:: TSEP_1 semantic presentation
begin
Lm1: for A being set
for B, C, D being Subset of A st B \ C = {} holds
B misses D \ C
proof
let A be set ; ::_thesis: for B, C, D being Subset of A st B \ C = {} holds
B misses D \ C
let B, C, D be Subset of A; ::_thesis: ( B \ C = {} implies B misses D \ C )
assume B \ C = {} ; ::_thesis: B misses D \ C
then A1: B c= C by XBOOLE_1:37;
C misses D \ C by XBOOLE_1:79;
hence B misses D \ C by A1, XBOOLE_1:63; ::_thesis: verum
end;
Lm2: for A, B, C being set holds (A /\ B) \ C = (A \ C) /\ (B \ C)
proof
let A, B, C be set ; ::_thesis: (A /\ B) \ C = (A \ C) /\ (B \ C)
A1: A \ C misses C by XBOOLE_1:79;
thus (A /\ B) \ C = (A \ C) /\ B by XBOOLE_1:49
.= (A \ C) \ ((A \ C) \ B) by XBOOLE_1:48
.= (A \ C) \ (A \ (C \/ B)) by XBOOLE_1:41
.= ((A \ C) \ A) \/ ((A \ C) /\ (C \/ B)) by XBOOLE_1:52
.= {} \/ ((A \ C) /\ (C \/ B)) by XBOOLE_1:37
.= (A \ C) /\ ((B \ C) \/ C) by XBOOLE_1:39
.= ((A \ C) /\ (B \ C)) \/ ((A \ C) /\ C) by XBOOLE_1:23
.= ((A \ C) /\ (B \ C)) \/ {} by A1, XBOOLE_0:def_7
.= (A \ C) /\ (B \ C) ; ::_thesis: verum
end;
theorem Th1: :: TSEP_1:1
for X being TopStruct
for X0 being SubSpace of X holds the carrier of X0 is Subset of X
proof
let X be TopStruct ; ::_thesis: for X0 being SubSpace of X holds the carrier of X0 is Subset of X
let X0 be SubSpace of X; ::_thesis: the carrier of X0 is Subset of X
reconsider A = [#] X0 as Subset of ([#] X) by PRE_TOPC:def_4;
A c= [#] X ;
hence the carrier of X0 is Subset of X ; ::_thesis: verum
end;
theorem Th2: :: TSEP_1:2
for X being TopStruct holds X is SubSpace of X
proof
let X be TopStruct ; ::_thesis: X is SubSpace of X
thus [#] X c= [#] X ; :: according to PRE_TOPC:def_4 ::_thesis: for b1 being Element of K10( the carrier of X) holds
( ( not b1 in the topology of X or ex b2 being Element of K10( the carrier of X) st
( b2 in the topology of X & b1 = b2 /\ ([#] X) ) ) & ( for b2 being Element of K10( the carrier of X) holds
( not b2 in the topology of X or not b1 = b2 /\ ([#] X) ) or b1 in the topology of X ) )
thus for P being Subset of X holds
( P in the topology of X iff ex Q being Subset of X st
( Q in the topology of X & P = Q /\ ([#] X) ) ) ::_thesis: verum
proof
let P be Subset of X; ::_thesis: ( P in the topology of X iff ex Q being Subset of X st
( Q in the topology of X & P = Q /\ ([#] X) ) )
thus ( P in the topology of X implies ex Q being Subset of X st
( Q in the topology of X & P = Q /\ ([#] X) ) ) ::_thesis: ( ex Q being Subset of X st
( Q in the topology of X & P = Q /\ ([#] X) ) implies P in the topology of X )
proof
assume A1: P in the topology of X ; ::_thesis: ex Q being Subset of X st
( Q in the topology of X & P = Q /\ ([#] X) )
take P ; ::_thesis: ( P in the topology of X & P = P /\ ([#] X) )
thus ( P in the topology of X & P = P /\ ([#] X) ) by A1, XBOOLE_1:28; ::_thesis: verum
end;
thus ( ex Q being Subset of X st
( Q in the topology of X & P = Q /\ ([#] X) ) implies P in the topology of X ) by XBOOLE_1:28; ::_thesis: verum
end;
end;
theorem :: TSEP_1:3
for X being strict TopStruct holds X | ([#] X) = X
proof
let X be strict TopStruct ; ::_thesis: X | ([#] X) = X
reconsider X0 = X as SubSpace of X by Th2;
reconsider P = [#] X0 as Subset of X ;
X | P = X0 by PRE_TOPC:def_5;
hence X | ([#] X) = X ; ::_thesis: verum
end;
theorem Th4: :: TSEP_1:4
for X being TopSpace
for X1, X2 being SubSpace of X holds
( the carrier of X1 c= the carrier of X2 iff X1 is SubSpace of X2 )
proof
let X be TopSpace; ::_thesis: for X1, X2 being SubSpace of X holds
( the carrier of X1 c= the carrier of X2 iff X1 is SubSpace of X2 )
let X1, X2 be SubSpace of X; ::_thesis: ( the carrier of X1 c= the carrier of X2 iff X1 is SubSpace of X2 )
set A1 = the carrier of X1;
set A2 = the carrier of X2;
A1: the carrier of X1 = [#] X1 ;
A2: the carrier of X2 = [#] X2 ;
thus ( the carrier of X1 c= the carrier of X2 implies X1 is SubSpace of X2 ) ::_thesis: ( X1 is SubSpace of X2 implies the carrier of X1 c= the carrier of X2 )
proof
assume A3: the carrier of X1 c= the carrier of X2 ; ::_thesis: X1 is SubSpace of X2
for P being Subset of X1 holds
( P in the topology of X1 iff ex Q being Subset of X2 st
( Q in the topology of X2 & P = Q /\ the carrier of X1 ) )
proof
let P be Subset of X1; ::_thesis: ( P in the topology of X1 iff ex Q being Subset of X2 st
( Q in the topology of X2 & P = Q /\ the carrier of X1 ) )
thus ( P in the topology of X1 implies ex Q being Subset of X2 st
( Q in the topology of X2 & P = Q /\ the carrier of X1 ) ) ::_thesis: ( ex Q being Subset of X2 st
( Q in the topology of X2 & P = Q /\ the carrier of X1 ) implies P in the topology of X1 )
proof
assume P in the topology of X1 ; ::_thesis: ex Q being Subset of X2 st
( Q in the topology of X2 & P = Q /\ the carrier of X1 )
then consider R being Subset of X such that
A4: R in the topology of X and
A5: P = R /\ the carrier of X1 by A1, PRE_TOPC:def_4;
reconsider Q = R /\ the carrier of X2 as Subset of X2 by XBOOLE_1:17;
take Q ; ::_thesis: ( Q in the topology of X2 & P = Q /\ the carrier of X1 )
thus Q in the topology of X2 by A2, A4, PRE_TOPC:def_4; ::_thesis: P = Q /\ the carrier of X1
Q /\ the carrier of X1 = R /\ ( the carrier of X2 /\ the carrier of X1) by XBOOLE_1:16
.= R /\ the carrier of X1 by A3, XBOOLE_1:28 ;
hence P = Q /\ the carrier of X1 by A5; ::_thesis: verum
end;
given Q being Subset of X2 such that A6: Q in the topology of X2 and
A7: P = Q /\ the carrier of X1 ; ::_thesis: P in the topology of X1
consider R being Subset of X such that
A8: R in the topology of X and
A9: Q = R /\ ([#] X2) by A6, PRE_TOPC:def_4;
P = R /\ ( the carrier of X2 /\ the carrier of X1) by A7, A9, XBOOLE_1:16
.= R /\ the carrier of X1 by A3, XBOOLE_1:28 ;
hence P in the topology of X1 by A1, A8, PRE_TOPC:def_4; ::_thesis: verum
end;
hence X1 is SubSpace of X2 by A1, A2, A3, PRE_TOPC:def_4; ::_thesis: verum
end;
thus ( X1 is SubSpace of X2 implies the carrier of X1 c= the carrier of X2 ) by A1, A2, PRE_TOPC:def_4; ::_thesis: verum
end;
Lm3: for X being TopStruct
for X0 being SubSpace of X holds TopStruct(# the carrier of X0, the topology of X0 #) is strict SubSpace of X
proof
let X be TopStruct ; ::_thesis: for X0 being SubSpace of X holds TopStruct(# the carrier of X0, the topology of X0 #) is strict SubSpace of X
let X0 be SubSpace of X; ::_thesis: TopStruct(# the carrier of X0, the topology of X0 #) is strict SubSpace of X
set S = TopStruct(# the carrier of X0, the topology of X0 #);
TopStruct(# the carrier of X0, the topology of X0 #) is SubSpace of X
proof
A1: [#] X0 = the carrier of X0 ;
hence [#] TopStruct(# the carrier of X0, the topology of X0 #) c= [#] X by PRE_TOPC:def_4; :: according to PRE_TOPC:def_4 ::_thesis: for b1 being Element of K10( the carrier of TopStruct(# the carrier of X0, the topology of X0 #)) holds
( ( not b1 in the topology of TopStruct(# the carrier of X0, the topology of X0 #) or ex b2 being Element of K10( the carrier of X) st
( b2 in the topology of X & b1 = b2 /\ ([#] TopStruct(# the carrier of X0, the topology of X0 #)) ) ) & ( for b2 being Element of K10( the carrier of X) holds
( not b2 in the topology of X or not b1 = b2 /\ ([#] TopStruct(# the carrier of X0, the topology of X0 #)) ) or b1 in the topology of TopStruct(# the carrier of X0, the topology of X0 #) ) )
let P be Subset of TopStruct(# the carrier of X0, the topology of X0 #); ::_thesis: ( ( not P in the topology of TopStruct(# the carrier of X0, the topology of X0 #) or ex b1 being Element of K10( the carrier of X) st
( b1 in the topology of X & P = b1 /\ ([#] TopStruct(# the carrier of X0, the topology of X0 #)) ) ) & ( for b1 being Element of K10( the carrier of X) holds
( not b1 in the topology of X or not P = b1 /\ ([#] TopStruct(# the carrier of X0, the topology of X0 #)) ) or P in the topology of TopStruct(# the carrier of X0, the topology of X0 #) ) )
thus ( P in the topology of TopStruct(# the carrier of X0, the topology of X0 #) implies ex Q being Subset of X st
( Q in the topology of X & P = Q /\ ([#] TopStruct(# the carrier of X0, the topology of X0 #)) ) ) by A1, PRE_TOPC:def_4; ::_thesis: ( for b1 being Element of K10( the carrier of X) holds
( not b1 in the topology of X or not P = b1 /\ ([#] TopStruct(# the carrier of X0, the topology of X0 #)) ) or P in the topology of TopStruct(# the carrier of X0, the topology of X0 #) )
given Q being Subset of X such that A2: ( Q in the topology of X & P = Q /\ ([#] TopStruct(# the carrier of X0, the topology of X0 #)) ) ; ::_thesis: P in the topology of TopStruct(# the carrier of X0, the topology of X0 #)
thus P in the topology of TopStruct(# the carrier of X0, the topology of X0 #) by A1, A2, PRE_TOPC:def_4; ::_thesis: verum
end;
hence TopStruct(# the carrier of X0, the topology of X0 #) is strict SubSpace of X ; ::_thesis: verum
end;
theorem Th5: :: TSEP_1:5
for X being TopStruct
for X1, X2 being SubSpace of X st the carrier of X1 = the carrier of X2 holds
TopStruct(# the carrier of X1, the topology of X1 #) = TopStruct(# the carrier of X2, the topology of X2 #)
proof
let X be TopStruct ; ::_thesis: for X1, X2 being SubSpace of X st the carrier of X1 = the carrier of X2 holds
TopStruct(# the carrier of X1, the topology of X1 #) = TopStruct(# the carrier of X2, the topology of X2 #)
let X1, X2 be SubSpace of X; ::_thesis: ( the carrier of X1 = the carrier of X2 implies TopStruct(# the carrier of X1, the topology of X1 #) = TopStruct(# the carrier of X2, the topology of X2 #) )
reconsider S1 = TopStruct(# the carrier of X1, the topology of X1 #), S2 = TopStruct(# the carrier of X2, the topology of X2 #) as strict SubSpace of X by Lm3;
set A1 = the carrier of X1;
set A2 = the carrier of X2;
assume A1: the carrier of X1 = the carrier of X2 ; ::_thesis: TopStruct(# the carrier of X1, the topology of X1 #) = TopStruct(# the carrier of X2, the topology of X2 #)
A2: the carrier of X1 = [#] S1 ;
A3: the carrier of X2 = [#] S2 ;
reconsider A1 = the carrier of X1 as Subset of X by BORSUK_1:1;
S1 = X | A1 by A2, PRE_TOPC:def_5;
hence TopStruct(# the carrier of X1, the topology of X1 #) = TopStruct(# the carrier of X2, the topology of X2 #) by A1, A3, PRE_TOPC:def_5; ::_thesis: verum
end;
theorem :: TSEP_1:6
for X being TopSpace
for X1, X2 being SubSpace of X st X1 is SubSpace of X2 & X2 is SubSpace of X1 holds
TopStruct(# the carrier of X1, the topology of X1 #) = TopStruct(# the carrier of X2, the topology of X2 #)
proof
let X be TopSpace; ::_thesis: for X1, X2 being SubSpace of X st X1 is SubSpace of X2 & X2 is SubSpace of X1 holds
TopStruct(# the carrier of X1, the topology of X1 #) = TopStruct(# the carrier of X2, the topology of X2 #)
let X1, X2 be SubSpace of X; ::_thesis: ( X1 is SubSpace of X2 & X2 is SubSpace of X1 implies TopStruct(# the carrier of X1, the topology of X1 #) = TopStruct(# the carrier of X2, the topology of X2 #) )
set A1 = the carrier of X1;
set A2 = the carrier of X2;
assume ( X1 is SubSpace of X2 & X2 is SubSpace of X1 ) ; ::_thesis: TopStruct(# the carrier of X1, the topology of X1 #) = TopStruct(# the carrier of X2, the topology of X2 #)
then ( the carrier of X1 c= the carrier of X2 & the carrier of X2 c= the carrier of X1 ) by Th4;
then the carrier of X1 = the carrier of X2 by XBOOLE_0:def_10;
hence TopStruct(# the carrier of X1, the topology of X1 #) = TopStruct(# the carrier of X2, the topology of X2 #) by Th5; ::_thesis: verum
end;
theorem Th7: :: TSEP_1:7
for X being TopSpace
for X1 being SubSpace of X
for X2 being SubSpace of X1 holds X2 is SubSpace of X
proof
let X be TopSpace; ::_thesis: for X1 being SubSpace of X
for X2 being SubSpace of X1 holds X2 is SubSpace of X
let X1 be SubSpace of X; ::_thesis: for X2 being SubSpace of X1 holds X2 is SubSpace of X
let X2 be SubSpace of X1; ::_thesis: X2 is SubSpace of X
A1: [#] X2 c= [#] X1 by PRE_TOPC:def_4;
[#] X1 c= [#] X by PRE_TOPC:def_4;
hence [#] X2 c= [#] X by A1, XBOOLE_1:1; :: according to PRE_TOPC:def_4 ::_thesis: for b1 being Element of K10( the carrier of X2) holds
