6;
(dist (b1,a)) + (dist (a,b3)) = (well_dist (a,X)) . (A,C) by A10, A12, A26, Def10;
hence (well_dist (a,X)) . (A,C) <= ((well_dist (a,X)) . (A,B)) + ((well_dist (a,X)) . (B,C)) by A30, A28, XREAL_1:7; ::_thesis: verum
end;
supposeA31: ( y1 <> y2 & y1 <> y3 & y2 = y3 ) ; ::_thesis: (well_dist (a,X)) . (A,C) <= ((well_dist (a,X)) . (A,B)) + ((well_dist (a,X)) . (B,C))
A32: dist (a,b3) <= (dist (a,b2)) + (dist (b2,b3)) by A9, METRIC_1:4;
(dist (b1,a)) + (dist (a,b3)) = (well_dist (a,X)) . (A,C) by A10, A12, A31, Def10;
then A33: (well_dist (a,X)) . (A,C) <= (dist (b1,a)) + ((dist (a,b2)) + (dist (b2,b3))) by A32, XREAL_1:7;
A34: dist (b2,b3) = (well_dist (a,X)) . (B,C) by A11, A12, A31, Def10;
(dist (b1,a)) + (dist (a,b2)) = (well_dist (a,X)) . (A,B) by A10, A11, A31, Def10;
hence (well_dist (a,X)) . (A,C) <= ((well_dist (a,X)) . (A,B)) + ((well_dist (a,X)) . (B,C)) by A34, A33; ::_thesis: verum
end;
end;
end;
hence (well_dist (a,X)) . (A,C) <= ((well_dist (a,X)) . (A,B)) + ((well_dist (a,X)) . (B,C)) ; ::_thesis: verum
end;
then well_dist (a,X) is triangle by METRIC_1:def_5;
hence WellSpace (a,X) is triangle by METRIC_1:def_9; ::_thesis: verum
end;
assume A35: ( M is triangle & M is symmetric & M is Reflexive & M is discerning ) ; ::_thesis: WellSpace (a,X) is discerning
now__::_thesis:_for_A,_B_being_Element_of_[:X,(_the_carrier_of_M_\_{a}):]_\/_{[X,a]}_st_(well_dist_(a,X))_._(A,B)_=_0_holds_
A_=_B
let A, B be Element of [:X,( the carrier of M \ {a}):] \/ {[X,a]}; ::_thesis: ( (well_dist (a,X)) . (A,B) = 0 implies A = B )
consider y1 being set , b1 being Point of M such that
A36: A = [y1,b1] and
A37: ( ( y1 in X & b1 <> a ) or ( y1 = X & b1 = a ) ) by Th38;
consider y2 being set , b2 being Point of M such that
A38: B = [y2,b2] and
A39: ( ( y2 in X & b2 <> a ) or ( y2 = X & b2 = a ) ) by Th38;
assume A40: (well_dist (a,X)) . (A,B) = 0 ; ::_thesis: A = B
now__::_thesis:_A_=_B
percases ( y1 = y2 or y1 <> y2 ) ;
supposeA41: y1 = y2 ; ::_thesis: A = B
then (well_dist (a,X)) . (A,B) = dist (b1,b2) by A36, A38, Def10;
hence A = B by A35, A40, A36, A38, A41, METRIC_1:2; ::_thesis: verum
end;
suppose y1 <> y2 ; ::_thesis: A = B
then A42: (well_dist (a,X)) . (A,B) = (dist (b1,a)) + (dist (a,b2)) by A36, A38, Def10;
A43: dist (b1,a) >= 0 by A35, METRIC_1:5;
dist (a,b2) >= 0 by A35, METRIC_1:5;
then dist (b1,a) = 0 by A40, A42, A43;
hence A = B by A35, A40, A36, A37, A38, A39, A42, METRIC_1:2; ::_thesis: verum
end;
end;
end;
hence A = B ; ::_thesis: verum
end;
then well_dist (a,X) is discerning by METRIC_1:def_3;
hence WellSpace (a,X) is discerning by METRIC_1:def_7; ::_thesis: verum
end;
registration
let M be Reflexive MetrStruct ;
let a be Point of M;
let X be set ;
cluster WellSpace (a,X) -> strict Reflexive ;
coherence
WellSpace (a,X) is Reflexive by Lm8;
end;
registration
let M be symmetric MetrStruct ;
let a be Point of M;
let X be set ;
cluster WellSpace (a,X) -> strict symmetric ;
coherence
WellSpace (a,X) is symmetric by Lm8;
end;
registration
let M be Reflexive symmetric triangle MetrStruct ;
let a be Point of M;
let X be set ;
cluster WellSpace (a,X) -> strict triangle ;
coherence
WellSpace (a,X) is triangle by Lm8;
end;
registration
let M be MetrSpace;
let a be Point of M;
let X be set ;
cluster WellSpace (a,X) -> strict discerning ;
coherence
WellSpace (a,X) is discerning by Lm8;
end;
theorem :: COMPL_SP:40
for M being non empty Reflexive triangle MetrStruct
for a being Point of M
for X being non empty set st WellSpace (a,X) is complete holds
M is complete
proof
let M be non empty Reflexive triangle MetrStruct ; ::_thesis: for a being Point of M
for X being non empty set st WellSpace (a,X) is complete holds
M is complete
let a be Point of M; ::_thesis: for X being non empty set st WellSpace (a,X) is complete holds
M is complete
let X be non empty set ; ::_thesis: ( WellSpace (a,X) is complete implies M is complete )
consider x0 being set such that
A1: x0 in X by XBOOLE_0:def_1;
set W = WellSpace (a,X);
assume A2: WellSpace (a,X) is complete ; ::_thesis: M is complete
let S be sequence of M; :: according to TBSP_1:def_5 ::_thesis: ( not S is Cauchy or S is convergent )
assume A3: S is Cauchy ; ::_thesis: S is convergent
defpred S1[ set , set ] means ( ( S . $1 <> a implies $2 = [x0,(S . $1)] ) & ( S . $1 = a implies $2 = [X,(S . $1)] ) );
A4: for x being set st x in NAT holds
ex y being set st
