:: 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;