( ( not b1 in the topology of X2 or ex b2 being Element of K10( the carrier of X) st
( b2 in the topology of X & b1 = b2 /\ ([#] X2) ) ) & ( for b2 being Element of K10( the carrier of X) holds
( not b2 in the topology of X or not b1 = b2 /\ ([#] X2) ) or b1 in the topology of X2 ) )
thus for P being Subset of X2 holds
( P in the topology of X2 iff ex Q being Subset of X st
( Q in the topology of X & P = Q /\ ([#] X2) ) ) ::_thesis: verum
proof
let P be Subset of X2; ::_thesis: ( P in the topology of X2 iff ex Q being Subset of X st
( Q in the topology of X & P = Q /\ ([#] X2) ) )
reconsider P1 = P as Subset of X2 ;
thus ( P in the topology of X2 implies ex Q being Subset of X st
( Q in the topology of X & P = Q /\ ([#] X2) ) ) ::_thesis: ( ex Q being Subset of X st
( Q in the topology of X & P = Q /\ ([#] X2) ) implies P in the topology of X2 )
proof
assume P in the topology of X2 ; ::_thesis: ex Q being Subset of X st
( Q in the topology of X & P = Q /\ ([#] X2) )
then consider R being Subset of X1 such that
A2: R in the topology of X1 and
A3: P = R /\ ([#] X2) by PRE_TOPC:def_4;
consider Q being Subset of X such that
A4: Q in the topology of X and
A5: R = Q /\ ([#] X1) by A2, PRE_TOPC:def_4;
Q /\ ([#] X2) = Q /\ (([#] X1) /\ ([#] X2)) by A1, XBOOLE_1:28
.= P by A3, A5, XBOOLE_1:16 ;
hence ex Q being Subset of X st
( Q in the topology of X & P = Q /\ ([#] X2) ) by A4; ::_thesis: verum
end;
given Q being Subset of X such that A6: Q in the topology of X and
A7: P = Q /\ ([#] X2) ; ::_thesis: P in the topology of X2
reconsider R = Q /\ ([#] X1) as Subset of X1 ;
reconsider Q1 = Q as Subset of X ;
Q1 is open by A6, PRE_TOPC:def_2;
then A8: R is open by TOPS_2:24;
R /\ ([#] X2) = Q /\ (([#] X1) /\ ([#] X2)) by XBOOLE_1:16
.= P by A1, A7, XBOOLE_1:28 ;
then P1 is open by A8, TOPS_2:24;
hence P in the topology of X2 by PRE_TOPC:def_2; ::_thesis: verum
end;
end;
theorem Th8: :: TSEP_1:8
for X being TopSpace
for X0 being SubSpace of X
for C, A being Subset of X
for B being Subset of X0 st C is closed & C c= the carrier of X0 & A c= C & A = B holds
( B is closed iff A is closed )
proof
let X be TopSpace; ::_thesis: for X0 being SubSpace of X
for C, A being Subset of X
for B being Subset of X0 st C is closed & C c= the carrier of X0 & A c= C & A = B holds
( B is closed iff A is closed )
let X0 be SubSpace of X; ::_thesis: for C, A being Subset of X
for B being Subset of X0 st C is closed & C c= the carrier of X0 & A c= C & A = B holds
( B is closed iff A is closed )
let C, A be Subset of X; ::_thesis: for B being Subset of X0 st C is closed & C c= the carrier of X0 & A c= C & A = B holds
( B is closed iff A is closed )
let B be Subset of X0; ::_thesis: ( C is closed & C c= the carrier of X0 & A c= C & A = B implies ( B is closed iff A is closed ) )
assume that
A1: C is closed and
A2: C c= the carrier of X0 and
A3: A c= C and
A4: A = B ; ::_thesis: ( B is closed iff A is closed )
thus ( B is closed implies A is closed ) ::_thesis: ( A is closed implies B is closed )
proof
assume B is closed ; ::_thesis: A is closed
then consider F being Subset of X such that
A5: F is closed and
A6: F /\ ([#] X0) = B by PRE_TOPC:13;
A c= F by A4, A6, XBOOLE_1:17;
then A7: A c= F /\ C by A3, XBOOLE_1:19;
F /\ C c= A by A2, A4, A6, XBOOLE_1:26;
hence A is closed by A1, A5, A7, XBOOLE_0:def_10; ::_thesis: verum
end;
thus ( A is closed implies B is closed ) by A4, TOPS_2:26; ::_thesis: verum
end;
theorem Th9: :: TSEP_1:9
for X being TopSpace
for X0 being SubSpace of X
for C, A being Subset of X
for B being Subset of X0 st C is open & C c= the carrier of X0 & A c= C & A = B holds
( B is open iff A is open )
proof
let X be TopSpace; ::_thesis: for X0 being SubSpace of X
for C, A being Subset of X
for B being Subset of X0 st C is open & C c= the carrier of X0 & A c= C & A = B holds
( B is open iff A is open )
let X0 be SubSpace of X; ::_thesis: for C, A being Subset of X
for B being Subset of X0 st C is open & C c= the carrier of X0 & A c= C & A = B holds
( B is open iff A is open )
let C, A be Subset of X; ::_thesis: for B being Subset of X0 st C is open & C c= the carrier of X0 & A c= C & A = B holds
( B is open iff A is open )
let B be Subset of X0; ::_thesis: ( C is open & C c= the carrier of X0 & A c= C & A = B implies ( B is open iff A is open ) )
assume that
A1: C is open and
A2: C c= the carrier of X0 and
A3: A c= C and
A4: A = B ; ::_thesis: ( B is open iff A is open )
thus ( B is open implies A is open ) ::_thesis: ( A is open implies B is open )
proof
assume B is open ; ::_thesis: A is open
then consider F being Subset of X such that
A5: F is open and
A6: F /\ ([#] X0) = B by TOPS_2:24;
A c= F by A4, A6, XBOOLE_1:17;
then A7: A c= F /\ C by A3, XBOOLE_1:19;
F /\ C c= A by A2, A4, A6, XBOOLE_1:26;
hence A is open by A1, A5, A7, XBOOLE_0:def_10; ::_thesis: verum
end;
thus ( A is open implies B is open ) by A4, TOPS_2:25; ::_thesis: verum
end;
theorem Th10: :: TSEP_1:10
for X being non empty TopStruct
for A0 being non empty Subset of X ex X0 being non empty strict SubSpace of X st A0 = the carrier of X0
proof
let X be non empty TopStruct ; ::_thesis: for A0 being non empty Subset of X ex X0 being non empty strict SubSpace of X st A0 = the carrier of X0
let A0 be non empty Subset of X; ::_thesis: ex X0 being non empty strict SubSpace of X st A0 = the carrier of X0
take X0 = X | A0; ::_thesis: A0 = the carrier of X0
A0 = [#] X0 by PRE_TOPC:def_5;
hence A0 = the carrier of X0 ; ::_thesis: verum
end;
theorem Th11: :: TSEP_1:11
for X being TopSpace
for X0 being SubSpace of X
for A being Subset of X st A = the carrier of X0 holds
( X0 is closed SubSpace of X iff A is closed )
proof
let X be TopSpace; ::_thesis: for X0 being SubSpace of X
for A being Subset of X st A = the carrier of X0 holds
( X0 is closed SubSpace of X iff A is closed )
let X0 be SubSpace of X; ::_thesis: for A being Subset of X st A = the carrier of X0 holds
( X0 is closed SubSpace of X iff A is closed )
let A be Subset of X; ::_thesis: ( A = the carrier of X0 implies ( X0 is closed SubSpace of X iff A is closed ) )
assume A1: A = the carrier of X0 ; ::_thesis: ( X0 is closed SubSpace of X iff A is closed )
hence ( X0 is closed SubSpace of X implies A is closed ) by BORSUK_1:def_11; ::_thesis: ( A is closed implies X0 is closed SubSpace of X )
thus ( A is closed implies X0 is closed SubSpace of X ) ::_thesis: verum
proof
assume A is closed ; ::_thesis: X0 is closed SubSpace of X
then for B being Subset of X st B = the carrier of X0 holds
B is closed by A1;
hence X0 is closed SubSpace of X by BORSUK_1:def_11; ::_thesis: verum
end;
end;
theorem :: TSEP_1:12
for X being TopSpace
for X0 being closed SubSpace of X
for A being Subset of X
for B being Subset of X0 st A = B holds
( B is closed iff A is closed )
proof
let X be TopSpace; ::_thesis: for X0 being closed SubSpace of X
for A being Subset of X
for B being Subset of X0 st A = B holds
( B is closed iff A is closed )
let X0 be closed SubSpace of X; ::_thesis: for A being Subset of X
for B being Subset of X0 st A = B holds
( B is closed iff A is closed )
let A be Subset of X; ::_thesis: for B being Subset of X0 st A = B holds
( B is closed iff A is closed )
let B be Subset of X0; ::_thesis: ( A = B implies ( B is closed iff A is closed ) )
assume A1: A = B ; ::_thesis: ( B is closed iff A is closed )
reconsider C = the carrier of X0 as Subset of X by Th1;
C is closed by Th11;
hence ( B is closed iff A is closed ) by A1, Th8; ::_thesis: verum
end;
theorem :: TSEP_1:13
for X being TopSpace
for X1 being closed SubSpace of X
for X2 being closed SubSpace of X1 holds X2 is closed SubSpace of X
proof
let X be TopSpace; ::_thesis: for X1 being closed SubSpace of X
for X2 being closed SubSpace of X1 holds X2 is closed SubSpace of X
let X1 be closed SubSpace of X; ::_thesis: for X2 being closed SubSpace of X1 holds X2 is closed SubSpace of X
let X2 be closed SubSpace of X1; ::_thesis: X2 is closed SubSpace of X
now__::_thesis:_for_B_being_Subset_of_X_st_B_=_the_carrier_of_X2_holds_
B_is_closed
reconsider C = [#] X1 as Subset of X by BORSUK_1:1;
let B be Subset of X; ::_thesis: ( B = the carrier of X2 implies B is closed )
assume A1: B = the carrier of X2 ; ::_thesis: B is closed
then reconsider BB = B as Subset of X1 by BORSUK_1:1;
BB is closed by A1, BORSUK_1:def_11;
then A2: ex A being Subset of X st
( A is closed & A /\ ([#] X1) = BB ) by PRE_TOPC:13;
C is closed by BORSUK_1:def_11;
hence B is closed by A2; ::_thesis: verum
end;
hence X2 is closed SubSpace of X by Th7, BORSUK_1:def_11; ::_thesis: verum
end;
theorem :: TSEP_1:14
for X being non empty TopSpace
for X1 being non empty closed SubSpace of X
for X2 being non empty SubSpace of X st the carrier of X1 c= the carrier of X2 holds
X1 is non empty closed SubSpace of X2
proof
let X be non empty TopSpace; ::_thesis: for X1 being non empty closed SubSpace of X
for X2 being non empty SubSpace of X st the carrier of X1 c= the carrier of X2 holds
X1 is non empty closed SubSpace of X2
let X1 be non empty closed SubSpace of X; ::_thesis: for X2 being non empty SubSpace of X st the carrier of X1 c= the carrier of X2 holds
X1 is non empty closed SubSpace of X2
let X2 be non empty SubSpace of X; ::_thesis: ( the carrier of X1 c= the carrier of X2 implies X1 is non empty closed SubSpace of X2 )
assume the carrier of X1 c= the carrier of X2 ; ::_thesis: X1 is non empty closed SubSpace of X2
then reconsider B = the carrier of X1 as Subset of X2 ;
now__::_thesis:_for_C_being_Subset_of_X2_st_C_=_the_carrier_of_X1_holds_
C_is_closed
let C be Subset of X2; ::_thesis: ( C = the carrier of X1 implies C is closed )
assume A1: C = the carrier of X1 ; ::_thesis: C is closed
then reconsider A = C as Subset of X by BORSUK_1:1;
A2: A /\ ([#] X2) = C by XBOOLE_1:28;
A is closed by A1, Th11;
hence C is closed by A2, PRE_TOPC:13; ::_thesis: verum
end;
then B is closed ;
hence X1 is non empty closed SubSpace of X2 by Th4, Th11; ::_thesis: verum
end;
theorem Th15: :: TSEP_1:15
for X being non empty TopSpace
for A0 being non empty Subset of X st A0 is closed holds
ex X0 being non empty strict closed SubSpace of X st A0 = the carrier of X0
proof
let X be non empty TopSpace; ::_thesis: for A0 being non empty Subset of X st A0 is closed holds
ex X0 being non empty strict closed SubSpace of X st A0 = the carrier of X0
let A0 be non empty Subset of X; ::_thesis: ( A0 is closed implies ex X0 being non empty strict closed SubSpace of X st A0 = the carrier of X0 )
assume A1: A0 is closed ; ::_thesis: ex X0 being non empty strict closed SubSpace of X st A0 = the carrier of X0
consider X0 being non empty strict SubSpace of X such that
A2: A0 = the carrier of X0 by Th10;
reconsider Y0 = X0 as non empty strict closed SubSpace of X by A1, A2, Th11;
take Y0 ; ::_thesis: A0 = the carrier of Y0
thus A0 = the carrier of Y0 by A2; ::_thesis: verum
end;
definition
let X be TopStruct ;
let IT be SubSpace of X;
attrIT is open means :Def1: :: TSEP_1:def 1
for A being Subset of X st A = the carrier of IT holds
A is open ;
end;
:: deftheorem Def1 defines open TSEP_1:def_1_:_
for X being TopStruct
for IT being SubSpace of X holds
( IT is open iff for A being Subset of X st A = the carrier of IT holds
A is open );
Lm4: for T being TopStruct holds TopStruct(# the carrier of T, the topology of T #) is SubSpace of T
proof
let T be TopStruct ; ::_thesis: TopStruct(# the carrier of T, the topology of T #) is SubSpace of T
set S = TopStruct(# the carrier of T, the topology of T #);
thus [#] TopStruct(# the carrier of T, the topology of T #) c= [#] T ; :: according to PRE_TOPC:def_4 ::_thesis: for b1 being Element of K10( the carrier of TopStruct(# the carrier of T, the topology of T #)) holds
( ( not b1 in the topology of TopStruct(# the carrier of T, the topology of T #) or ex b2 being Element of K10( the carrier of T) st
( b2 in the topology of T & b1 = b2 /\ ([#] TopStruct(# the carrier of T, the topology of T #)) ) ) & ( for b2 being Element of K10( the carrier of T) holds