( y in the carrier of (WellSpace (a,X)) & S1[x,y] )
proof
let x be set ; ::_thesis: ( x in NAT implies ex y being set st
( y in the carrier of (WellSpace (a,X)) & S1[x,y] ) )
assume x in NAT ; ::_thesis: ex y being set st
( y in the carrier of (WellSpace (a,X)) & S1[x,y] )
then reconsider i = x as Element of NAT ;
percases ( S . i <> a or S . x = a ) ;
supposeA5: S . i <> a ; ::_thesis: ex y being set st
( y in the carrier of (WellSpace (a,X)) & S1[x,y] )
take [x0,(S . i)] ; ::_thesis: ( [x0,(S . i)] in the carrier of (WellSpace (a,X)) & S1[x,[x0,(S . i)]] )
thus ( [x0,(S . i)] in the carrier of (WellSpace (a,X)) & S1[x,[x0,(S . i)]] ) by A1, A5, Th38; ::_thesis: verum
end;
supposeA6: S . x = a ; ::_thesis: ex y being set st
( y in the carrier of (WellSpace (a,X)) & S1[x,y] )
take [X,a] ; ::_thesis: ( [X,a] in the carrier of (WellSpace (a,X)) & S1[x,[X,a]] )
thus ( [X,a] in the carrier of (WellSpace (a,X)) & S1[x,[X,a]] ) by A6, Th38; ::_thesis: verum
end;
end;
end;
consider S9 being sequence of (WellSpace (a,X)) such that
A7: for x being set st x in NAT holds
S1[x,S9 . x] from FUNCT_2:sch_1(A4);
S9 is Cauchy
proof
let r be Real; :: according to TBSP_1:def_4 ::_thesis: ( r <= 0 or ex b1 being Element of NAT st
for b2, b3 being Element of NAT holds
( not b1 <= b2 or not b1 <= b3 or not r <= dist ((S9 . b2),(S9 . b3)) ) )
assume r > 0 ; ::_thesis: ex b1 being Element of NAT st
for b2, b3 being Element of NAT holds
( not b1 <= b2 or not b1 <= b3 or not r <= dist ((S9 . b2),(S9 . b3)) )
then consider p being Element of NAT such that
A8: for n, m being Element of NAT st p <= n & p <= m holds
dist ((S . n),(S . m)) < r by A3, TBSP_1:def_4;
take p ; ::_thesis: for b1, b2 being Element of NAT holds
( not p <= b1 or not p <= b2 or not r <= dist ((S9 . b1),(S9 . b2)) )
let n, m be Element of NAT ; ::_thesis: ( not p <= n or not p <= m or not r <= dist ((S9 . n),(S9 . m)) )
assume that
A9: p <= n and
A10: p <= m ; ::_thesis: not r <= dist ((S9 . n),(S9 . m))
percases ( ( S . n = a & S . m = a ) or ( S . n <> a & S . m = a ) or ( S . n = a & S . m <> a ) or ( S . n <> a & S . m <> a ) ) ;
supposeA11: ( S . n = a & S . m = a ) ; ::_thesis: not r <= dist ((S9 . n),(S9 . m))
then A12: [X,(S . m)] = S9 . m by A7;
[X,(S . n)] = S9 . n by A7, A11;
then dist ((S9 . n),(S9 . m)) = dist ((S . n),(S . m)) by A12, Def10;
hence not r <= dist ((S9 . n),(S9 . m)) by A8, A9, A10; ::_thesis: verum
end;
supposeA13: ( S . n <> a & S . m = a ) ; ::_thesis: not r <= dist ((S9 . n),(S9 . m))
then A14: [X,(S . m)] = S9 . m by A7;
A15: dist ((S . m),(S . m)) = 0 by METRIC_1:1;
A16: X <> x0 by A1;
[x0,(S . n)] = S9 . n by A7, A13;
then dist ((S9 . n),(S9 . m)) = (dist ((S . n),(S . m))) + (dist ((S . m),(S . m))) by A13, A14, A16, Def10;
hence not r <= dist ((S9 . n),(S9 . m)) by A8, A9, A10, A15; ::_thesis: verum
end;
supposeA17: ( S . n = a & S . m <> a ) ; ::_thesis: not r <= dist ((S9 . n),(S9 . m))
then A18: [x0,(S . m)] = S9 . m by A7;
A19: dist ((S . n),(S . n)) = 0 by METRIC_1:1;
A20: X <> x0 by A1;
[X,(S . n)] = S9 . n by A7, A17;
then dist ((S9 . n),(S9 . m)) = (dist ((S . n),(S . n))) + (dist ((S . n),(S . m))) by A17, A18, A20, Def10;
hence not r <= dist ((S9 . n),(S9 . m)) by A8, A9, A10, A19; ::_thesis: verum
end;
supposeA21: ( S . n <> a & S . m <> a ) ; ::_thesis: not r <= dist ((S9 . n),(S9 . m))
then A22: [x0,(S . m)] = S9 . m by A7;
[x0,(S . n)] = S9 . n by A7, A21;
then dist ((S9 . n),(S9 . m)) = dist ((S . n),(S . m)) by A22, Def10;
hence not r <= dist ((S9 . n),(S9 . m)) by A8, A9, A10; ::_thesis: verum
end;
end;
end;
then S9 is convergent by A2, TBSP_1:def_5;
then consider L being Element of (WellSpace (a,X)) such that
A23: for r being Real st r > 0 holds
ex n being Element of NAT st
for m being Element of NAT st n <= m holds
dist ((S9 . m),L) < r by TBSP_1:def_2;
consider L1 being set , L2 being Point of M such that
A24: L = [L1,L2] and
( ( L1 in X & L2 <> a ) or ( L1 = X & L2 = a ) ) by Th38;
take L2 ; :: according to TBSP_1:def_2 ::_thesis: for b1 being Element of REAL holds
( b1 <= 0 or ex b2 being Element of NAT st
for b3 being Element of NAT holds
( not b2 <= b3 or not b1 <= dist ((S . b3),L2) ) )
let r be Real; ::_thesis: ( r <= 0 or ex b1 being Element of NAT st
for b2 being Element of NAT holds
( not b1 <= b2 or not r <= dist ((S . b2),L2) ) )
assume r > 0 ; ::_thesis: ex b1 being Element of NAT st
for b2 being Element of NAT holds