( not b2 in the topology of T or not b1 = b2 /\ ([#] TopStruct(# the carrier of T, the topology of T #)) ) or b1 in the topology of TopStruct(# the carrier of T, the topology of T #) ) )
let P be Subset of TopStruct(# the carrier of T, the topology of T #); ::_thesis: ( ( not P in the topology of TopStruct(# the carrier of T, the topology of T #) or ex b1 being Element of K10( the carrier of T) st
( b1 in the topology of T & P = b1 /\ ([#] TopStruct(# the carrier of T, the topology of T #)) ) ) & ( for b1 being Element of K10( the carrier of T) holds
( not b1 in the topology of T or not P = b1 /\ ([#] TopStruct(# the carrier of T, the topology of T #)) ) or P in the topology of TopStruct(# the carrier of T, the topology of T #) ) )
hereby ::_thesis: ( for b1 being Element of K10( the carrier of T) holds
( not b1 in the topology of T or not P = b1 /\ ([#] TopStruct(# the carrier of T, the topology of T #)) ) or P in the topology of TopStruct(# the carrier of T, the topology of T #) )
reconsider Q = P as Subset of T ;
assume A1: P in the topology of TopStruct(# the carrier of T, the topology of T #) ; ::_thesis: ex Q being Subset of T st
( Q in the topology of T & P = Q /\ ([#] TopStruct(# the carrier of T, the topology of T #)) )
take Q = Q; ::_thesis: ( Q in the topology of T & P = Q /\ ([#] TopStruct(# the carrier of T, the topology of T #)) )
thus Q in the topology of T by A1; ::_thesis: P = Q /\ ([#] TopStruct(# the carrier of T, the topology of T #))
thus P = Q /\ ([#] TopStruct(# the carrier of T, the topology of T #)) by XBOOLE_1:28; ::_thesis: verum
end;
given Q being Subset of T such that A2: ( Q in the topology of T & P = Q /\ ([#] TopStruct(# the carrier of T, the topology of T #)) ) ; ::_thesis: P in the topology of TopStruct(# the carrier of T, the topology of T #)
thus P in the topology of TopStruct(# the carrier of T, the topology of T #) by A2, XBOOLE_1:28; ::_thesis: verum
end;
registration
let X be TopSpace;
cluster strict TopSpace-like open for SubSpace of X;
existence
ex b1 being SubSpace of X st
( b1 is strict & b1 is open )
proof
reconsider Y = TopStruct(# the carrier of X, the topology of X #) as strict SubSpace of X by Lm4;
take Y ; ::_thesis: ( Y is strict & Y is open )
Y is open
proof
let A be Subset of X; :: according to TSEP_1:def_1 ::_thesis: ( A = the carrier of Y implies A is open )
assume A = the carrier of Y ; ::_thesis: A is open
then A = [#] X ;
hence A is open ; ::_thesis: verum
end;
hence ( Y is strict & Y is open ) ; ::_thesis: verum
end;
end;
registration
let X be non empty TopSpace;
cluster non empty strict TopSpace-like open for SubSpace of X;
existence
ex b1 being SubSpace of X st
( b1 is strict & b1 is open & not b1 is empty )
proof
X | ([#] X) is open
proof
let A be Subset of X; :: according to TSEP_1:def_1 ::_thesis: ( A = the carrier of (X | ([#] X)) implies A is open )
assume A = the carrier of (X | ([#] X)) ; ::_thesis: A is open
then A = [#] (X | ([#] X))
.= [#] X by PRE_TOPC:def_5 ;
hence A is open ; ::_thesis: verum
end;
hence ex b1 being SubSpace of X st
( b1 is strict & b1 is open & not b1 is empty ) ; ::_thesis: verum
end;
end;
theorem Th16: :: TSEP_1:16
for X being TopSpace
for X0 being SubSpace of X
for A being Subset of X st A = the carrier of X0 holds
( X0 is open SubSpace of X iff A is open )
proof
let X be TopSpace; ::_thesis: for X0 being SubSpace of X
for A being Subset of X st A = the carrier of X0 holds
( X0 is open SubSpace of X iff A is open )
let X0 be SubSpace of X; ::_thesis: for A being Subset of X st A = the carrier of X0 holds
( X0 is open SubSpace of X iff A is open )
let A be Subset of X; ::_thesis: ( A = the carrier of X0 implies ( X0 is open SubSpace of X iff A is open ) )
assume A1: A = the carrier of X0 ; ::_thesis: ( X0 is open SubSpace of X iff A is open )
hence ( X0 is open SubSpace of X implies A is open ) by Def1; ::_thesis: ( A is open implies X0 is open SubSpace of X )
thus ( A is open implies X0 is open SubSpace of X ) ::_thesis: verum
proof
assume A is open ; ::_thesis: X0 is open SubSpace of X
then for B being Subset of X st B = the carrier of X0 holds
B is open by A1;
hence X0 is open SubSpace of X by Def1; ::_thesis: verum
end;
end;
theorem :: TSEP_1:17
for X being TopSpace
for X0 being open SubSpace of X
for A being Subset of X
for B being Subset of X0 st A = B holds
( B is open iff A is open )
proof
let X be TopSpace; ::_thesis: for X0 being open SubSpace of X
for A being Subset of X
for B being Subset of X0 st A = B holds
( B is open iff A is open )
let X0 be open SubSpace of X; ::_thesis: for A being Subset of X
for B being Subset of X0 st A = B holds
( B is open iff A is open )
let A be Subset of X; ::_thesis: for B being Subset of X0 st A = B holds
( B is open iff A is open )
let B be Subset of X0; ::_thesis: ( A = B implies ( B is open iff A is open ) )
assume A1: A = B ; ::_thesis: ( B is open iff A is open )
reconsider C = the carrier of X0 as Subset of X by Th1;
C is open by Th16;
hence ( B is open iff A is open ) by A1, Th9; ::_thesis: verum
end;
theorem :: TSEP_1:18
for X being TopSpace
for X1 being open SubSpace of X
for X2 being open SubSpace of X1 holds X2 is open SubSpace of X
proof
let X be TopSpace; ::_thesis: for X1 being open SubSpace of X
for X2 being open SubSpace of X1 holds X2 is open SubSpace of X
let X1 be open SubSpace of X; ::_thesis: for X2 being open SubSpace of X1 holds X2 is open SubSpace of X
let X2 be open SubSpace of X1; ::_thesis: X2 is open SubSpace of X
now__::_thesis:_for_B_being_Subset_of_X_st_B_=_the_carrier_of_X2_holds_
B_is_open
reconsider C = [#] X1 as Subset of X by BORSUK_1:1;
let B be Subset of X; ::_thesis: ( B = the carrier of X2 implies B is open )
assume A1: B = the carrier of X2 ; ::_thesis: B is open
then reconsider BB = B as Subset of X1 by BORSUK_1:1;
BB is open by A1, Def1;
then A2: ex A being Subset of X st
( A is open & A /\ ([#] X1) = BB ) by TOPS_2:24;
C is open by Def1;
hence B is open by A2; ::_thesis: verum
end;
hence X2 is open SubSpace of X by Def1, Th7; ::_thesis: verum
end;
theorem :: TSEP_1:19
for X being non empty TopSpace
for X1 being open SubSpace of X
for X2 being non empty SubSpace of X st the carrier of X1 c= the carrier of X2 holds
X1 is open SubSpace of X2
proof
let X be non empty TopSpace; ::_thesis: for X1 being open SubSpace of X
for X2 being non empty SubSpace of X st the carrier of X1 c= the carrier of X2 holds
X1 is open SubSpace of X2
let X1 be open SubSpace of X; ::_thesis: for X2 being non empty SubSpace of X st the carrier of X1 c= the carrier of X2 holds
X1 is open SubSpace of X2
let X2 be non empty SubSpace of X; ::_thesis: ( the carrier of X1 c= the carrier of X2 implies X1 is open SubSpace of X2 )
assume the carrier of X1 c= the carrier of X2 ; ::_thesis: X1 is open SubSpace of X2
then reconsider B = the carrier of X1 as Subset of X2 ;
now__::_thesis:_for_C_being_Subset_of_X2_st_C_=_the_carrier_of_X1_holds_
C_is_open
let C be Subset of X2; ::_thesis: ( C = the carrier of X1 implies C is open )
assume A1: C = the carrier of X1 ; ::_thesis: C is open
then reconsider A = C as Subset of X by BORSUK_1:1;
A2: A /\ ([#] X2) = C by XBOOLE_1:28;
A is open by A1, Th16;
hence C is open by A2, TOPS_2:24; ::_thesis: verum
end;
then B is open ;
hence X1 is open SubSpace of X2 by Th4, Th16; ::_thesis: verum
end;
theorem Th20: :: TSEP_1:20
for X being non empty TopSpace
for A0 being non empty Subset of X st A0 is open holds
ex X0 being non empty strict open SubSpace of X st A0 = the carrier of X0
proof
let X be non empty TopSpace; ::_thesis: for A0 being non empty Subset of X st A0 is open holds
ex X0 being non empty strict open SubSpace of X st A0 = the carrier of X0
let A0 be non empty Subset of X; ::_thesis: ( A0 is open implies ex X0 being non empty strict open SubSpace of X st A0 = the carrier of X0 )
assume A1: A0 is open ; ::_thesis: ex X0 being non empty strict open SubSpace of X st A0 = the carrier of X0
consider X0 being non empty strict SubSpace of X such that
A2: A0 = the carrier of X0 by Th10;
reconsider Y0 = X0 as non empty strict open SubSpace of X by A1, A2, Th16;
take Y0 ; ::_thesis: A0 = the carrier of Y0
thus A0 = the carrier of Y0 by A2; ::_thesis: verum
end;
begin
definition
let X be non empty TopStruct ;
let X1, X2 be non empty SubSpace of X;
funcX1 union X2 -> non empty strict SubSpace of X means :Def2: :: TSEP_1:def 2
the carrier of it = the carrier of X1 \/ the carrier of X2;
existence
ex b1 being non empty strict SubSpace of X st the carrier of b1 = the carrier of X1 \/ the carrier of X2
proof
reconsider A1 = the carrier of X1, A2 = the carrier of X2 as Subset of X by Th1;
set A = A1 \/ A2;
reconsider A = A1 \/ A2 as non empty Subset of X ;
take X | A ; ::_thesis: the carrier of (X | A) = the carrier of X1 \/ the carrier of X2
thus the carrier of (X | A) = [#] (X | A)
.= the carrier of X1 \/ the carrier of X2 by PRE_TOPC:def_5 ; ::_thesis: verum
end;
uniqueness
for b1, b2 being non empty strict SubSpace of X st the carrier of b1 = the carrier of X1 \/ the carrier of X2 & the carrier of b2 = the carrier of X1 \/ the carrier of X2 holds
b1 = b2 by Th5;
commutativity
for b1 being non empty strict SubSpace of X
for X1, X2 being non empty SubSpace of X st the carrier of b1 = the carrier of X1 \/ the carrier of X2 holds
the carrier of b1 = the carrier of X2 \/ the carrier of X1 ;
end;
:: deftheorem Def2 defines union TSEP_1:def_2_:_
for X being non empty TopStruct
for X1, X2 being non empty SubSpace of X
for b4 being non empty strict SubSpace of X holds
( b4 = X1 union X2 iff the carrier of b4 = the carrier of X1 \/ the carrier of X2 );
theorem :: TSEP_1:21
for X being non empty TopSpace
for X1, X2, X3 being non empty SubSpace of X holds (X1 union X2) union X3 = X1 union (X2 union X3)
proof
let X be non empty TopSpace; ::_thesis: for X1, X2, X3 being non empty SubSpace of X holds (X1 union X2) union X3 = X1 union (X2 union X3)
let X1, X2, X3 be non empty SubSpace of X; ::_thesis: (X1 union X2) union X3 = X1 union (X2 union X3)
the carrier of ((X1 union X2) union X3) = the carrier of (X1 union X2) \/ the carrier of X3 by Def2
.= ( the carrier of X1 \/ the carrier of X2) \/ the carrier of X3 by Def2
.= the carrier of X1 \/ ( the carrier of X2 \/ the carrier of X3) by XBOOLE_1:4
.= the carrier of X1 \/ the carrier of (X2 union X3) by Def2
.= the carrier of (X1 union (X2 union X3)) by Def2 ;
hence (X1 union X2) union X3 = X1 union (X2 union X3) by Th5; ::_thesis: verum
end;
theorem Th22: :: TSEP_1:22
for X being non empty TopSpace
for X1, X2 being non empty SubSpace of X holds X1 is SubSpace of X1 union X2
proof
let X be non empty TopSpace; ::_thesis: for X1, X2 being non empty SubSpace of X holds X1 is SubSpace of X1 union X2
let X1, X2 be non empty SubSpace of X; ::_thesis: X1 is SubSpace of X1 union X2
set A1 = the carrier of X1;
set A2 = the carrier of X2;
set A = the carrier of (X1 union X2);
the carrier of (X1 union X2) = the carrier of X1 \/ the carrier of X2 by Def2;
then the carrier of X1 c= the carrier of (X1 union X2) by XBOOLE_1:7;
hence X1 is SubSpace of X1 union X2 by Th4; ::_thesis: verum
end;
theorem :: TSEP_1:23
for X being non empty TopSpace
for X1, X2 being non empty SubSpace of X holds
( X1 is SubSpace of X2 iff X1 union X2 = TopStruct(# the carrier of X2, the topology of X2 #) )
proof
let X be non empty TopSpace; ::_thesis: for X1, X2 being non empty SubSpace of X holds
( X1 is SubSpace of X2 iff X1 union X2 = TopStruct(# the carrier of X2, the topology of X2 #) )
let X1, X2 be non empty SubSpace of X; ::_thesis: ( X1 is SubSpace of X2 iff X1 union X2 = TopStruct(# the carrier of X2, the topology of X2 #) )
set A1 = the carrier of X1;
set A2 = the carrier of X2;
thus ( X1 is SubSpace of X2 iff X1 union X2 = TopStruct(# the carrier of X2, the topology of X2 #) ) ::_thesis: verum
proof
thus ( X1 is SubSpace of X2 implies X1 union X2 = TopStruct(# the carrier of X2, the topology of X2 #) ) ::_thesis: ( X1 union X2 = TopStruct(# the carrier of X2, the topology of X2 #) implies X1 is SubSpace of X2 )
proof
assume X1 is SubSpace of X2 ; ::_thesis: X1 union X2 = TopStruct(# the carrier of X2, the topology of X2 #)
then A1: the carrier of X1 \/ the carrier of X2 = the carrier of TopStruct(# the carrier of X2, the topology of X2 #) by BORSUK_1:1, XBOOLE_1:12;
TopStruct(# the carrier of X2, the topology of X2 #) is strict SubSpace of X by Lm3;
hence X1 union X2 = TopStruct(# the carrier of X2, the topology of X2 #) by A1, Def2; ::_thesis: verum
end;
assume X1 union X2 = TopStruct(# the carrier of X2, the topology of X2 #) ; ::_thesis: X1 is SubSpace of X2
then the carrier of X1 \/ the carrier of X2 = the carrier of X2 by Def2;
then the carrier of X1 c= the carrier of X2 by XBOOLE_1:7;
hence X1 is SubSpace of X2 by Th4; ::_thesis: verum
end;
end;
theorem :: TSEP_1:24
for X being non empty TopSpace
for X1, X2 being non empty closed SubSpace of X holds X1 union X2 is closed SubSpace of X
proof
let X be non empty TopSpace; ::_thesis: for X1, X2 being non empty closed SubSpace of X holds X1 union X2 is closed SubSpace of X
let X1, X2 be non empty closed SubSpace of X; ::_thesis: X1 union X2 is closed SubSpace of X
reconsider A2 = the carrier of X2 as Subset of X by Th1;
reconsider A1 = the carrier of X1 as Subset of X by Th1;
reconsider A = the carrier of (X1 union X2) as Subset of X by Th1;
( A1 is closed & A2 is closed ) by Th11;
then A1 \/ A2 is closed ;
then A is closed by Def2;
hence X1 union X2 is closed SubSpace of X by Th11; ::_thesis: verum
end;
theorem :: TSEP_1:25
for X being non empty TopSpace
for X1, X2 being non empty open SubSpace of X holds X1 union X2 is open SubSpace of X
proof
let X be non empty TopSpace; ::_thesis: for X1, X2 being non empty open SubSpace of X holds X1 union X2 is open SubSpace of X
let X1, X2 be non empty open SubSpace of X; ::_thesis: X1 union X2 is open SubSpace of X
reconsider A2 = the carrier of X2 as Subset of X by Th1;
reconsider A1 = the carrier of X1 as Subset of X by Th1;
reconsider A = the carrier of (X1 union X2) as Subset of X by Th1;
( A1 is open & A2 is open ) by Th16;
then A1 \/ A2 is open ;
then A is open by Def2;
hence X1 union X2 is open SubSpace of X by Th16; ::_thesis: verum
end;
definition
let X1, X2 be 1-sorted ;
predX1 misses X2 means :Def3: :: TSEP_1:def 3
the carrier of X1 misses the carrier of X2;
correctness
;
symmetry
for X1, X2 being 1-sorted st the carrier of X1 misses the carrier of X2 holds
the carrier of X2 misses the carrier of X1 ;
end;
:: deftheorem Def3 defines misses TSEP_1:def_3_:_
for X1, X2 being 1-sorted holds
( X1 misses X2 iff the carrier of X1 misses the carrier of X2 );
notation
let X1, X2 be 1-sorted ;
antonym X1 meets X2 for X1 misses X2;
end;
definition
let X be non empty TopStruct ;
let X1, X2 be non empty SubSpace of X;
assume A1: X1 meets X2 ;
funcX1 meet X2 -> non empty strict SubSpace of X means :Def4: :: TSEP_1:def 4
the carrier of it = the carrier of X1 /\ the carrier of X2;
existence
ex b1 being non empty strict SubSpace of X st the carrier of b1 = the carrier of X1 /\ the carrier of X2
proof
reconsider A1 = the carrier of X1, A2 = the carrier of X2 as Subset of X by Th1;
set A = A1 /\ A2;
A1 meets A2 by A1, Def3;
then reconsider A = A1 /\ A2 as non empty Subset of X by XBOOLE_0:def_7;
take X | A ; ::_thesis: the carrier of (X | A) = the carrier of X1 /\ the carrier of X2
thus the carrier of (X | A) = [#] (X | A)
.= the carrier of X1 /\ the carrier of X2 by PRE_TOPC:def_5 ; ::_thesis: verum
end;
uniqueness
for b1, b2 being non empty strict SubSpace of X st the carrier of b1 = the carrier of X1 /\ the carrier of X2 & the carrier of b2 = the carrier of X1 /\ the carrier of X2 holds
b1 = b2 by Th5;
end;
:: deftheorem Def4 defines meet TSEP_1:def_4_:_
for X being non empty TopStruct
for X1, X2 being non empty SubSpace of X st X1 meets X2 holds
for b4 being non empty strict SubSpace of X holds
( b4 = X1 meet X2 iff the carrier of b4 = the carrier of X1 /\ the carrier of X2 );
theorem Th26: :: TSEP_1:26
for X being non empty TopSpace
for X1, X2, X3 being non empty SubSpace of X holds
( ( X1 meets X2 implies X1 meet X2 = X2 meet X1 ) & ( ( ( X1 meets X2 & X1 meet X2 meets X3 ) or ( X2 meets X3 & X1 meets X2 meet X3 ) ) implies (X1 meet X2) meet X3 = X1 meet (X2 meet X3) ) )
proof
let X be non empty TopSpace; ::_thesis: for X1, X2, X3 being non empty SubSpace of X holds
( ( X1 meets X2 implies X1 meet X2 = X2 meet X1 ) & ( ( ( X1 meets X2 & X1 meet X2 meets X3 ) or ( X2 meets X3 & X1 meets X2 meet X3 ) ) implies (X1 meet X2) meet X3 = X1 meet (X2 meet X3) ) )
let X1, X2, X3 be non empty SubSpace of X; ::_thesis: ( ( X1 meets X2 implies X1 meet X2 = X2 meet X1 ) & ( ( ( X1 meets X2 & X1 meet X2 meets X3 ) or ( X2 meets X3 & X1 meets X2 meet X3 ) ) implies (X1 meet X2) meet X3 = X1 meet (X2 meet X3) ) )
thus ( X1 meets X2 implies X1 meet X2 = X2 meet X1 ) ::_thesis: ( ( ( X1 meets X2 & X1 meet X2 meets X3 ) or ( X2 meets X3 & X1 meets X2 meet X3 ) ) implies (X1 meet X2) meet X3 = X1 meet (X2 meet X3) )
proof
assume A1: X1 meets X2 ; ::_thesis: X1 meet X2 = X2 meet X1
then the carrier of (X1 meet X2) = the carrier of X2 /\ the carrier of X1 by Def4
.= the carrier of (X2 meet X1) by A1, Def4 ;
hence X1 meet X2 = X2 meet X1 by Th5; ::_thesis: verum
end;
now__::_thesis:_(_(_(_X1_meets_X2_&_X1_meet_X2_meets_X3_)_or_(_X2_meets_X3_&_X1_meets_X2_meet_X3_)_)_implies_(X1_meet_X2)_meet_X3_=_X1_meet_(X2_meet_X3)_)
A2: now__::_thesis:_(_X1_meets_X2_&_X1_meet_X2_meets_X3_implies_(_X1_meets_X2_&_X1_meet_X2_meets_X3_&_X2_meets_X3_&_X1_meets_X2_meet_X3_)_)
assume that
A3: X1 meets X2 and
A4: X1 meet X2 meets X3 ; ::_thesis: ( X1 meets X2 & X1 meet X2 meets X3 & X2 meets X3 & X1 meets X2 meet X3 )
the carrier of (X1 meet X2) meets the carrier of X3 by A4, Def3;
then the carrier of (X1 meet X2) /\ the carrier of X3 <> {} by XBOOLE_0:def_7;
then ( the carrier of X1 /\ the carrier of X2) /\ the carrier of X3 <> {} by A3, Def4;
then A5: the carrier of X1 /\ ( the carrier of X2 /\ the carrier of X3) <> {} by XBOOLE_1:16;
then the carrier of X2 /\ the carrier of X3 <> {} ;
then A6: the carrier of X2 meets the carrier of X3 by XBOOLE_0:def_7;
then X2 meets X3 by Def3;
then the carrier of X1 /\ the carrier of (X2 meet X3) <> {} by A5, Def4;
then the carrier of X1 meets the carrier of (X2 meet X3) by XBOOLE_0:def_7;
hence ( X1 meets X2 & X1 meet X2 meets X3 & X2 meets X3 & X1 meets X2 meet X3 ) by A3, A4, A6, Def3; ::_thesis: verum
end;
assume A7: ( ( X1 meets X2 & X1 meet X2 meets X3 ) or ( X2 meets X3 & X1 meets X2 meet X3 ) ) ; ::_thesis: (X1 meet X2) meet X3 = X1 meet (X2 meet X3)
A8: now__::_thesis:_(_X2_meets_X3_&_X1_meets_X2_meet_X3_implies_(_X1_meets_X2_&_X1_meet_X2_meets_X3_&_X2_meets_X3_&_X1_meets_X2_meet_X3_)_)
assume that
A9: X2 meets X3 and
A10: X1 meets X2 meet X3 ; ::_thesis: ( X1 meets X2 & X1 meet X2 meets X3 & X2 meets X3 & X1 meets X2 meet X3 )
the carrier of X1 meets the carrier of (X2 meet X3) by A10, Def3;
then the carrier of X1 /\ the carrier of (X2 meet X3) <> {} by XBOOLE_0:def_7;
then the carrier of X1 /\ ( the carrier of X2 /\ the carrier of X3) <> {} by A9, Def4;
then A11: ( the carrier of X1 /\ the carrier of X2) /\ the carrier of X3 <> {} by XBOOLE_1:16;
then the carrier of X1 /\ the carrier of X2 <> {} ;
then A12: the carrier of X1 meets the carrier of X2 by XBOOLE_0:def_7;
then X1 meets X2 by Def3;
then the carrier of (X1 meet X2) /\ the carrier of X3 <> {} by A11, Def4;
then the carrier of (X1 meet X2) meets the carrier of X3 by XBOOLE_0:def_7;
hence ( X1 meets X2 & X1 meet X2 meets X3 & X2 meets X3 & X1 meets X2 meet X3 ) by A9, A10, A12, Def3; ::_thesis: verum
end;
then the carrier of ((X1 meet X2) meet X3) = the carrier of (X1 meet X2) /\ the carrier of X3 by A7, Def4
.= ( the carrier of X1 /\ the carrier of X2) /\ the carrier of X3 by A7, A8, Def4
.= the carrier of X1 /\ ( the carrier of X2 /\ the carrier of X3) by XBOOLE_1:16
.= the carrier of X1 /\ the carrier of (X2 meet X3) by A7, A2, Def4
.= the carrier of (X1 meet (X2 meet X3)) by A7, A2, Def4 ;
hence (X1 meet X2) meet X3 = X1 meet (X2 meet X3) by Th5; ::_thesis: verum
end;
hence ( ( ( X1 meets X2 & X1 meet X2 meets X3 ) or ( X2 meets X3 & X1 meets X2 meet X3 ) ) implies (X1 meet X2) meet X3 = X1 meet (X2 meet X3) ) ; ::_thesis: verum
end;
theorem Th27: :: TSEP_1:27
for X being non empty TopSpace
for X1, X2 being non empty SubSpace of X st X1 meets X2 holds
( X1 meet X2 is SubSpace of X1 & X1 meet X2 is SubSpace of X2 )
proof
let X be non empty TopSpace; ::_thesis: for X1, X2 being non empty SubSpace of X st X1 meets X2 holds
( X1 meet X2 is SubSpace of X1 & X1 meet X2 is SubSpace of X2 )
let X1, X2 be non empty SubSpace of X; ::_thesis: ( X1 meets X2 implies ( X1 meet X2 is SubSpace of X1 & X1 meet X2 is SubSpace of X2 ) )
assume X1 meets X2 ; ::_thesis: ( X1 meet X2 is SubSpace of X1 & X1 meet X2 is SubSpace of X2 )
then the carrier of (X1 meet X2) = the carrier of X1 /\ the carrier of X2 by Def4;
then ( the carrier of (X1 meet X2) c= the carrier of X1 & the carrier of (X1 meet X2) c= the carrier of X2 ) by XBOOLE_1:17;
hence ( X1 meet X2 is SubSpace of X1 & X1 meet X2 is SubSpace of X2 ) by Th4; ::_thesis: verum
end;
theorem :: TSEP_1:28
for X being non empty TopSpace
for X1, X2 being non empty SubSpace of X st X1 meets X2 holds
( ( X1 is SubSpace of X2 implies X1 meet X2 = TopStruct(# the carrier of X1, the topology of X1 #) ) & ( X1 meet X2 = TopStruct(# the carrier of X1, the topology of X1 #) implies X1 is SubSpace of X2 ) & ( X2 is SubSpace of X1 implies X1 meet X2 = TopStruct(# the carrier of X2, the topology of X2 #) ) & ( X1 meet X2 = TopStruct(# the carrier of X2, the topology of X2 #) implies X2 is SubSpace of X1 ) )
proof
let X be non empty TopSpace; ::_thesis: for X1, X2 being non empty SubSpace of X st X1 meets X2 holds
( ( X1 is SubSpace of X2 implies X1 meet X2 = TopStruct(# the carrier of X1, the topology of X1 #) ) & ( X1 meet X2 = TopStruct(# the carrier of X1, the topology of X1 #) implies X1 is SubSpace of X2 ) & ( X2 is SubSpace of X1 implies X1 meet X2 = TopStruct(# the carrier of X2, the topology of X2 #) ) & ( X1 meet X2 = TopStruct(# the carrier of X2, the topology of X2 #) implies X2 is SubSpace of X1 ) )
let X1, X2 be non empty SubSpace of X; ::_thesis: ( X1 meets X2 implies ( ( X1 is SubSpace of X2 implies X1 meet X2 = TopStruct(# the carrier of X1, the topology of X1 #) ) & ( X1 meet X2 = TopStruct(# the carrier of X1, the topology of X1 #) implies X1 is SubSpace of X2 ) & ( X2 is SubSpace of X1 implies X1 meet X2 = TopStruct(# the carrier of X2, the topology of X2 #) ) & ( X1 meet X2 = TopStruct(# the carrier of X2, the topology of X2 #) implies X2 is SubSpace of X1 ) ) )
set A1 = the carrier of X1;
set A2 = the carrier of X2;
assume A1: X1 meets X2 ; ::_thesis: ( ( X1 is SubSpace of X2 implies X1 meet X2 = TopStruct(# the carrier of X1, the topology of X1 #) ) & ( X1 meet X2 = TopStruct(# the carrier of X1, the topology of X1 #) implies X1 is SubSpace of X2 ) & ( X2 is SubSpace of X1 implies X1 meet X2 = TopStruct(# the carrier of X2, the topology of X2 #) ) & ( X1 meet X2 = TopStruct(# the carrier of X2, the topology of X2 #) implies X2 is SubSpace of X1 ) )
thus ( X1 is SubSpace of X2 iff X1 meet X2 = TopStruct(# the carrier of X1, the topology of X1 #) ) ::_thesis: ( X2 is SubSpace of X1 iff X1 meet X2 = TopStruct(# the carrier of X2, the topology of X2 #) )
proof
thus ( X1 is SubSpace of X2 implies X1 meet X2 = TopStruct(# the carrier of X1, the topology of X1 #) ) ::_thesis: ( X1 meet X2 = TopStruct(# the carrier of X1, the topology of X1 #) implies X1 is SubSpace of X2 )
proof
assume X1 is SubSpace of X2 ; ::_thesis: X1 meet X2 = TopStruct(# the carrier of X1, the topology of X1 #)
then A2: the carrier of X1 /\ the carrier of X2 = the carrier of TopStruct(# the carrier of X1, the topology of X1 #) by BORSUK_1:1, XBOOLE_1:28;
TopStruct(# the carrier of X1, the topology of X1 #) is strict SubSpace of X by Lm3;
hence X1 meet X2 = TopStruct(# the carrier of X1, the topology of X1 #) by A1, A2, Def4; ::_thesis: verum