( not b1 <= b2 or not r <= dist ((S . b2),L2) )
then consider n being Element of NAT such that
A25: for m being Element of NAT st n <= m holds
dist ((S9 . m),L) < r by A23;
take n ; ::_thesis: for b1 being Element of NAT holds
( not n <= b1 or not r <= dist ((S . b1),L2) )
let m be Element of NAT ; ::_thesis: ( not n <= m or not r <= dist ((S . m),L2) )
assume A26: n <= m ; ::_thesis: not r <= dist ((S . m),L2)
percases ( ( S . m = a & L1 = X ) or ( S . m = a & L1 <> X ) or ( S . m <> a & L1 = x0 ) or ( S . m <> a & L1 <> x0 ) ) ;
supposeA27: ( S . m = a & L1 = X ) ; ::_thesis: not r <= dist ((S . m),L2)
then S9 . m = [X,a] by A7;
then dist ((S9 . m),L) = dist ((S . m),L2) by A24, A27, Def10;
hence not r <= dist ((S . m),L2) by A25, A26; ::_thesis: verum
end;
supposeA28: ( S . m = a & L1 <> X ) ; ::_thesis: not r <= dist ((S . m),L2)
then S9 . m = [X,a] by A7;
then A29: dist ((S9 . m),L) = (dist ((S . m),(S . m))) + (dist ((S . m),L2)) by A24, A28, Def10;
dist ((S . m),(S . m)) = 0 by METRIC_1:1;
hence not r <= dist ((S . m),L2) by A25, A26, A29; ::_thesis: verum
end;
supposeA30: ( S . m <> a & L1 = x0 ) ; ::_thesis: not r <= dist ((S . m),L2)
then S9 . m = [x0,(S . m)] by A7;
then dist ((S9 . m),L) = dist ((S . m),L2) by A24, A30, Def10;
hence not r <= dist ((S . m),L2) by A25, A26; ::_thesis: verum
end;
supposeA31: ( S . m <> a & L1 <> x0 ) ; ::_thesis: not r <= dist ((S . m),L2)
then S9 . m = [x0,(S . m)] by A7;
then A32: dist ((S9 . m),L) = (dist ((S . m),a)) + (dist (a,L2)) by A24, A31, Def10;
A33: (dist ((S . m),a)) + (dist (a,L2)) >= dist ((S . m),L2) by METRIC_1:4;
dist ((S9 . m),L) < r by A25, A26;
hence not r <= dist ((S . m),L2) by A32, A33, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
theorem Th41: :: COMPL_SP:41
for X being set
for M being non empty Reflexive symmetric triangle MetrStruct
for a being Point of M
for S being sequence of (WellSpace (a,X)) holds
( not S is Cauchy or for Xa being Point of (WellSpace (a,X)) st Xa = [X,a] holds
for r being Real st r > 0 holds
ex n being Nat st
for m being Nat st m >= n holds
dist ((S . m),Xa) < r or ex n being Nat ex Y being set st
for m being Nat st m >= n holds
ex p being Point of M st S . m = [Y,p] )
proof
let X be set ; ::_thesis: for M being non empty Reflexive symmetric triangle MetrStruct
for a being Point of M
for S being sequence of (WellSpace (a,X)) holds
( not S is Cauchy or for Xa being Point of (WellSpace (a,X)) st Xa = [X,a] holds
for r being Real st r > 0 holds
ex n being Nat st
for m being Nat st m >= n holds
dist ((S . m),Xa) < r or ex n being Nat ex Y being set st
for m being Nat st m >= n holds
ex p being Point of M st S . m = [Y,p] )
let M be non empty Reflexive symmetric triangle MetrStruct ; ::_thesis: for a being Point of M
for S being sequence of (WellSpace (a,X)) holds
( not S is Cauchy or for Xa being Point of (WellSpace (a,X)) st Xa = [X,a] holds
for r being Real st r > 0 holds
ex n being Nat st
for m being Nat st m >= n holds
dist ((S . m),Xa) < r or ex n being Nat ex Y being set st
for m being Nat st m >= n holds
ex p being Point of M st S . m = [Y,p] )
let a be Point of M; ::_thesis: for S being sequence of (WellSpace (a,X)) holds
( not S is Cauchy or for Xa being Point of (WellSpace (a,X)) st Xa = [X,a] holds
for r being Real st r > 0 holds
ex n being Nat st
for m being Nat st m >= n holds
dist ((S . m),Xa) < r or ex n being Nat ex Y being set st
for m being Nat st m >= n holds
ex p being Point of M st S . m = [Y,p] )
set W = WellSpace (a,X);
reconsider Xa = [X,a] as Point of (WellSpace (a,X)) by Th38;
let S be sequence of (WellSpace (a,X)); ::_thesis: ( not S is Cauchy or for Xa being Point of (WellSpace (a,X)) st Xa = [X,a] holds
for r being Real st r > 0 holds
ex n being Nat st
for m being Nat st m >= n holds
dist ((S . m),Xa) < r or ex n being Nat ex Y being set st
for m being Nat st m >= n holds
ex p being Point of M st S . m = [Y,p] )
assume A1: S is Cauchy ; ::_thesis: ( for Xa being Point of (WellSpace (a,X)) st Xa = [X,a] holds
for r being Real st r > 0 holds
ex n being Nat st
for m being Nat st m >= n holds
dist ((S . m),Xa) < r or ex n being Nat ex Y being set st
for m being Nat st m >= n holds
ex p being Point of M st S . m = [Y,p] )
percases ( for r being Real st r > 0 holds
ex n being Nat st
for m being Nat st m >= n holds
dist ((S . m),Xa) < r or ex r being Real st
( r > 0 & ( for n being Nat ex m being Nat st