end;
assume X1 meet X2 = TopStruct(# the carrier of X1, the topology of X1 #) ; ::_thesis: X1 is SubSpace of X2
then the carrier of X1 /\ the carrier of X2 = the carrier of X1 by A1, Def4;
then the carrier of X1 c= the carrier of X2 by XBOOLE_1:17;
hence X1 is SubSpace of X2 by Th4; ::_thesis: verum
end;
thus ( X2 is SubSpace of X1 iff X1 meet X2 = TopStruct(# the carrier of X2, the topology of X2 #) ) ::_thesis: verum
proof
thus ( X2 is SubSpace of X1 implies X1 meet X2 = TopStruct(# the carrier of X2, the topology of X2 #) ) ::_thesis: ( X1 meet X2 = TopStruct(# the carrier of X2, the topology of X2 #) implies X2 is SubSpace of X1 )
proof
assume X2 is SubSpace of X1 ; ::_thesis: X1 meet X2 = TopStruct(# the carrier of X2, the topology of X2 #)
then A3: the carrier of X1 /\ the carrier of X2 = the carrier of TopStruct(# the carrier of X2, the topology of X2 #) by BORSUK_1:1, XBOOLE_1:28;
TopStruct(# the carrier of X2, the topology of X2 #) is strict SubSpace of X by Lm3;
hence X1 meet X2 = TopStruct(# the carrier of X2, the topology of X2 #) by A1, A3, Def4; ::_thesis: verum
end;
assume X1 meet X2 = TopStruct(# the carrier of X2, the topology of X2 #) ; ::_thesis: X2 is SubSpace of X1
then the carrier of X1 /\ the carrier of X2 = the carrier of X2 by A1, Def4;
then the carrier of X2 c= the carrier of X1 by XBOOLE_1:17;
hence X2 is SubSpace of X1 by Th4; ::_thesis: verum
end;
end;
theorem :: TSEP_1:29
for X being non empty TopSpace
for X1, X2 being non empty closed SubSpace of X st X1 meets X2 holds
X1 meet X2 is closed SubSpace of X
proof
let X be non empty TopSpace; ::_thesis: for X1, X2 being non empty closed SubSpace of X st X1 meets X2 holds
X1 meet X2 is closed SubSpace of X
let X1, X2 be non empty closed SubSpace of X; ::_thesis: ( X1 meets X2 implies X1 meet X2 is closed SubSpace of X )
assume A1: X1 meets X2 ; ::_thesis: X1 meet X2 is closed SubSpace of X
reconsider A2 = the carrier of X2 as Subset of X by Th1;
reconsider A1 = the carrier of X1 as Subset of X by Th1;
reconsider A = the carrier of (X1 meet X2) as Subset of X by Th1;
( A1 is closed & A2 is closed ) by Th11;
then A1 /\ A2 is closed ;
then A is closed by A1, Def4;
hence X1 meet X2 is closed SubSpace of X by Th11; ::_thesis: verum
end;
theorem :: TSEP_1:30
for X being non empty TopSpace
for X1, X2 being non empty open SubSpace of X st X1 meets X2 holds
X1 meet X2 is open SubSpace of X
proof
let X be non empty TopSpace; ::_thesis: for X1, X2 being non empty open SubSpace of X st X1 meets X2 holds
X1 meet X2 is open SubSpace of X
let X1, X2 be non empty open SubSpace of X; ::_thesis: ( X1 meets X2 implies X1 meet X2 is open SubSpace of X )
assume A1: X1 meets X2 ; ::_thesis: X1 meet X2 is open SubSpace of X
reconsider A2 = the carrier of X2 as Subset of X by Th1;
reconsider A1 = the carrier of X1 as Subset of X by Th1;
reconsider A = the carrier of (X1 meet X2) as Subset of X by Th1;
( A1 is open & A2 is open ) by Th16;
then A1 /\ A2 is open ;
then A is open by A1, Def4;
hence X1 meet X2 is open SubSpace of X by Th16; ::_thesis: verum
end;
theorem :: TSEP_1:31
for X being non empty TopSpace
for X1, X2 being non empty SubSpace of X st X1 meets X2 holds
X1 meet X2 is SubSpace of X1 union X2
proof
let X be non empty TopSpace; ::_thesis: for X1, X2 being non empty SubSpace of X st X1 meets X2 holds
X1 meet X2 is SubSpace of X1 union X2
let X1, X2 be non empty SubSpace of X; ::_thesis: ( X1 meets X2 implies X1 meet X2 is SubSpace of X1 union X2 )
assume X1 meets X2 ; ::_thesis: X1 meet X2 is SubSpace of X1 union X2
then A1: X1 meet X2 is SubSpace of X1 by Th27;
X1 is SubSpace of X1 union X2 by Th22;
hence X1 meet X2 is SubSpace of X1 union X2 by A1, Th7; ::_thesis: verum
end;
theorem :: TSEP_1:32
for X being non empty TopSpace
for X1, X2, Y being non empty SubSpace of X st X1 meets Y & Y meets X2 holds
( (X1 union X2) meet Y = (X1 meet Y) union (X2 meet Y) & Y meet (X1 union X2) = (Y meet X1) union (Y meet X2) )
proof
let X be non empty TopSpace; ::_thesis: for X1, X2, Y being non empty SubSpace of X st X1 meets Y & Y meets X2 holds
( (X1 union X2) meet Y = (X1 meet Y) union (X2 meet Y) & Y meet (X1 union X2) = (Y meet X1) union (Y meet X2) )
let X1, X2, Y be non empty SubSpace of X; ::_thesis: ( X1 meets Y & Y meets X2 implies ( (X1 union X2) meet Y = (X1 meet Y) union (X2 meet Y) & Y meet (X1 union X2) = (Y meet X1) union (Y meet X2) ) )
assume that
A1: X1 meets Y and
A2: Y meets X2 ; ::_thesis: ( (X1 union X2) meet Y = (X1 meet Y) union (X2 meet Y) & Y meet (X1 union X2) = (Y meet X1) union (Y meet X2) )
X1 is SubSpace of X1 union X2 by Th22;
then A3: the carrier of X1 c= the carrier of (X1 union X2) by Th4;
the carrier of X1 meets the carrier of Y by A1, Def3;
then the carrier of X1 /\ the carrier of Y <> {} by XBOOLE_0:def_7;
then the carrier of (X1 union X2) /\ the carrier of Y <> {} by A3, XBOOLE_1:3, XBOOLE_1:26;
then the carrier of (X1 union X2) meets the carrier of Y by XBOOLE_0:def_7;
then A4: X1 union X2 meets Y by Def3;
then the carrier of ((X1 union X2) meet Y) = the carrier of (X1 union X2) /\ the carrier of Y by Def4
.= ( the carrier of X1 \/ the carrier of X2) /\ the carrier of Y by Def2
.= ( the carrier of X1 /\ the carrier of Y) \/ ( the carrier of X2 /\ the carrier of Y) by XBOOLE_1:23
.= the carrier of (X1 meet Y) \/ ( the carrier of X2 /\ the carrier of Y) by A1, Def4
.= the carrier of (X1 meet Y) \/ the carrier of (X2 meet Y) by A2, Def4
.= the carrier of ((X1 meet Y) union (X2 meet Y)) by Def2 ;
hence (X1 union X2) meet Y = (X1 meet Y) union (X2 meet Y) by Th5; ::_thesis: Y meet (X1 union X2) = (Y meet X1) union (Y meet X2)
hence Y meet (X1 union X2) = (X1 meet Y) union (X2 meet Y) by A4, Th26
.= (Y meet X1) union (X2 meet Y) by A1, Th26
.= (Y meet X1) union (Y meet X2) by A2, Th26 ;
::_thesis: verum
end;
theorem :: TSEP_1:33
for X being non empty TopSpace
for X1, X2, Y being non empty SubSpace of X st X1 meets X2 holds
( (X1 meet X2) union Y = (X1 union Y) meet (X2 union Y) & Y union (X1 meet X2) = (Y union X1) meet (Y union X2) )
proof
let X be non empty TopSpace; ::_thesis: for X1, X2, Y being non empty SubSpace of X st X1 meets X2 holds
( (X1 meet X2) union Y = (X1 union Y) meet (X2 union Y) & Y union (X1 meet X2) = (Y union X1) meet (Y union X2) )
let X1, X2, Y be non empty SubSpace of X; ::_thesis: ( X1 meets X2 implies ( (X1 meet X2) union Y = (X1 union Y) meet (X2 union Y) & Y union (X1 meet X2) = (Y union X1) meet (Y union X2) ) )
assume A1: X1 meets X2 ; ::_thesis: ( (X1 meet X2) union Y = (X1 union Y) meet (X2 union Y) & Y union (X1 meet X2) = (Y union X1) meet (Y union X2) )
Y is SubSpace of X2 union Y by Th22;
then A2: the carrier of Y c= the carrier of (X2 union Y) by Th4;
Y is SubSpace of X1 union Y by Th22;
then the carrier of Y c= the carrier of (X1 union Y) by Th4;
then the carrier of (X1 union Y) /\ the carrier of (X2 union Y) <> {} by A2, XBOOLE_1:3, XBOOLE_1:19;
then the carrier of (X1 union Y) meets the carrier of (X2 union Y) by XBOOLE_0:def_7;
then A3: X1 union Y meets X2 union Y by Def3;
A4: the carrier of ((X1 meet X2) union Y) = the carrier of (X1 meet X2) \/ the carrier of Y by Def2
.= ( the carrier of X1 /\ the carrier of X2) \/ the carrier of Y by A1, Def4
.= ( the carrier of X1 \/ the carrier of Y) /\ ( the carrier of X2 \/ the carrier of Y) by XBOOLE_1:24
.= the carrier of (X1 union Y) /\ ( the carrier of X2 \/ the carrier of Y) by Def2
.= the carrier of (X1 union Y) /\ the carrier of (X2 union Y) by Def2
.= the carrier of ((X1 union Y) meet (X2 union Y)) by A3, Def4 ;
hence (X1 meet X2) union Y = (X1 union Y) meet (X2 union Y) by Th5; ::_thesis: Y union (X1 meet X2) = (Y union X1) meet (Y union X2)
thus Y union (X1 meet X2) = (Y union X1) meet (Y union X2) by A4, Th5; ::_thesis: verum
end;
begin
notation
let X be TopStruct ;
let A1, A2 be Subset of X;
antonym A1,A2 are_not_separated for A1,A2 are_separated ;
end;
theorem Th34: :: TSEP_1:34
for X being TopSpace
for A1, A2 being Subset of X st A1 is closed & A2 is closed holds
( A1 misses A2 iff A1,A2 are_separated )
proof
let X be TopSpace; ::_thesis: for A1, A2 being Subset of X st A1 is closed & A2 is closed holds
( A1 misses A2 iff A1,A2 are_separated )
let A1, A2 be Subset of X; ::_thesis: ( A1 is closed & A2 is closed implies ( A1 misses A2 iff A1,A2 are_separated ) )
assume A1: ( A1 is closed & A2 is closed ) ; ::_thesis: ( A1 misses A2 iff A1,A2 are_separated )
thus ( A1 misses A2 implies A1,A2 are_separated ) ::_thesis: ( A1,A2 are_separated implies A1 misses A2 )
proof
assume A2: A1 misses A2 ; ::_thesis: A1,A2 are_separated
( Cl A1 = A1 & Cl A2 = A2 ) by A1, PRE_TOPC:22;
hence A1,A2 are_separated by A2, CONNSP_1:def_1; ::_thesis: verum
end;
thus ( A1,A2 are_separated implies A1 misses A2 ) by CONNSP_1:1; ::_thesis: verum
end;
theorem Th35: :: TSEP_1:35
for X being TopSpace
for A1, A2 being Subset of X st A1 \/ A2 is closed & A1,A2 are_separated holds
( A1 is closed & A2 is closed )
proof
let X be TopSpace; ::_thesis: for A1, A2 being Subset of X st A1 \/ A2 is closed & A1,A2 are_separated holds
( A1 is closed & A2 is closed )
let A1, A2 be Subset of X; ::_thesis: ( A1 \/ A2 is closed & A1,A2 are_separated implies ( A1 is closed & A2 is closed ) )
assume A1: A1 \/ A2 is closed ; ::_thesis: ( not A1,A2 are_separated or ( A1 is closed & A2 is closed ) )
then Cl A1 c= A1 \/ A2 by TOPS_1:5, XBOOLE_1:7;
then A2: (Cl A1) \ A2 c= A1 by XBOOLE_1:43;
assume A3: A1,A2 are_separated ; ::_thesis: ( A1 is closed & A2 is closed )
then Cl A1 misses A2 by CONNSP_1:def_1;
then A4: Cl A1 c= A1 by A2, XBOOLE_1:83;
Cl A2 c= A1 \/ A2 by A1, TOPS_1:5, XBOOLE_1:7;
then A5: (Cl A2) \ A1 c= A2 by XBOOLE_1:43;
Cl A2 misses A1 by A3, CONNSP_1:def_1;
then A6: Cl A2 c= A2 by A5, XBOOLE_1:83;
( A1 c= Cl A1 & A2 c= Cl A2 ) by PRE_TOPC:18;
hence ( A1 is closed & A2 is closed ) by A6, A4, XBOOLE_0:def_10; ::_thesis: verum
end;
theorem Th36: :: TSEP_1:36
for X being TopSpace
for A1, A2 being Subset of X st A1 misses A2 & A1 is open holds
A1 misses Cl A2
proof
let X be TopSpace; ::_thesis: for A1, A2 being Subset of X st A1 misses A2 & A1 is open holds
A1 misses Cl A2
let A1, A2 be Subset of X; ::_thesis: ( A1 misses A2 & A1 is open implies A1 misses Cl A2 )
assume A1: A1 misses A2 ; ::_thesis: ( not A1 is open or A1 misses Cl A2 )
thus ( A1 is open implies A1 misses Cl A2 ) ::_thesis: verum
proof
assume A2: A1 is open ; ::_thesis: A1 misses Cl A2
A2 c= A1 ` by A1, SUBSET_1:23;
then Cl A2 c= A1 ` by A2, TOPS_1:5;
then Cl A2 misses (A1 `) ` by SUBSET_1:24;
hence A1 misses Cl A2 ; ::_thesis: verum
end;
end;
theorem Th37: :: TSEP_1:37
for X being TopSpace
for A1, A2 being Subset of X st A1 is open & A2 is open holds
( A1 misses A2 iff A1,A2 are_separated )
proof
let X be TopSpace; ::_thesis: for A1, A2 being Subset of X st A1 is open & A2 is open holds
( A1 misses A2 iff A1,A2 are_separated )
let A1, A2 be Subset of X; ::_thesis: ( A1 is open & A2 is open implies ( A1 misses A2 iff A1,A2 are_separated ) )
assume A1: ( A1 is open & A2 is open ) ; ::_thesis: ( A1 misses A2 iff A1,A2 are_separated )
thus ( A1 misses A2 implies A1,A2 are_separated ) ::_thesis: ( A1,A2 are_separated implies A1 misses A2 )
proof
assume A1 misses A2 ; ::_thesis: A1,A2 are_separated
then ( A1 misses Cl A2 & Cl A1 misses A2 ) by A1, Th36;
hence A1,A2 are_separated by CONNSP_1:def_1; ::_thesis: verum
end;
thus ( A1,A2 are_separated implies A1 misses A2 ) by CONNSP_1:1; ::_thesis: verum
end;
theorem Th38: :: TSEP_1:38
for X being TopSpace
for A1, A2 being Subset of X st A1 \/ A2 is open & A1,A2 are_separated holds
( A1 is open & A2 is open )
proof
let X be TopSpace; ::_thesis: for A1, A2 being Subset of X st A1 \/ A2 is open & A1,A2 are_separated holds
( A1 is open & A2 is open )
let A1, A2 be Subset of X; ::_thesis: ( A1 \/ A2 is open & A1,A2 are_separated implies ( A1 is open & A2 is open ) )
assume A1: A1 \/ A2 is open ; ::_thesis: ( not A1,A2 are_separated or ( A1 is open & A2 is open ) )
A2: A1 c= Cl A1 by PRE_TOPC:18;