( m >= n & dist ((S . m),Xa) >= r ) ) ) ) ;
suppose for r being Real st r > 0 holds
ex n being Nat st
for m being Nat st m >= n holds
dist ((S . m),Xa) < r ; ::_thesis: ( for Xa being Point of (WellSpace (a,X)) st Xa = [X,a] holds
for r being Real st r > 0 holds
ex n being Nat st
for m being Nat st m >= n holds
dist ((S . m),Xa) < r or ex n being Nat ex Y being set st
for m being Nat st m >= n holds
ex p being Point of M st S . m = [Y,p] )
hence ( for Xa being Point of (WellSpace (a,X)) st Xa = [X,a] holds
for r being Real st r > 0 holds
ex n being Nat st
for m being Nat st m >= n holds
dist ((S . m),Xa) < r or ex n being Nat ex Y being set st
for m being Nat st m >= n holds
ex p being Point of M st S . m = [Y,p] ) ; ::_thesis: verum
end;
suppose ex r being Real st
( r > 0 & ( for n being Nat ex m being Nat st
( m >= n & dist ((S . m),Xa) >= r ) ) ) ; ::_thesis: ( for Xa being Point of (WellSpace (a,X)) st Xa = [X,a] holds
for r being Real st r > 0 holds
ex n being Nat st
for m being Nat st m >= n holds
dist ((S . m),Xa) < r or ex n being Nat ex Y being set st
for m being Nat st m >= n holds
ex p being Point of M st S . m = [Y,p] )
then consider r being Real such that
A2: r > 0 and
A3: for n being Nat ex m being Nat st
( m >= n & dist ((S . m),Xa) >= r ) ;
consider p being Element of NAT such that
A4: for n, m being Element of NAT st n >= p & m >= p holds
dist ((S . n),(S . m)) < r by A1, A2, TBSP_1:def_4;
consider p9 being Nat such that
A5: p9 >= p and
A6: dist ((S . p9),Xa) >= r by A3;
consider Y being set , y being Point of M such that
A7: S . p9 = [Y,y] and
( ( Y in X & y <> a ) or ( Y = X & y = a ) ) by Th38;
ex n being Nat ex Y being set st
for m being Nat st m >= n holds
ex p being Point of M st S . m = [Y,p]
proof
take p9 ; ::_thesis: ex Y being set st
for m being Nat st m >= p9 holds
ex p being Point of M st S . m = [Y,p]
take Y ; ::_thesis: for m being Nat st m >= p9 holds
ex p being Point of M st S . m = [Y,p]
let m be Nat; ::_thesis: ( m >= p9 implies ex p being Point of M st S . m = [Y,p] )
assume A8: m >= p9 ; ::_thesis: ex p being Point of M st S . m = [Y,p]
consider Z being set , z being Point of M such that
A9: S . m = [Z,z] and
( ( Z in X & z <> a ) or ( Z = X & z = a ) ) by Th38;
Y = Z
proof
A10: p9 in NAT by ORDINAL1:def_12;
A11: dist (a,z) >= 0 by METRIC_1:5;
A12: dist (a,a) = 0 by METRIC_1:1;
( X = Y or X <> Y ) ;
then ( dist ((S . p9),Xa) = dist (y,a) or dist ((S . p9),Xa) = (dist (y,a)) + 0 ) by A7, A12, Def10;
then A13: (dist (y,a)) + (dist (a,z)) >= r + 0 by A6, A11, XREAL_1:7;
assume Y <> Z ; ::_thesis: contradiction
then A14: dist ((S . p9),(S . m)) >= r by A7, A9, A13, Def10;
A15: m in NAT by ORDINAL1:def_12;
m >= p by A5, A8, XXREAL_0:2;
hence contradiction by A4, A5, A15, A10, A14; ::_thesis: verum
end;
hence ex p being Point of M st S . m = [Y,p] by A9; ::_thesis: verum
end;
hence ( for Xa being Point of (WellSpace (a,X)) st Xa = [X,a] holds
for r being Real st r > 0 holds
ex n being Nat st
for m being Nat st m >= n holds
dist ((S . m),Xa) < r or ex n being Nat ex Y being set st
for m being Nat st m >= n holds
ex p being Point of M st S . m = [Y,p] ) ; ::_thesis: verum
end;
end;
end;
theorem Th42: :: COMPL_SP:42
for X being set
for M being non empty Reflexive symmetric triangle MetrStruct
for a being Point of M st M is complete holds
WellSpace (a,X) is complete
proof
let X be set ; ::_thesis: for M being non empty Reflexive symmetric triangle MetrStruct
for a being Point of M st M is complete holds
WellSpace (a,X) is complete
let M be non empty Reflexive symmetric triangle MetrStruct ; ::_thesis: for a being Point of M st M is complete holds
WellSpace (a,X) is complete
let a be Point of M; ::_thesis: ( M is complete implies WellSpace (a,X) is complete )
set W = WellSpace (a,X);
reconsider Xa = [X,a] as Point of (WellSpace (a,X)) by Th38;
assume A1: M is complete ; ::_thesis: WellSpace (a,X) is complete
let S9 be sequence of (WellSpace (a,X)); :: according to TBSP_1:def_5 ::_thesis: ( not S9 is Cauchy or S9 is convergent )
assume A2: S9 is Cauchy ; ::_thesis: S9 is convergent
defpred S1[ set , set ] means ex x being set st S9 . $1 = [x,$2];
A3: for x being set st x in NAT holds
ex y being set st
( y in the carrier of M & S1[x,y] )
proof
let x be set ; ::_thesis: ( x in NAT implies ex y being set st
( y in the carrier of M & S1[x,y] ) )
assume x in NAT ; ::_thesis: ex y being set st
( y in the carrier of M & S1[x,y] )