assume A3: A1,A2 are_separated ; ::_thesis: ( A1 is open & A2 is open )
then A2 misses Cl A1 by CONNSP_1:def_1;
then A4: A2 c= (Cl A1) ` by SUBSET_1:23;
A1 misses Cl A2 by A3, CONNSP_1:def_1;
then A5: A1 c= (Cl A2) ` by SUBSET_1:23;
A6: A2 c= Cl A2 by PRE_TOPC:18;
A7: (A1 \/ A2) /\ ((Cl A2) `) = (A1 \/ A2) \ (Cl A2) by SUBSET_1:13
.= (A1 \ (Cl A2)) \/ (A2 \ (Cl A2)) by XBOOLE_1:42
.= (A1 \ (Cl A2)) \/ {} by A6, XBOOLE_1:37
.= A1 /\ ((Cl A2) `) by SUBSET_1:13
.= A1 by A5, XBOOLE_1:28 ;
(A1 \/ A2) /\ ((Cl A1) `) = (A1 \/ A2) \ (Cl A1) by SUBSET_1:13
.= (A1 \ (Cl A1)) \/ (A2 \ (Cl A1)) by XBOOLE_1:42
.= {} \/ (A2 \ (Cl A1)) by A2, XBOOLE_1:37
.= A2 /\ ((Cl A1) `) by SUBSET_1:13
.= A2 by A4, XBOOLE_1:28 ;
hence ( A1 is open & A2 is open ) by A1, A7; ::_thesis: verum
end;
theorem Th39: :: TSEP_1:39
for X being TopSpace
for A1, A2, C being Subset of X st A1,A2 are_separated holds
A1 /\ C,A2 /\ C are_separated
proof
let X be TopSpace; ::_thesis: for A1, A2, C being Subset of X st A1,A2 are_separated holds
A1 /\ C,A2 /\ C are_separated
let A1, A2, C be Subset of X; ::_thesis: ( A1,A2 are_separated implies A1 /\ C,A2 /\ C are_separated )
A1: ( A1 /\ C c= A1 & A2 /\ C c= A2 ) by XBOOLE_1:17;
assume A1,A2 are_separated ; ::_thesis: A1 /\ C,A2 /\ C are_separated
hence A1 /\ C,A2 /\ C are_separated by A1, CONNSP_1:7; ::_thesis: verum
end;
theorem Th40: :: TSEP_1:40
for X being TopSpace
for A1, A2, B being Subset of X st ( A1,B are_separated or A2,B are_separated ) holds
A1 /\ A2,B are_separated
proof
let X be TopSpace; ::_thesis: for A1, A2, B being Subset of X st ( A1,B are_separated or A2,B are_separated ) holds
A1 /\ A2,B are_separated
let A1, A2, B be Subset of X; ::_thesis: ( ( A1,B are_separated or A2,B are_separated ) implies A1 /\ A2,B are_separated )
thus ( ( A1,B are_separated or A2,B are_separated ) implies A1 /\ A2,B are_separated ) ::_thesis: verum
proof
A1: now__::_thesis:_(_A2,B_are_separated_&_(_A1,B_are_separated_or_A2,B_are_separated_)_implies_A1_/\_A2,B_are_separated_)
A2: A1 /\ A2 c= A2 by XBOOLE_1:17;
assume A2,B are_separated ; ::_thesis: ( ( A1,B are_separated or A2,B are_separated ) implies A1 /\ A2,B are_separated )
hence ( ( A1,B are_separated or A2,B are_separated ) implies A1 /\ A2,B are_separated ) by A2, CONNSP_1:7; ::_thesis: verum
end;
A3: now__::_thesis:_(_A1,B_are_separated_&_(_A1,B_are_separated_or_A2,B_are_separated_)_implies_A1_/\_A2,B_are_separated_)
A4: A1 /\ A2 c= A1 by XBOOLE_1:17;
assume A1,B are_separated ; ::_thesis: ( ( A1,B are_separated or A2,B are_separated ) implies A1 /\ A2,B are_separated )
hence ( ( A1,B are_separated or A2,B are_separated ) implies A1 /\ A2,B are_separated ) by A4, CONNSP_1:7; ::_thesis: verum
end;
assume ( A1,B are_separated or A2,B are_separated ) ; ::_thesis: A1 /\ A2,B are_separated
hence A1 /\ A2,B are_separated by A3, A1; ::_thesis: verum
end;
end;
theorem Th41: :: TSEP_1:41
for X being TopSpace
for A1, A2, B being Subset of X holds
( ( A1,B are_separated & A2,B are_separated ) iff A1 \/ A2,B are_separated )
proof
let X be TopSpace; ::_thesis: for A1, A2, B being Subset of X holds
( ( A1,B are_separated & A2,B are_separated ) iff A1 \/ A2,B are_separated )
let A1, A2, B be Subset of X; ::_thesis: ( ( A1,B are_separated & A2,B are_separated ) iff A1 \/ A2,B are_separated )
( A1 \/ A2,B are_separated implies ( A1,B are_separated & A2,B are_separated ) )
proof
A1: ( A1 c= A1 \/ A2 & A2 c= A1 \/ A2 ) by XBOOLE_1:7;
assume A1 \/ A2,B are_separated ; ::_thesis: ( A1,B are_separated & A2,B are_separated )
hence ( A1,B are_separated & A2,B are_separated ) by A1, CONNSP_1:7; ::_thesis: verum
end;
hence ( ( A1,B are_separated & A2,B are_separated ) iff A1 \/ A2,B are_separated ) by CONNSP_1:8; ::_thesis: verum
end;
theorem Th42: :: TSEP_1:42
for X being TopSpace
for A1, A2 being Subset of X holds
( A1,A2 are_separated iff ex C1, C2 being Subset of X st
( A1 c= C1 & A2 c= C2 & C1 misses A2 & C2 misses A1 & C1 is closed & C2 is closed ) )
proof
let X be TopSpace; ::_thesis: for A1, A2 being Subset of X holds
( A1,A2 are_separated iff ex C1, C2 being Subset of X st
( A1 c= C1 & A2 c= C2 & C1 misses A2 & C2 misses A1 & C1 is closed & C2 is closed ) )
let A1, A2 be Subset of X; ::_thesis: ( A1,A2 are_separated iff ex C1, C2 being Subset of X st
( A1 c= C1 & A2 c= C2 & C1 misses A2 & C2 misses A1 & C1 is closed & C2 is closed ) )
thus ( A1,A2 are_separated implies ex C1, C2 being Subset of X st
( A1 c= C1 & A2 c= C2 & C1 misses A2 & C2 misses A1 & C1 is closed & C2 is closed ) ) ::_thesis: ( ex C1, C2 being Subset of X st
( A1 c= C1 & A2 c= C2 & C1 misses A2 & C2 misses A1 & C1 is closed & C2 is closed ) implies A1,A2 are_separated )
proof
set C1 = Cl A1;
set C2 = Cl A2;
assume A1: A1,A2 are_separated ; ::_thesis: ex C1, C2 being Subset of X st
( A1 c= C1 & A2 c= C2 & C1 misses A2 & C2 misses A1 & C1 is closed & C2 is closed )
take Cl A1 ; ::_thesis: ex C2 being Subset of X st
( A1 c= Cl A1 & A2 c= C2 & Cl A1 misses A2 & C2 misses A1 & Cl A1 is closed & C2 is closed )
take Cl A2 ; ::_thesis: ( A1 c= Cl A1 & A2 c= Cl A2 & Cl A1 misses A2 & Cl A2 misses A1 & Cl A1 is closed & Cl A2 is closed )
thus ( A1 c= Cl A1 & A2 c= Cl A2 & Cl A1 misses A2 & Cl A2 misses A1 & Cl A1 is closed & Cl A2 is closed ) by A1, CONNSP_1:def_1, PRE_TOPC:18; ::_thesis: verum
end;
given C1, C2 being Subset of X such that A2: ( A1 c= C1 & A2 c= C2 & C1 misses A2 & C2 misses A1 & C1 is closed & C2 is closed ) ; ::_thesis: A1,A2 are_separated
( Cl A1 misses A2 & A1 misses Cl A2 ) by A2, TOPS_1:5, XBOOLE_1:63;
hence A1,A2 are_separated by CONNSP_1:def_1; ::_thesis: verum
end;
theorem Th43: :: TSEP_1:43
for X being TopSpace
for A1, A2 being Subset of X holds
( A1,A2 are_separated iff ex C1, C2 being Subset of X st
( A1 c= C1 & A2 c= C2 & C1 /\ C2 misses A1 \/ A2 & C1 is closed & C2 is closed ) )
proof
let X be TopSpace; ::_thesis: for A1, A2 being Subset of X holds
( A1,A2 are_separated iff ex C1, C2 being Subset of X st
( A1 c= C1 & A2 c= C2 & C1 /\ C2 misses A1 \/ A2 & C1 is closed & C2 is closed ) )
let A1, A2 be Subset of X; ::_thesis: ( A1,A2 are_separated iff ex C1, C2 being Subset of X st
( A1 c= C1 & A2 c= C2 & C1 /\ C2 misses A1 \/ A2 & C1 is closed & C2 is closed ) )
thus ( A1,A2 are_separated implies ex C1, C2 being Subset of X st
( A1 c= C1 & A2 c= C2 & C1 /\ C2 misses A1 \/ A2 & C1 is closed & C2 is closed ) ) ::_thesis: ( ex C1, C2 being Subset of X st
( A1 c= C1 & A2 c= C2 & C1 /\ C2 misses A1 \/ A2 & C1 is closed & C2 is closed ) implies A1,A2 are_separated )
proof
assume A1,A2 are_separated ; ::_thesis: ex C1, C2 being Subset of X st
( A1 c= C1 & A2 c= C2 & C1 /\ C2 misses A1 \/ A2 & C1 is closed & C2 is closed )
then consider C1, C2 being Subset of X such that
A1: ( A1 c= C1 & A2 c= C2 ) and
A2: ( C1 misses A2 & C2 misses A1 ) and
A3: ( C1 is closed & C2 is closed ) by Th42;
take C1 ; ::_thesis: ex C2 being Subset of X st
( A1 c= C1 & A2 c= C2 & C1 /\ C2 misses A1 \/ A2 & C1 is closed & C2 is closed )
take C2 ; ::_thesis: ( A1 c= C1 & A2 c= C2 & C1 /\ C2 misses A1 \/ A2 & C1 is closed & C2 is closed )
( C1 /\ C2 misses A1 & C1 /\ C2 misses A2 ) by A2, XBOOLE_1:17, XBOOLE_1:63;
hence ( A1 c= C1 & A2 c= C2 & C1 /\ C2 misses A1 \/ A2 & C1 is closed & C2 is closed ) by A1, A3, XBOOLE_1:70; ::_thesis: verum
end;
given C1, C2 being Subset of X such that A4: A1 c= C1 and
A5: A2 c= C2 and
A6: C1 /\ C2 misses A1 \/ A2 and
A7: ( C1 is closed & C2 is closed ) ; ::_thesis: A1,A2 are_separated
ex C1, C2 being Subset of X st
( A1 c= C1 & A2 c= C2 & C1 misses A2 & C2 misses A1 & C1 is closed & C2 is closed )
proof
take C1 ; ::_thesis: ex C2 being Subset of X st
( A1 c= C1 & A2 c= C2 & C1 misses A2 & C2 misses A1 & C1 is closed & C2 is closed )
take C2 ; ::_thesis: ( A1 c= C1 & A2 c= C2 & C1 misses A2 & C2 misses A1 & C1 is closed & C2 is closed )
A8: now__::_thesis:_not_C2_meets_A1
( A1 /\ C2 c= C1 /\ C2 & A1 /\ C2 c= A1 ) by A4, XBOOLE_1:17, XBOOLE_1:26;
then A9: A1 /\ C2 c= (C1 /\ C2) /\ A1 by XBOOLE_1:19;
assume C2 meets A1 ; ::_thesis: contradiction
then A10: A1 /\ C2 <> {} by XBOOLE_0:def_7;
(C1 /\ C2) /\ A1 c= (C1 /\ C2) /\ (A1 \/ A2) by XBOOLE_1:7, XBOOLE_1:26;
then (C1 /\ C2) /\ (A1 \/ A2) <> {} by A10, A9, XBOOLE_1:1, XBOOLE_1:3;
hence contradiction by A6, XBOOLE_0:def_7; ::_thesis: verum
end;
now__::_thesis:_not_C1_meets_A2
( C1 /\ A2 c= C1 /\ C2 & C1 /\ A2 c= A2 ) by A5, XBOOLE_1:17, XBOOLE_1:26;
then A11: C1 /\ A2 c= (C1 /\ C2) /\ A2 by XBOOLE_1:19;
assume C1 meets A2 ; ::_thesis: contradiction
then A12: C1 /\ A2 <> {} by XBOOLE_0:def_7;
(C1 /\ C2) /\ A2 c= (C1 /\ C2) /\ (A1 \/ A2) by XBOOLE_1:7, XBOOLE_1:26;
then (C1 /\ C2) /\ (A1 \/ A2) <> {} by A12, A11, XBOOLE_1:1, XBOOLE_1:3;
hence contradiction by A6, XBOOLE_0:def_7; ::_thesis: verum
end;
hence ( A1 c= C1 & A2 c= C2 & C1 misses A2 & C2 misses A1 & C1 is closed & C2 is closed ) by A4, A5, A7, A8; ::_thesis: verum
end;
hence A1,A2 are_separated by Th42; ::_thesis: verum
end;
theorem Th44: :: TSEP_1:44
for X being TopSpace
for A1, A2 being Subset of X holds
( A1,A2 are_separated iff ex C1, C2 being Subset of X st
( A1 c= C1 & A2 c= C2 & C1 misses A2 & C2 misses A1 & C1 is open & C2 is open ) )
proof
let X be TopSpace; ::_thesis: for A1, A2 being Subset of X holds
( A1,A2 are_separated iff ex C1, C2 being Subset of X st
( A1 c= C1 & A2 c= C2 & C1 misses A2 & C2 misses A1 & C1 is open & C2 is open ) )
let A1, A2 be Subset of X; ::_thesis: ( A1,A2 are_separated iff ex C1, C2 being Subset of X st
( A1 c= C1 & A2 c= C2 & C1 misses A2 & C2 misses A1 & C1 is open & C2 is open ) )
thus ( A1,A2 are_separated implies ex C1, C2 being Subset of X st
( A1 c= C1 & A2 c= C2 & C1 misses A2 & C2 misses A1 & C1 is open & C2 is open ) ) ::_thesis: ( ex C1, C2 being Subset of X st
( A1 c= C1 & A2 c= C2 & C1 misses A2 & C2 misses A1 & C1 is open & C2 is open ) implies A1,A2 are_separated )
proof
assume A1,A2 are_separated ; ::_thesis: ex C1, C2 being Subset of X st
( A1 c= C1 & A2 c= C2 & C1 misses A2 & C2 misses A1 & C1 is open & C2 is open )
then consider C1, C2 being Subset of X such that
A1: ( A1 c= C1 & A2 c= C2 & C1 misses A2 & C2 misses A1 & C1 is closed & C2 is closed ) by Th42;
take C2 ` ; ::_thesis: ex C2 being Subset of X st
( A1 c= C2 ` & A2 c= C2 & C2 ` misses A2 & C2 misses A1 & C2 ` is open & C2 is open )
take C1 ` ; ::_thesis: ( A1 c= C2 ` & A2 c= C1 ` & C2 ` misses A2 & C1 ` misses A1 & C2 ` is open & C1 ` is open )
thus ( A1 c= C2 ` & A2 c= C1 ` & C2 ` misses A2 & C1 ` misses A1 & C2 ` is open & C1 ` is open ) by A1, SUBSET_1:23, SUBSET_1:24; ::_thesis: verum
end;
given C1, C2 being Subset of X such that A2: ( A1 c= C1 & A2 c= C2 & C1 misses A2 & C2 misses A1 & C1 is open & C2 is open ) ; ::_thesis: A1,A2 are_separated
ex C1, C2 being Subset of X st
( A1 c= C1 & A2 c= C2 & C1 misses A2 & C2 misses A1 & C1 is closed & C2 is closed )
proof
take C2 ` ; ::_thesis: ex C2 being Subset of X st
( A1 c= C2 ` & A2 c= C2 & C2 ` misses A2 & C2 misses A1 & C2 ` is closed & C2 is closed )
take C1 ` ; ::_thesis: ( A1 c= C2 ` & A2 c= C1 ` & C2 ` misses A2 & C1 ` misses A1 & C2 ` is closed & C1 ` is closed )
thus ( A1 c= C2 ` & A2 c= C1 ` & C2 ` misses A2 & C1 ` misses A1 & C2 ` is closed & C1 ` is closed ) by A2, SUBSET_1:23, SUBSET_1:24; ::_thesis: verum
end;
hence A1,A2 are_separated by Th42; ::_thesis: verum
end;
theorem Th45: :: TSEP_1:45
for X being TopSpace
for A1, A2 being Subset of X holds
( A1,A2 are_separated iff ex C1, C2 being Subset of X st
( A1 c= C1 & A2 c= C2 & C1 /\ C2 misses A1 \/ A2 & C1 is open & C2 is open ) )
proof
let X be TopSpace; ::_thesis: for A1, A2 being Subset of X holds
( A1,A2 are_separated iff ex C1, C2 being Subset of X st
( A1 c= C1 & A2 c= C2 & C1 /\ C2 misses A1 \/ A2 & C1 is open & C2 is open ) )
let A1, A2 be Subset of X; ::_thesis: ( A1,A2 are_separated iff ex C1, C2 being Subset of X st
( A1 c= C1 & A2 c= C2 & C1 /\ C2 misses A1 \/ A2 & C1 is open & C2 is open ) )
thus ( A1,A2 are_separated implies ex C1, C2 being Subset of X st
( A1 c= C1 & A2 c= C2 & C1 /\ C2 misses A1 \/ A2 & C1 is open & C2 is open ) ) ::_thesis: ( ex C1, C2 being Subset of X st
( A1 c= C1 & A2 c= C2 & C1 /\ C2 misses A1 \/ A2 & C1 is open & C2 is open ) implies A1,A2 are_separated )
proof
assume A1,A2 are_separated ; ::_thesis: ex C1, C2 being Subset of X st
( A1 c= C1 & A2 c= C2 & C1 /\ C2 misses A1 \/ A2 & C1 is open & C2 is open )
then consider C1, C2 being Subset of X such that
A1: ( A1 c= C1 & A2 c= C2 ) and
A2: ( C1 misses A2 & C2 misses A1 ) and
A3: ( C1 is open & C2 is open ) by Th44;
take C1 ; ::_thesis: ex C2 being Subset of X st
( A1 c= C1 & A2 c= C2 & C1 /\ C2 misses A1 \/ A2 & C1 is open & C2 is open )
take C2 ; ::_thesis: ( A1 c= C1 & A2 c= C2 & C1 /\ C2 misses A1 \/ A2 & C1 is open & C2 is open )
( C1 /\ C2 misses A1 & C1 /\ C2 misses A2 ) by A2, XBOOLE_1:17, XBOOLE_1:63;
hence ( A1 c= C1 & A2 c= C2 & C1 /\ C2 misses A1 \/ A2 & C1 is open & C2 is open ) by A1, A3, XBOOLE_1:70; ::_thesis: verum
end;
given C1, C2 being Subset of X such that A4: A1 c= C1 and
A5: A2 c= C2 and
A6: C1 /\ C2 misses A1 \/ A2 and
A7: ( C1 is open & C2 is open ) ; ::_thesis: A1,A2 are_separated
ex C1, C2 being Subset of X st
( A1 c= C1 & A2 c= C2 & C1 misses A2 & C2 misses A1 & C1 is open & C2 is open )
proof
take C1 ; ::_thesis: ex C2 being Subset of X st
( A1 c= C1 & A2 c= C2 & C1 misses A2 & C2 misses A1 & C1 is open & C2 is open )
take C2 ; ::_thesis: ( A1 c= C1 & A2 c= C2 & C1 misses A2 & C2 misses A1 & C1 is open & C2 is open )
A8: now__::_thesis:_not_C2_meets_A1
( A1 /\ C2 c= C1 /\ C2 & A1 /\ C2 c= A1 ) by A4, XBOOLE_1:17, XBOOLE_1:26;
then A9: A1 /\ C2 c= (C1 /\ C2) /\ A1 by XBOOLE_1:19;
assume C2 meets A1 ; ::_thesis: contradiction
then A10: A1 /\ C2 <> {} by XBOOLE_0:def_7;
(C1 /\ C2) /\ A1 c= (C1 /\ C2) /\ (A1 \/ A2) by XBOOLE_1:7, XBOOLE_1:26;
then (C1 /\ C2) /\ (A1 \/ A2) <> {} by A10, A9, XBOOLE_1:1, XBOOLE_1:3;
hence contradiction by A6, XBOOLE_0:def_7; ::_thesis: verum
end;
now__::_thesis:_not_C1_meets_A2
( C1 /\ A2 c= C1 /\ C2 & C1 /\ A2 c= A2 ) by A5, XBOOLE_1:17, XBOOLE_1:26;
then A11: C1 /\ A2 c= (C1 /\ C2) /\ A2 by XBOOLE_1:19;
assume C1 meets A2 ; ::_thesis: contradiction
then A12: C1 /\ A2 <> {} by XBOOLE_0:def_7;
(C1 /\ C2) /\ A2 c= (C1 /\ C2) /\ (A1 \/ A2) by XBOOLE_1:7, XBOOLE_1:26;
then (C1 /\ C2) /\ (A1 \/ A2) <> {} by A12, A11, XBOOLE_1:1, XBOOLE_1:3;
hence contradiction by A6, XBOOLE_0:def_7; ::_thesis: verum
end;
hence ( A1 c= C1 & A2 c= C2 & C1 misses A2 & C2 misses A1 & C1 is open & C2 is open ) by A4, A5, A7, A8; ::_thesis: verum
end;
hence A1,A2 are_separated by Th44; ::_thesis: verum
end;
definition
let X be TopStruct ;
let A1, A2 be Subset of X;
predA1,A2 are_weakly_separated means :Def5: :: TSEP_1:def 5
A1 \ A2,A2 \ A1 are_separated ;
symmetry
for A1, A2 being Subset of X st A1 \ A2,A2 \ A1 are_separated holds
A2 \ A1,A1 \ A2 are_separated ;
end;
:: deftheorem Def5 defines are_weakly_separated TSEP_1:def_5_:_
for X being TopStruct
for A1, A2 being Subset of X holds
( A1,A2 are_weakly_separated iff A1 \ A2,A2 \ A1 are_separated );
notation
let X be TopStruct ;
let A1, A2 be Subset of X;
antonym A1,A2 are_not_weakly_separated for A1,A2 are_weakly_separated ;
end;
theorem Th46: :: TSEP_1:46
for X being TopSpace
for A1, A2 being Subset of X holds
( ( A1 misses A2 & A1,A2 are_weakly_separated ) iff A1,A2 are_separated )
proof
let X be TopSpace; ::_thesis: for A1, A2 being Subset of X holds
( ( A1 misses A2 & A1,A2 are_weakly_separated ) iff A1,A2 are_separated )
let A1, A2 be Subset of X; ::_thesis: ( ( A1 misses A2 & A1,A2 are_weakly_separated ) iff A1,A2 are_separated )
thus ( A1 misses A2 & A1,A2 are_weakly_separated implies A1,A2 are_separated ) ::_thesis: ( A1,A2 are_separated implies ( A1 misses A2 & A1,A2 are_weakly_separated ) )
proof
assume that
A1: A1 misses A2 and
A2: A1,A2 are_weakly_separated ; ::_thesis: A1,A2 are_separated
( A1 \ A2 = A1 & A2 \ A1 = A2 ) by A1, XBOOLE_1:83;
hence A1,A2 are_separated by A2, Def5; ::_thesis: verum
end;
assume A3: A1,A2 are_separated ; ::_thesis: ( A1 misses A2 & A1,A2 are_weakly_separated )
then A1 misses A2 by CONNSP_1:1;
then ( A1 \ A2 = A1 & A2 \ A1 = A2 ) by XBOOLE_1:83;
hence ( A1 misses A2 & A1,A2 are_weakly_separated ) by A3, Def5, CONNSP_1:1; ::_thesis: verum
end;
theorem Th47: :: TSEP_1:47
for X being TopSpace
for A1, A2 being Subset of X st A1 c= A2 holds
A1,A2 are_weakly_separated
proof
let X be TopSpace; ::_thesis: for A1, A2 being Subset of X st A1 c= A2 holds
A1,A2 are_weakly_separated
let A1, A2 be Subset of X; ::_thesis: ( A1 c= A2 implies A1,A2 are_weakly_separated )
A1: now__::_thesis:_(_A1_c=_A2_&_A1_c=_A2_implies_A1,A2_are_weakly_separated_)
assume A1 c= A2 ; ::_thesis: ( A1 c= A2 implies A1,A2 are_weakly_separated )
then A2: A1 \ A2 = {} by XBOOLE_1:37;
then Cl (A1 \ A2) = {} by PRE_TOPC:22;
then (Cl (A1 \ A2)) /\ (A2 \ A1) = {} ;
then A3: Cl (A1 \ A2) misses A2 \ A1 by XBOOLE_0:def_7;
(A1 \ A2) /\ (Cl (A2 \ A1)) = {} by A2;
then A1 \ A2 misses Cl (A2 \ A1) by XBOOLE_0:def_7;
then A1 \ A2,A2 \ A1 are_separated by A3, CONNSP_1:def_1;
hence ( A1 c= A2 implies A1,A2 are_weakly_separated ) by Def5; ::_thesis: verum
end;
assume A1 c= A2 ; ::_thesis: A1,A2 are_weakly_separated
hence A1,A2 are_weakly_separated by A1; ::_thesis: verum
end;
theorem Th48: :: TSEP_1:48
for X being TopSpace
for A1, A2 being Subset of X st A1 is closed & A2 is closed holds
A1,A2 are_weakly_separated
proof
let X be TopSpace; ::_thesis: for A1, A2 being Subset of X st A1 is closed & A2 is closed holds
A1,A2 are_weakly_separated
let A1, A2 be Subset of X; ::_thesis: ( A1 is closed & A2 is closed implies A1,A2 are_weakly_separated )
assume that
A1: A1 is closed and
A2: A2 is closed ; ::_thesis: A1,A2 are_weakly_separated
Cl (A2 \ A1) c= A2 by A2, TOPS_1:5, XBOOLE_1:36;
then (Cl (A2 \ A1)) \ A2 = {} by XBOOLE_1:37;
then A3: A1 \ A2 misses Cl (A2 \ A1) by Lm1;
Cl (A1 \ A2) c= A1 by A1, TOPS_1:5, XBOOLE_1:36;
then (Cl (A1 \ A2)) \ A1 = {} by XBOOLE_1:37;
then Cl (A1 \ A2) misses A2 \ A1 by Lm1;
then A1 \ A2,A2 \ A1 are_separated by A3, CONNSP_1:def_1;
hence A1,A2 are_weakly_separated by Def5; ::_thesis: verum
end;
theorem Th49: :: TSEP_1:49
for X being TopSpace
for A1, A2 being Subset of X st A1 is open & A2 is open holds
A1,A2 are_weakly_separated
proof
let X be TopSpace; ::_thesis: for A1, A2 being Subset of X st A1 is open & A2 is open holds
A1,A2 are_weakly_separated
let A1, A2 be Subset of X; ::_thesis: ( A1 is open & A2 is open implies A1,A2 are_weakly_separated )
assume that
A1: A1 is open and
A2: A2 is open ; ::_thesis: A1,A2 are_weakly_separated
A2 \ A1 misses A1 by XBOOLE_1:79;
then Cl (A2 \ A1) misses A1 by A1, Th36;
then A3: A1 \ A2 misses Cl (A2 \ A1) by XBOOLE_1:36, XBOOLE_1:63;
A2 misses A1 \ A2 by XBOOLE_1:79;
then A2 misses Cl (A1 \ A2) by A2, Th36;
then Cl (A1 \ A2) misses A2 \ A1 by XBOOLE_1:36, XBOOLE_1:63;
then A1 \ A2,A2 \ A1 are_separated by A3, CONNSP_1:def_1;
hence A1,A2 are_weakly_separated by Def5; ::_thesis: verum
end;
theorem Th50: :: TSEP_1:50
for X being TopSpace
for A1, A2, C being Subset of X st A1,A2 are_weakly_separated holds
A1 \/ C,A2 \/ C are_weakly_separated
proof
let X be TopSpace; ::_thesis: for A1, A2, C being Subset of X st A1,A2 are_weakly_separated holds
A1 \/ C,A2 \/ C are_weakly_separated
let A1, A2, C be Subset of X; ::_thesis: ( A1,A2 are_weakly_separated implies A1 \/ C,A2 \/ C are_weakly_separated )
(A1 \/ C) \ (A2 \/ C) = (A1 \ (A2 \/ C)) \/ (C \ (A2 \/ C)) by XBOOLE_1:42
.= (A1 \ (A2 \/ C)) \/ {} by XBOOLE_1:46
.= (A1 \ A2) /\ (A1 \ C) by XBOOLE_1:53 ;
then A1: (A1 \/ C) \ (A2 \/ C) c= A1 \ A2 by XBOOLE_1:17;
(A2 \/ C) \ (A1 \/ C) = (A2 \ (A1 \/ C)) \/ (C \ (A1 \/ C)) by XBOOLE_1:42
.= (A2 \ (A1 \/ C)) \/ {} by XBOOLE_1:46
.= (A2 \ A1) /\ (A2 \ C) by XBOOLE_1:53 ;
then A2: (A2 \/ C) \ (A1 \/ C) c= A2 \ A1 by XBOOLE_1:17;
assume A1,A2 are_weakly_separated ; ::_thesis: A1 \/ C,A2 \/ C are_weakly_separated
then A1 \ A2,A2 \ A1 are_separated by Def5;
then (A1 \/ C) \ (A2 \/ C),(A2 \/ C) \ (A1 \/ C) are_separated by A1, A2, CONNSP_1:7;
hence A1 \/ C,A2 \/ C are_weakly_separated by Def5; ::_thesis: verum
end;
theorem Th51: :: TSEP_1:51
for X being TopSpace
for A2, A1, B1, B2 being Subset of X st B1 c= A2 & B2 c= A1 & A1,A2 are_weakly_separated holds
A1 \/ B1,A2 \/ B2 are_weakly_separated
proof
let X be TopSpace; ::_thesis: for A2, A1, B1, B2 being Subset of X st B1 c= A2 & B2 c= A1 & A1,A2 are_weakly_separated holds
A1 \/ B1,A2 \/ B2 are_weakly_separated
let A2, A1 be Subset of X; ::_thesis: for B1, B2 being Subset of X st B1 c= A2 & B2 c= A1 & A1,A2 are_weakly_separated holds
A1 \/ B1,A2 \/ B2 are_weakly_separated
let B1, B2 be Subset of X; ::_thesis: ( B1 c= A2 & B2 c= A1 & A1,A2 are_weakly_separated implies A1 \/ B1,A2 \/ B2 are_weakly_separated )
assume that
A1: B1 c= A2 and
A2: B2 c= A1 ; ::_thesis: ( not A1,A2 are_weakly_separated or A1 \/ B1,A2 \/ B2 are_weakly_separated )
A2 c= A2 \/ B2 by XBOOLE_1:7;
then B1 c= A2 \/ B2 by A1, XBOOLE_1:1;
then A3: B1 \ (A2 \/ B2) = {} by XBOOLE_1:37;
A1 c= A1 \/ B1 by XBOOLE_1:7;
then B2 c= A1 \/ B1 by A2, XBOOLE_1:1;
then A4: B2 \ (A1 \/ B1) = {} by XBOOLE_1:37;
(A2 \/ B2) \ (A1 \/ B1) = (A2 \ (A1 \/ B1)) \/ (B2 \ (A1 \/ B1)) by XBOOLE_1:42;
then A5: (A2 \/ B2) \ (A1 \/ B1) c= A2 \ A1 by A4, XBOOLE_1:7, XBOOLE_1:34;
(A1 \/ B1) \ (A2 \/ B2) = (A1 \ (A2 \/ B2)) \/ (B1 \ (A2 \/ B2)) by XBOOLE_1:42;
then A6: (A1 \/ B1) \ (A2 \/ B2) c= A1 \ A2 by A3, XBOOLE_1:7, XBOOLE_1:34;
assume A1,A2 are_weakly_separated ; ::_thesis: A1 \/ B1,A2 \/ B2 are_weakly_separated
then A1 \ A2,A2 \ A1 are_separated by Def5;
then (A1 \/ B1) \ (A2 \/ B2),(A2 \/ B2) \ (A1 \/ B1) are_separated by A6, A5, CONNSP_1:7;
hence A1 \/ B1,A2 \/ B2 are_weakly_separated by Def5; ::_thesis: verum
end;
theorem Th52: :: TSEP_1:52
for X being TopSpace
for A1, A2, B being Subset of X st A1,B are_weakly_separated & A2,B are_weakly_separated holds
A1 /\ A2,B are_weakly_separated
proof
let X be TopSpace; ::_thesis: for A1, A2, B being Subset of X st A1,B are_weakly_separated & A2,B are_weakly_separated holds
A1 /\ A2,B are_weakly_separated
let A1, A2, B be Subset of X; ::_thesis: ( A1,B are_weakly_separated & A2,B are_weakly_separated implies A1 /\ A2,B are_weakly_separated )
thus ( A1,B are_weakly_separated & A2,B are_weakly_separated implies A1 /\ A2,B are_weakly_separated ) ::_thesis: verum
proof
assume that
A1: A1,B are_weakly_separated and
A2: A2,B are_weakly_separated ; ::_thesis: A1 /\ A2,B are_weakly_separated
A2 \ B,B \ A2 are_separated by A2, Def5;
then A3: (A1 \ B) /\ (A2 \ B),B \ A2 are_separated by Th40;
A1 \ B,B \ A1 are_separated by A1, Def5;
then (A1 \ B) /\ (A2 \ B),B \ A1 are_separated by Th40;
then (A1 \ B) /\ (A2 \ B),(B \ A1) \/ (B \ A2) are_separated by A3, Th41;
then (A1 /\ A2) \ B,(B \ A1) \/ (B \ A2) are_separated by Lm2;
then (A1 /\ A2) \ B,B \ (A1 /\ A2) are_separated by XBOOLE_1:54;
hence A1 /\ A2,B are_weakly_separated by Def5; ::_thesis: verum
end;
end;
theorem Th53: :: TSEP_1:53
for X being TopSpace
for A1, A2, B being Subset of X st A1,B are_weakly_separated & A2,B are_weakly_separated holds
A1 \/ A2,B are_weakly_separated
proof
let X be TopSpace; ::_thesis: for A1, A2, B being Subset of X st A1,B are_weakly_separated & A2,B are_weakly_separated holds
A1 \/ A2,B are_weakly_separated
let A1, A2, B be Subset of X; ::_thesis: ( A1,B are_weakly_separated & A2,B are_weakly_separated implies A1 \/ A2,B are_weakly_separated )
thus ( A1,B are_weakly_separated & A2,B are_weakly_separated implies A1 \/ A2,B are_weakly_separated ) ::_thesis: verum
proof
assume that
A1: A1,B are_weakly_separated and
A2: A2,B are_weakly_separated ; ::_thesis: A1 \/ A2,B are_weakly_separated
A2 \ B,B \ A2 are_separated by A2, Def5;
then A3: A2 \ B,(B \ A1) /\ (B \ A2) are_separated by Th40;
A1 \ B,B \ A1 are_separated by A1, Def5;
then A1 \ B,(B \ A1) /\ (B \ A2) are_separated by Th40;
then (A1 \ B) \/ (A2 \ B),(B \ A1) /\ (B \ A2) are_separated by A3, Th41;
then (A1 \/ A2) \ B,(B \ A1) /\ (B \ A2) are_separated by XBOOLE_1:42;
then (A1 \/ A2) \ B,B \ (A1 \/ A2) are_separated by XBOOLE_1:53;