then reconsider i = x as Element of NAT ;
consider s1 being set , s2 being Point of M such that
A4: S9 . i = [s1,s2] and
( ( s1 in X & s2 <> a ) or ( s1 = X & s2 = a ) ) by Th38;
take s2 ; ::_thesis: ( s2 in the carrier of M & S1[x,s2] )
thus ( s2 in the carrier of M & S1[x,s2] ) by A4; ::_thesis: verum
end;
consider S being sequence of M such that
A5: for x being set st x in NAT holds
S1[x,S . x] from FUNCT_2:sch_1(A3);
S is Cauchy
proof
let r be Real; :: according to TBSP_1:def_4 ::_thesis: ( r <= 0 or ex b1 being Element of NAT st
for b2, b3 being Element of NAT holds
( not b1 <= b2 or not b1 <= b3 or not r <= dist ((S . b2),(S . b3)) ) )
assume r > 0 ; ::_thesis: ex b1 being Element of NAT st
for b2, b3 being Element of NAT holds
( not b1 <= b2 or not b1 <= b3 or not r <= dist ((S . b2),(S . b3)) )
then consider p being Element of NAT such that
A6: for n, m being Element of NAT st p <= n & p <= m holds
dist ((S9 . n),(S9 . m)) < r by A2, TBSP_1:def_4;
take p ; ::_thesis: for b1, b2 being Element of NAT holds
( not p <= b1 or not p <= b2 or not r <= dist ((S . b1),(S . b2)) )
let n, m be Element of NAT ; ::_thesis: ( not p <= n or not p <= m or not r <= dist ((S . n),(S . m)) )
assume that
A7: p <= n and
A8: p <= m ; ::_thesis: not r <= dist ((S . n),(S . m))
consider x being set such that
A9: S9 . n = [x,(S . n)] by A5;
consider y being set such that
A10: S9 . m = [y,(S . m)] by A5;
percases ( x = y or x <> y ) ;
suppose x = y ; ::_thesis: not r <= dist ((S . n),(S . m))
then dist ((S9 . n),(S9 . m)) = dist ((S . n),(S . m)) by A9, A10, Def10;
hence not r <= dist ((S . n),(S . m)) by A6, A7, A8; ::_thesis: verum
end;
supposeA11: x <> y ; ::_thesis: not r <= dist ((S . n),(S . m))
A12: dist ((S . n),(S . m)) <= (dist ((S . n),a)) + (dist (a,(S . m))) by METRIC_1:4;
A13: dist ((S9 . n),(S9 . m)) < r by A6, A7, A8;
dist ((S9 . n),(S9 . m)) = (dist ((S . n),a)) + (dist (a,(S . m))) by A9, A10, A11, Def10;
hence not r <= dist ((S . n),(S . m)) by A12, A13, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
then S is convergent by A1, TBSP_1:def_5;
then consider L being Element of M such that
A14: for r being Real st r > 0 holds
ex n being Element of NAT st
for m being Element of NAT st n <= m holds
dist ((S . m),L) < r by TBSP_1:def_2;
percases ( L = a or for Xa being Point of (WellSpace (a,X)) st Xa = [X,a] holds
for r being Real st r > 0 holds
ex n being Nat st
for m being Nat st m >= n holds
dist ((S9 . m),Xa) < r or ( a <> L & ex n being Nat ex Y being set st
for m being Nat st m >= n holds
ex p being Point of M st S9 . m = [Y,p] ) ) by A2, Th41;
supposeA15: L = a ; ::_thesis: S9 is convergent
take Xa ; :: according to TBSP_1:def_2 ::_thesis: for b1 being Element of REAL holds
( b1 <= 0 or ex b2 being Element of NAT st
for b3 being Element of NAT holds
( not b2 <= b3 or not b1 <= dist ((S9 . b3),Xa) ) )
A16: dist (a,a) = 0 by METRIC_1:1;
let r be Real; ::_thesis: ( r <= 0 or ex b1 being Element of NAT st
for b2 being Element of NAT holds
( not b1 <= b2 or not r <= dist ((S9 . b2),Xa) ) )
assume r > 0 ; ::_thesis: ex b1 being Element of NAT st
for b2 being Element of NAT holds
( not b1 <= b2 or not r <= dist ((S9 . b2),Xa) )
then consider n being Element of NAT such that
A17: for m being Element of NAT st n <= m holds
dist ((S . m),L) < r by A14;
take n ; ::_thesis: for b1 being Element of NAT holds
( not n <= b1 or not r <= dist ((S9 . b1),Xa) )
let m be Element of NAT ; ::_thesis: ( not n <= m or not r <= dist ((S9 . m),Xa) )
assume A18: m >= n ; ::_thesis: not r <= dist ((S9 . m),Xa)
consider x being set such that
A19: S9 . m = [x,(S . m)] by A5;
( x = X or x <> X ) ;
then ( dist ((S9 . m),Xa) = dist ((S . m),L) or dist ((S9 . m),Xa) = (dist ((S . m),L)) + 0 ) by A15, A19, A16, Def10;
hence dist ((S9 . m),Xa) < r by A17, A18; ::_thesis: verum
end;
supposeA20: for Xa being Point of (WellSpace (a,X)) st Xa = [X,a] holds
for r being Real st r > 0 holds
ex n being Nat st
for m being Nat st m >= n holds
dist ((S9 . m),Xa) < r ; ::_thesis: S9 is convergent
take Xa ; :: according to TBSP_1:def_2 ::_thesis: for b1 being Element of REAL holds
( b1 <= 0 or ex b2 being Element of NAT st
for b3 being Element of NAT holds
( not b2 <= b3 or not b1 <= dist ((S9 . b3),Xa) ) )
let r be Real; ::_thesis: ( r <= 0 or ex b1 being Element of NAT st
for b2 being Element of NAT holds
( not b1 <= b2 or not r <= dist ((S9 . b2),Xa) ) )
assume r > 0 ; ::_thesis: ex b1 being Element of NAT st
for b2 being Element of NAT holds
( not b1 <= b2 or not r <= dist ((S9 . b2),Xa) )
then consider n being Nat such that
A21: for m being Nat st m >= n holds
dist ((S9 . m),Xa) < r by A20;
reconsider n = n as Element of NAT by ORDINAL1:def_12;
take n ; ::_thesis: for b1 being Element of NAT holds
( not n <= b1 or not r <= dist ((S9 . b1),Xa) )
let m be Element of NAT ; ::_thesis: ( not n <= m or not r <= dist ((S9 . m),Xa) )
assume m >= n ; ::_thesis: not r <= dist ((S9 . m),Xa)
hence dist ((S9 . m),Xa) < r by A21; ::_thesis: verum
end;
supposeA22: ( a <> L & ex n being Nat ex Y being set st
for m being Nat st m >= n holds
ex p being Point of M st S9 . m = [Y,p] ) ; ::_thesis: S9 is convergent
then consider n being Nat, Y being set such that
A23: for m being Nat st m >= n holds
ex p being Point of M st S9 . m = [Y,p] ;
A24: ex s3 being Point of M st S9 . n = [Y,s3] by A23;
A25: ex s1 being set ex s2 being Point of M st
( S9 . n = [s1,s2] & ( ( s1 in X & s2 <> a ) or ( s1 = X & s2 = a ) ) ) by Th38;
percases ( Y in X or Y = X ) by A25, A24, XTUPLE_0:1;
suppose Y in X ; ::_thesis: S9 is convergent
then reconsider YL = [Y,L] as Point of (WellSpace (a,X)) by A22, Th38;
take YL ; :: according to TBSP_1:def_2 ::_thesis: for b1 being Element of REAL holds
( b1 <= 0 or ex b2 being Element of NAT st
for b3 being Element of NAT holds
( not b2 <= b3 or not b1 <= dist ((S9 . b3),YL) ) )
let r be Real; ::_thesis: ( r <= 0 or ex b1 being Element of NAT st
for b2 being Element of NAT holds
( not b1 <= b2 or not r <= dist ((S9 . b2),YL) ) )
assume r > 0 ; ::_thesis: ex b1 being Element of NAT st
for b2 being Element of NAT holds
( not b1 <= b2 or not r <= dist ((S9 . b2),YL) )
then consider p being Element of NAT such that
A26: for m being Element of NAT st p <= m holds
dist ((S . m),L) < r by A14;
reconsider mm = max (p,n) as Element of NAT by ORDINAL1:def_12;
take mm ; ::_thesis: for b1 being Element of NAT holds
( not mm <= b1 or not r <= dist ((S9 . b1),YL) )
let m be Element of NAT ; ::_thesis: ( not mm <= m or not r <= dist ((S9 . m),YL) )
assume A27: m >= mm ; ::_thesis: not r <= dist ((S9 . m),YL)
consider x being set such that
A28: S9 . m = [x,(S . m)] by A5;
mm >= n by XXREAL_0:25;
then ex pm being Point of M st S9 . m = [Y,pm] by A23, A27, XXREAL_0:2;
then x = Y by A28, XTUPLE_0:1;
then A29: dist ((S9 . m),YL) = dist ((S . m),L) by A28, Def10;
mm >= p by XXREAL_0:25;
then m >= p by A27, XXREAL_0:2;
hence dist ((S9 . m),YL) < r by A26, A29; ::_thesis: verum
end;
supposeA30: Y = X ; ::_thesis: S9 is convergent
reconsider n = n as Element of NAT by ORDINAL1:def_12;
take Xa ; :: according to TBSP_1:def_2 ::_thesis: for b1 being Element of REAL holds
( b1 <= 0 or ex b2 being Element of NAT st
for b3 being Element of NAT holds
( not b2 <= b3 or not b1 <= dist ((S9 . b3),Xa) ) )
let r be Real; ::_thesis: ( r <= 0 or ex b1 being Element of NAT st
for b2 being Element of NAT holds
( not b1 <= b2 or not r <= dist ((S9 . b2),Xa) ) )
assume A31: r > 0 ; ::_thesis: ex b1 being Element of NAT st
for b2 being Element of NAT holds
( not b1 <= b2 or not r <= dist ((S9 . b2),Xa) )
take n ; ::_thesis: for b1 being Element of NAT holds
( not n <= b1 or not r <= dist ((S9 . b1),Xa) )
let m be Element of NAT ; ::_thesis: ( not n <= m or not r <= dist ((S9 . m),Xa) )
assume m >= n ; ::_thesis: not r <= dist ((S9 . m),Xa)
then A32: ex t3 being Point of M st S9 . m = [Y,t3] by A23;
consider t1 being set , t2 being Point of M such that
A33: S9 . m = [t1,t2] and
A34: ( ( t1 in X & t2 <> a ) or ( t1 = X & t2 = a ) ) by Th38;
Y = t1 by A33, A32, XTUPLE_0:1;
hence dist ((S9 . m),Xa) < r by A30, A31, A33, A34, METRIC_1:1; ::_thesis: verum
end;
end;
end;
end;
end;
theorem Th43: :: COMPL_SP:43
for M being non empty Reflexive symmetric triangle MetrStruct st M is complete holds
for a being Point of M st ex b being Point of M st dist (a,b) <> 0 holds
for X being infinite set holds
( WellSpace (a,X) is complete & ex S being non-empty pointwise_bounded SetSequence of (WellSpace (a,X)) st
( S is closed & S is V172() & meet S is empty ) )
proof
let M be non empty Reflexive symmetric triangle MetrStruct ; ::_thesis: ( M is complete implies for a being Point of M st ex b being Point of M st dist (a,b) <> 0 holds
for X being infinite set holds
( WellSpace (a,X) is complete & ex S being non-empty pointwise_bounded SetSequence of (WellSpace (a,X)) st
( S is closed & S is V172() & meet S is empty ) ) )
assume A1: M is complete ; ::_thesis: for a being Point of M st ex b being Point of M st dist (a,b) <> 0 holds
for X being infinite set holds
( WellSpace (a,X) is complete & ex S being non-empty pointwise_bounded SetSequence of (WellSpace (a,X)) st
( S is closed & S is V172() & meet S is empty ) )
let a be Point of M; ::_thesis: ( ex b being Point of M st dist (a,b) <> 0 implies for X being infinite set holds
( WellSpace (a,X) is complete & ex S being non-empty pointwise_bounded SetSequence of (WellSpace (a,X)) st
( S is closed & S is V172() & meet S is empty ) ) )
assume ex b being Point of M st dist (a,b) <> 0 ; ::_thesis: for X being infinite set holds
( WellSpace (a,X) is complete & ex S being non-empty pointwise_bounded SetSequence of (WellSpace (a,X)) st
( S is closed & S is V172() & meet S is empty ) )
then consider b being Point of M such that
A2: dist (a,b) <> 0 ;
let X be infinite set ; ::_thesis: ( WellSpace (a,X) is complete & ex S being non-empty pointwise_bounded SetSequence of (WellSpace (a,X)) st
( S is closed & S is V172() & meet S is empty ) )
set W = WellSpace (a,X);
thus WellSpace (a,X) is complete by A1, Th42; ::_thesis: ex S being non-empty pointwise_bounded SetSequence of (WellSpace (a,X)) st
( S is closed & S is V172() & meet S is empty )
set TW = TopSpaceMetr (WellSpace (a,X));
consider f being Function of NAT,X such that
A3: f is one-to-one by DICKSON:3;
defpred S1[ set , set ] means $2 = [(f . $1),b];
A4: b <> a by A2, METRIC_1:1;
A5: for x being set st x in NAT holds
ex y being set st
( y in the carrier of (TopSpaceMetr (WellSpace (a,X))) & S1[x,y] )
proof
let x be set ; ::_thesis: ( x in NAT implies ex y being set st
( y in the carrier of (TopSpaceMetr (WellSpace (a,X))) & S1[x,y] ) )
assume x in NAT ; ::_thesis: ex y being set st
( y in the carrier of (TopSpaceMetr (WellSpace (a,X))) & S1[x,y] )
then x in dom f by FUNCT_2:def_1;
then A6: f . x in rng f by FUNCT_1:def_3;
take [(f . x),b] ; ::_thesis: ( [(f . x),b] in the carrier of (TopSpaceMetr (WellSpace (a,X))) & S1[x,[(f . x),b]] )
thus ( [(f . x),b] in the carrier of (TopSpaceMetr (WellSpace (a,X))) & S1[x,[(f . x),b]] ) by A4, A6, Th38; ::_thesis: verum
end;
consider s being Function of NAT, the carrier of (TopSpaceMetr (WellSpace (a,X))) such that
A7: for x being set st x in NAT holds
S1[x,s . x] from FUNCT_2:sch_1(A5);
deffunc H1( set ) -> set = {(s . $1)};
A8: for x being set st x in NAT holds
H1(x) in bool the carrier of (TopSpaceMetr (WellSpace (a,X)))
proof
A9: dom s = NAT by FUNCT_2:def_1;
let x be set ; ::_thesis: ( x in NAT implies H1(x) in bool the carrier of (TopSpaceMetr (WellSpace (a,X))) )
assume x in NAT ; ::_thesis: H1(x) in bool the carrier of (TopSpaceMetr (WellSpace (a,X)))
then s . x in rng s by A9, FUNCT_1:def_3;
then H1(x) is Subset of (WellSpace (a,X)) by SUBSET_1:33;
hence H1(x) in bool the carrier of (TopSpaceMetr (WellSpace (a,X))) ; ::_thesis: verum
end;
consider S being SetSequence of (TopSpaceMetr (WellSpace (a,X))) such that
A10: for x being set st x in NAT holds
S . x = H1(x) from FUNCT_2:sch_2(A8);
A11: now__::_thesis:_for_x1,_x2_being_set_st_x1_in_NAT_&_x2_in_NAT_&_S_._x1_=_S_._x2_holds_
x1_=_x2
let x1, x2 be set ; ::_thesis: ( x1 in NAT & x2 in NAT & S . x1 = S . x2 implies x1 = x2 )
assume that
A12: x1 in NAT and
A13: x2 in NAT and
A14: S . x1 = S . x2 ; ::_thesis: x1 = x2
A15: S . x2 = {(s . x2)} by A10, A13;
A16: s . x1 = [(f . x1),b] by A7, A12;
A17: s . x1 in {(s . x1)} by TARSKI:def_1;
A18: s . x2 = [(f . x2),b] by A7, A13;
S . x1 = {(s . x1)} by A10, A12;
then s . x1 = s . x2 by A14, A15, A17, TARSKI:def_1;
then f . x1 = f . x2 by A16, A18, XTUPLE_0:1;
hence x1 = x2 by A3, A12, A13, FUNCT_2:19; ::_thesis: verum
end;
reconsider rngs = rng s as Subset of (TopSpaceMetr (WellSpace (a,X))) ;
set F = { {x} where x is Element of (TopSpaceMetr (WellSpace (a,X))) : x in rngs } ;
reconsider F = { {x} where x is Element of (TopSpaceMetr (WellSpace (a,X))) : x in rngs } as Subset-Family of (TopSpaceMetr (WellSpace (a,X))) by RELSET_2:16;
dist (a,b) > 0 by A2, METRIC_1:5;
then A19: 2 * (dist (a,b)) > 0 by XREAL_1:129;
A20: rng S c= F
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng S or x in F )
assume x in rng S ; ::_thesis: x in F
then consider y being set such that
A21: y in dom S and
A22: S . y = x by FUNCT_1:def_3;
dom s = NAT by FUNCT_2:def_1;
then A23: s . y in rngs by A21, FUNCT_1:def_3;
x = {(s . y)} by A10, A21, A22;
hence x in F by A23; ::_thesis: verum
end;
now__::_thesis:_for_x_being_set_st_x_in_dom_S_holds_
not_S_._x_is_empty
let x be set ; ::_thesis: ( x in dom S implies not S . x is empty )
assume x in dom S ; ::_thesis: not S . x is empty
then S . x = {(s . x)} by A10;
hence not S . x is empty ; ::_thesis: verum
end;
then S is non-empty by FUNCT_1:def_9;
then consider R being non-empty closed SetSequence of (TopSpaceMetr (WellSpace (a,X))) such that
A24: R is V172() and
A25: ( F is locally_finite & S is one-to-one implies meet R = {} ) and
A26: for i being Nat ex Si being Subset-Family of (TopSpaceMetr (WellSpace (a,X))) st
( R . i = Cl (union Si) & Si = { (S . j) where j is Element of NAT : j >= i } ) by A20, Th23;
reconsider R9 = R as non-empty SetSequence of (WellSpace (a,X)) ;
A27: now__::_thesis:_for_x,_y_being_Point_of_(WellSpace_(a,X))_st_x_in_rngs_&_y_in_rngs_&_x_<>_y_holds_
dist_(x,y)_=_2_*_(dist_(a,b))
let x, y be Point of (WellSpace (a,X)); ::_thesis: ( x in rngs & y in rngs & x <> y implies dist (x,y) = 2 * (dist (a,b)) )
assume that
A28: x in rngs and
A29: y in rngs and
A30: x <> y ; ::_thesis: dist (x,y) = 2 * (dist (a,b))
consider y1 being set such that
A31: y1 in dom s and
A32: s . y1 = y by A29, FUNCT_1:def_3;
A33: y = [(f . y1),b] by A7, A31, A32;
consider x1 being set such that
A34: x1 in dom s and
A35: s . x1 = x by A28, FUNCT_1:def_3;
x = [(f . x1),b] by A7, A34, A35;
then (well_dist (a,X)) . (x,y) = (dist (b,a)) + (dist (a,b)) by A30, A33, Def10;
hence dist (x,y) = 2 * (dist (a,b)) ; ::_thesis: verum
end;
now__::_thesis:_for_i_being_Nat_holds_R9_._i_is_bounded
let i be Nat; ::_thesis: R9 . i is bounded
consider Si being Subset-Family of (TopSpaceMetr (WellSpace (a,X))) such that
A36: R . i = Cl (union Si) and
A37: Si = { (S . j) where j is Element of NAT : j >= i } by A26;
reconsider SI = union Si as Subset of (WellSpace (a,X)) ;
now__::_thesis:_for_x,_y_being_Point_of_(WellSpace_(a,X))_st_x_in_SI_&_y_in_SI_holds_
dist_(x,y)_<=_2_*_(dist_(a,b))
let x, y be Point of (WellSpace (a,X)); ::_thesis: ( x in SI & y in SI implies dist (x,y) <= 2 * (dist (a,b)) )
assume that
A38: x in SI and
A39: y in SI ; ::_thesis: dist (x,y) <= 2 * (dist (a,b))
consider xS being set such that
A40: x in xS and
A41: xS in Si by A38, TARSKI:def_4;
consider j1 being Element of NAT such that
A42: xS = S . j1 and
j1 >= i by A37, A41;
A43: S . j1 = {(s . j1)} by A10;
A44: dom s = NAT by FUNCT_2:def_1;
then s . j1 in rngs by FUNCT_1:def_3;
then A45: x in rngs by A40, A42, A43, TARSKI:def_1;
consider yS being set such that
A46: y in yS and
A47: yS in Si by A39, TARSKI:def_4;
consider j2 being Element of NAT such that
A48: yS = S . j2 and
j2 >= i by A37, A47;
A49: S . j2 = {(s . j2)} by A10;
s . j2 in rngs by A44, FUNCT_1:def_3;
then A50: y in rngs by A46, A48, A49, TARSKI:def_1;
( x = y or x <> y ) ;
hence dist (x,y) <= 2 * (dist (a,b)) by A19, A27, A45, A50, METRIC_1:1; ::_thesis: verum
end;
then SI is bounded by A19, TBSP_1:def_7;
hence R9 . i is bounded by A36, Th8; ::_thesis: verum
end;
then reconsider R9 = R9 as non-empty pointwise_bounded SetSequence of (WellSpace (a,X)) by Def1;
take R9 ; ::_thesis: ( R9 is closed & R9 is V172() & meet R9 is empty )
thus ( R9 is closed & R9 is V172() ) by A24, Th7; ::_thesis: meet R9 is empty
for x, y being Point of (WellSpace (a,X)) st x <> y & x in rngs & y in rngs holds
dist (x,y) >= 2 * (dist (a,b)) by A27;
hence meet R9 is empty by A25, A19, A11, Lm7, FUNCT_2:19; ::_thesis: verum
end;
theorem :: COMPL_SP:44
ex M being non empty MetrSpace st
( M is complete & ex S being non-empty pointwise_bounded SetSequence of M st
( S is closed & S is V172() & meet S is empty ) )
proof
reconsider D = DiscreteSpace 2 as non empty MetrSpace ;
reconsider a = 0 , b = 1 as Point of D by NAT_1:44;
TopSpaceMetr D is compact by COMPTS_1:18;
then A1: D is complete by TBSP_1:8;
A2: 1 = dist (a,b) by METRIC_1:def_10;
then A3: ex S being non-empty pointwise_bounded SetSequence of (WellSpace (a,NAT)) st
( S is closed & S is V172() & meet S is empty ) by A1, Th43;
WellSpace (a,NAT) is complete by A2, A1, Th43;
hence ex M being non empty MetrSpace st
( M is complete & ex S being non-empty pointwise_bounded SetSequence of M st
( S is closed & S is V172() & meet S is empty ) ) by A3; ::_thesis: verum
end;