hence A1 \/ A2,B are_weakly_separated by Def5; ::_thesis: verum
end;
end;
theorem Th54: :: TSEP_1:54
for X being TopSpace
for A1, A2 being Subset of X holds
( A1,A2 are_weakly_separated iff ex C1, C2, C being Subset of X st
( C1 /\ (A1 \/ A2) c= A1 & C2 /\ (A1 \/ A2) c= A2 & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (C1 \/ C2) \/ C & C1 is closed & C2 is closed & C is open ) )
proof
let X be TopSpace; ::_thesis: for A1, A2 being Subset of X holds
( A1,A2 are_weakly_separated iff ex C1, C2, C being Subset of X st
( C1 /\ (A1 \/ A2) c= A1 & C2 /\ (A1 \/ A2) c= A2 & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (C1 \/ C2) \/ C & C1 is closed & C2 is closed & C is open ) )
let A1, A2 be Subset of X; ::_thesis: ( A1,A2 are_weakly_separated iff ex C1, C2, C being Subset of X st
( C1 /\ (A1 \/ A2) c= A1 & C2 /\ (A1 \/ A2) c= A2 & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (C1 \/ C2) \/ C & C1 is closed & C2 is closed & C is open ) )
set B1 = A1 \ A2;
set B2 = A2 \ A1;
A1: (A1 \/ A2) ` misses A1 \/ A2 by XBOOLE_1:79;
thus ( A1,A2 are_weakly_separated implies ex C1, C2, C being Subset of X st
( C1 /\ (A1 \/ A2) c= A1 & C2 /\ (A1 \/ A2) c= A2 & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (C1 \/ C2) \/ C & C1 is closed & C2 is closed & C is open ) ) ::_thesis: ( ex C1, C2, C being Subset of X st
( C1 /\ (A1 \/ A2) c= A1 & C2 /\ (A1 \/ A2) c= A2 & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (C1 \/ C2) \/ C & C1 is closed & C2 is closed & C is open ) implies A1,A2 are_weakly_separated )
proof
assume A1,A2 are_weakly_separated ; ::_thesis: ex C1, C2, C being Subset of X st
( C1 /\ (A1 \/ A2) c= A1 & C2 /\ (A1 \/ A2) c= A2 & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (C1 \/ C2) \/ C & C1 is closed & C2 is closed & C is open )
then A1 \ A2,A2 \ A1 are_separated by Def5;
then consider C1, C2 being Subset of X such that
A2: ( A1 \ A2 c= C1 & A2 \ A1 c= C2 ) and
A3: C1 misses A2 \ A1 and
A4: C2 misses A1 \ A2 and
A5: ( C1 is closed & C2 is closed ) by Th42;
C1 /\ (A2 \ A1) = {} by A3, XBOOLE_0:def_7;
then (C1 /\ A2) \ (C1 /\ A1) = {} by XBOOLE_1:50;
then A6: C1 /\ A2 c= C1 /\ A1 by XBOOLE_1:37;
take C1 ; ::_thesis: ex C2, C being Subset of X st
( C1 /\ (A1 \/ A2) c= A1 & C2 /\ (A1 \/ A2) c= A2 & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (C1 \/ C2) \/ C & C1 is closed & C2 is closed & C is open )
take C2 ; ::_thesis: ex C being Subset of X st
( C1 /\ (A1 \/ A2) c= A1 & C2 /\ (A1 \/ A2) c= A2 & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (C1 \/ C2) \/ C & C1 is closed & C2 is closed & C is open )
take C = (C1 \/ C2) ` ; ::_thesis: ( C1 /\ (A1 \/ A2) c= A1 & C2 /\ (A1 \/ A2) c= A2 & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (C1 \/ C2) \/ C & C1 is closed & C2 is closed & C is open )
(A1 \ A2) \/ (A2 \ A1) c= C1 \/ C2 by A2, XBOOLE_1:13;
then C c= ((A1 \ A2) \/ (A2 \ A1)) ` by SUBSET_1:12;
then C c= (A1 \+\ A2) ` by XBOOLE_0:def_6;
then C c= ((A1 \/ A2) \ (A1 /\ A2)) ` by XBOOLE_1:101;
then C c= ((A1 \/ A2) `) \/ (A1 /\ A2) by SUBSET_1:14;
then C /\ (A1 \/ A2) c= (((A1 \/ A2) `) \/ (A1 /\ A2)) /\ (A1 \/ A2) by XBOOLE_1:26;
then C /\ (A1 \/ A2) c= (((A1 \/ A2) `) /\ (A1 \/ A2)) \/ ((A1 /\ A2) /\ (A1 \/ A2)) by XBOOLE_1:23;
then A7: C /\ (A1 \/ A2) c= ({} X) \/ ((A1 /\ A2) /\ (A1 \/ A2)) by A1, XBOOLE_0:def_7;
C2 /\ (A1 \ A2) = {} by A4, XBOOLE_0:def_7;
then (C2 /\ A1) \ (C2 /\ A2) = {} by XBOOLE_1:50;
then A8: C2 /\ A1 c= C2 /\ A2 by XBOOLE_1:37;
C2 /\ (A1 \/ A2) = (C2 /\ A1) \/ (C2 /\ A2) by XBOOLE_1:23;
then A9: C2 /\ (A1 \/ A2) = C2 /\ A2 by A8, XBOOLE_1:12;
C1 /\ (A1 \/ A2) = (C1 /\ A1) \/ (C1 /\ A2) by XBOOLE_1:23;
then A10: C1 /\ (A1 \/ A2) = C1 /\ A1 by A6, XBOOLE_1:12;
( [#] X = C \/ (C `) & (A1 /\ A2) /\ (A1 \/ A2) c= A1 /\ A2 ) by PRE_TOPC:2, XBOOLE_1:17;
hence ( C1 /\ (A1 \/ A2) c= A1 & C2 /\ (A1 \/ A2) c= A2 & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (C1 \/ C2) \/ C & C1 is closed & C2 is closed & C is open ) by A5, A10, A9, A7, XBOOLE_1:1, XBOOLE_1:17; ::_thesis: verum
end;
given C1, C2, C being Subset of X such that A11: C1 /\ (A1 \/ A2) c= A1 and
A12: C2 /\ (A1 \/ A2) c= A2 and
A13: C /\ (A1 \/ A2) c= A1 /\ A2 and
A14: the carrier of X = (C1 \/ C2) \/ C and
A15: ( C1 is closed & C2 is closed ) and
C is open ; ::_thesis: A1,A2 are_weakly_separated
ex C1, C2 being Subset of X st
( A1 \ A2 c= C1 & A2 \ A1 c= C2 & C1 /\ C2 misses (A1 \ A2) \/ (A2 \ A1) & C1 is closed & C2 is closed )
proof
(C1 /\ (A1 \/ A2)) /\ (C2 /\ (A1 \/ A2)) c= A1 /\ A2 by A11, A12, XBOOLE_1:27;
then (C1 /\ ((A1 \/ A2) /\ C2)) /\ (A1 \/ A2) c= A1 /\ A2 by XBOOLE_1:16;
then ((C1 /\ C2) /\ (A1 \/ A2)) /\ (A1 \/ A2) c= A1 /\ A2 by XBOOLE_1:16;
then (C1 /\ C2) /\ ((A1 \/ A2) /\ (A1 \/ A2)) c= A1 /\ A2 by XBOOLE_1:16;
then ((C1 /\ C2) /\ (A1 \/ A2)) \ (A1 /\ A2) = {} by XBOOLE_1:37;
then (C1 /\ C2) /\ ((A1 \/ A2) \ (A1 /\ A2)) = {} by XBOOLE_1:49;
then A16: (C1 /\ C2) /\ ((A1 \ A2) \/ (A2 \ A1)) = {} by XBOOLE_1:55;
A1 /\ A2 c= A2 by XBOOLE_1:17;
then C /\ (A1 \/ A2) c= A2 by A13, XBOOLE_1:1;
then (C2 /\ (A1 \/ A2)) \/ (C /\ (A1 \/ A2)) c= A2 by A12, XBOOLE_1:8;
then A17: (C2 \/ C) /\ (A1 \/ A2) c= A2 by XBOOLE_1:23;
A1 c= A1 \/ A2 by XBOOLE_1:7;
then A1 \ A2 c= (A1 \/ A2) \ ((C2 \/ C) /\ (A1 \/ A2)) by A17, XBOOLE_1:35;
then A18: A1 \ A2 c= (A1 \/ A2) \ (C2 \/ C) by XBOOLE_1:47;
A1 /\ A2 c= A1 by XBOOLE_1:17;
then C /\ (A1 \/ A2) c= A1 by A13, XBOOLE_1:1;
then (C /\ (A1 \/ A2)) \/ (C1 /\ (A1 \/ A2)) c= A1 by A11, XBOOLE_1:8;
then A19: (C \/ C1) /\ (A1 \/ A2) c= A1 by XBOOLE_1:23;
A2 c= A1 \/ A2 by XBOOLE_1:7;
then A2 \ A1 c= (A1 \/ A2) \ ((C \/ C1) /\ (A1 \/ A2)) by A19, XBOOLE_1:35;
then A20: A2 \ A1 c= (A1 \/ A2) \ (C \/ C1) by XBOOLE_1:47;
take C1 ; ::_thesis: ex C2 being Subset of X st
( A1 \ A2 c= C1 & A2 \ A1 c= C2 & C1 /\ C2 misses (A1 \ A2) \/ (A2 \ A1) & C1 is closed & C2 is closed )
take C2 ; ::_thesis: ( A1 \ A2 c= C1 & A2 \ A1 c= C2 & C1 /\ C2 misses (A1 \ A2) \/ (A2 \ A1) & C1 is closed & C2 is closed )
A21: A1 \/ A2 c= [#] X ;
then A1 \/ A2 c= (C2 \/ C) \/ C1 by A14, XBOOLE_1:4;
then A22: (A1 \/ A2) \ (C2 \/ C) c= C1 by XBOOLE_1:43;
A1 \/ A2 c= (C \/ C1) \/ C2 by A14, A21, XBOOLE_1:4;
then (A1 \/ A2) \ (C \/ C1) c= C2 by XBOOLE_1:43;
hence ( A1 \ A2 c= C1 & A2 \ A1 c= C2 & C1 /\ C2 misses (A1 \ A2) \/ (A2 \ A1) & C1 is closed & C2 is closed ) by A15, A20, A18, A22, A16, XBOOLE_0:def_7, XBOOLE_1:1; ::_thesis: verum
end;
then A1 \ A2,A2 \ A1 are_separated by Th43;
hence A1,A2 are_weakly_separated by Def5; ::_thesis: verum
end;
theorem Th55: :: TSEP_1:55
for X being non empty TopSpace
for A1, A2 being Subset of X st A1,A2 are_weakly_separated & not A1 c= A2 & not A2 c= A1 holds
ex C1, C2 being non empty Subset of X st
( C1 is closed & C2 is closed & C1 /\ (A1 \/ A2) c= A1 & C2 /\ (A1 \/ A2) c= A2 & ( A1 \/ A2 c= C1 \/ C2 or ex C being non empty Subset of X st
( C is open & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (C1 \/ C2) \/ C ) ) )
proof
let X be non empty TopSpace; ::_thesis: for A1, A2 being Subset of X st A1,A2 are_weakly_separated & not A1 c= A2 & not A2 c= A1 holds
ex C1, C2 being non empty Subset of X st
( C1 is closed & C2 is closed & C1 /\ (A1 \/ A2) c= A1 & C2 /\ (A1 \/ A2) c= A2 & ( A1 \/ A2 c= C1 \/ C2 or ex C being non empty Subset of X st
( C is open & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (C1 \/ C2) \/ C ) ) )
let A1, A2 be Subset of X; ::_thesis: ( A1,A2 are_weakly_separated & not A1 c= A2 & not A2 c= A1 implies ex C1, C2 being non empty Subset of X st
( C1 is closed & C2 is closed & C1 /\ (A1 \/ A2) c= A1 & C2 /\ (A1 \/ A2) c= A2 & ( A1 \/ A2 c= C1 \/ C2 or ex C being non empty Subset of X st
( C is open & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (C1 \/ C2) \/ C ) ) ) )
assume that
A1: A1,A2 are_weakly_separated and
A2: not A1 c= A2 and
A3: not A2 c= A1 ; ::_thesis: ex C1, C2 being non empty Subset of X st
( C1 is closed & C2 is closed & C1 /\ (A1 \/ A2) c= A1 & C2 /\ (A1 \/ A2) c= A2 & ( A1 \/ A2 c= C1 \/ C2 or ex C being non empty Subset of X st
( C is open & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (C1 \/ C2) \/ C ) ) )
set B1 = A1 \ A2;
set B2 = A2 \ A1;
A4: A1 \ A2 <> {} by A2, XBOOLE_1:37;
A5: A2 \ A1 <> {} by A3, XBOOLE_1:37;
A6: A1 c= A1 \/ A2 by XBOOLE_1:7;
A7: A2 c= A1 \/ A2 by XBOOLE_1:7;
consider C1, C2, C being Subset of X such that
A8: C1 /\ (A1 \/ A2) c= A1 and
A9: C2 /\ (A1 \/ A2) c= A2 and
A10: C /\ (A1 \/ A2) c= A1 /\ A2 and
A11: the carrier of X = (C1 \/ C2) \/ C and
A12: ( C1 is closed & C2 is closed ) and
A13: C is open by A1, Th54;
A1 /\ A2 c= A1 by XBOOLE_1:17;
then C /\ (A1 \/ A2) c= A1 by A10, XBOOLE_1:1;
then (C /\ (A1 \/ A2)) \/ (C1 /\ (A1 \/ A2)) c= A1 by A8, XBOOLE_1:8;
then (C \/ C1) /\ (A1 \/ A2) c= A1 by XBOOLE_1:23;
then A2 \ A1 c= (A1 \/ A2) \ ((C \/ C1) /\ (A1 \/ A2)) by A7, XBOOLE_1:35;
then A14: A2 \ A1 c= (A1 \/ A2) \ (C \/ C1) by XBOOLE_1:47;
A1 /\ A2 c= A2 by XBOOLE_1:17;
then C /\ (A1 \/ A2) c= A2 by A10, XBOOLE_1:1;
then (C2 /\ (A1 \/ A2)) \/ (C /\ (A1 \/ A2)) c= A2 by A9, XBOOLE_1:8;
then (C2 \/ C) /\ (A1 \/ A2) c= A2 by XBOOLE_1:23;
then A1 \ A2 c= (A1 \/ A2) \ ((C2 \/ C) /\ (A1 \/ A2)) by A6, XBOOLE_1:35;
then A15: A1 \ A2 c= (A1 \/ A2) \ (C2 \/ C) by XBOOLE_1:47;
A16: A1 \/ A2 c= [#] X ;
then A1 \/ A2 c= (C \/ C1) \/ C2 by A11, XBOOLE_1:4;
then (A1 \/ A2) \ (C \/ C1) c= C2 by XBOOLE_1:43;
then reconsider D2 = C2 as non empty Subset of X by A14, A5, XBOOLE_1:1, XBOOLE_1:3;
A1 \/ A2 c= (C2 \/ C) \/ C1 by A11, A16, XBOOLE_1:4;
then (A1 \/ A2) \ (C2 \/ C) c= C1 by XBOOLE_1:43;
then reconsider D1 = C1 as non empty Subset of X by A15, A4, XBOOLE_1:1, XBOOLE_1:3;
take D1 ; ::_thesis: ex C2 being non empty Subset of X st
( D1 is closed & C2 is closed & D1 /\ (A1 \/ A2) c= A1 & C2 /\ (A1 \/ A2) c= A2 & ( A1 \/ A2 c= D1 \/ C2 or ex C being non empty Subset of X st
( C is open & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (D1 \/ C2) \/ C ) ) )
take D2 ; ::_thesis: ( D1 is closed & D2 is closed & D1 /\ (A1 \/ A2) c= A1 & D2 /\ (A1 \/ A2) c= A2 & ( A1 \/ A2 c= D1 \/ D2 or ex C being non empty Subset of X st
( C is open & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (D1 \/ D2) \/ C ) ) )
now__::_thesis:_(_not_A1_\/_A2_c=_C1_\/_C2_implies_ex_C_being_non_empty_Subset_of_X_st_
(_the_carrier_of_X_=_(C1_\/_C2)_\/_C_&_C_/\_(A1_\/_A2)_c=_A1_/\_A2_&_C_is_open_)_)
assume A17: not A1 \/ A2 c= C1 \/ C2 ; ::_thesis: ex C being non empty Subset of X st
( the carrier of X = (C1 \/ C2) \/ C & C /\ (A1 \/ A2) c= A1 /\ A2 & C is open )
thus ex C being non empty Subset of X st
( the carrier of X = (C1 \/ C2) \/ C & C /\ (A1 \/ A2) c= A1 /\ A2 & C is open ) ::_thesis: verum
proof
reconsider C0 = C as non empty Subset of X by A11, A17;
take C0 ; ::_thesis: ( the carrier of X = (C1 \/ C2) \/ C0 & C0 /\ (A1 \/ A2) c= A1 /\ A2 & C0 is open )
thus ( the carrier of X = (C1 \/ C2) \/ C0 & C0 /\ (A1 \/ A2) c= A1 /\ A2 & C0 is open ) by A10, A11, A13; ::_thesis: verum
end;
end;
hence ( D1 is closed & D2 is closed & D1 /\ (A1 \/ A2) c= A1 & D2 /\ (A1 \/ A2) c= A2 & ( A1 \/ A2 c= D1 \/ D2 or ex C being non empty Subset of X st
( C is open & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (D1 \/ D2) \/ C ) ) ) by A8, A9, A12; ::_thesis: verum
end;
theorem Th56: :: TSEP_1:56
for X being non empty TopSpace
for A1, A2 being Subset of X st A1 \/ A2 = the carrier of X holds
( A1,A2 are_weakly_separated iff ex C1, C2, C being Subset of X st
( A1 \/ A2 = (C1 \/ C2) \/ C & C1 c= A1 & C2 c= A2 & C c= A1 /\ A2 & C1 is closed & C2 is closed & C is open ) )
proof
let X be non empty TopSpace; ::_thesis: for A1, A2 being Subset of X st A1 \/ A2 = the carrier of X holds
( A1,A2 are_weakly_separated iff ex C1, C2, C being Subset of X st
( A1 \/ A2 = (C1 \/ C2) \/ C & C1 c= A1 & C2 c= A2 & C c= A1 /\ A2 & C1 is closed & C2 is closed & C is open ) )
let A1, A2 be Subset of X; ::_thesis: ( A1 \/ A2 = the carrier of X implies ( A1,A2 are_weakly_separated iff ex C1, C2, C being Subset of X st
( A1 \/ A2 = (C1 \/ C2) \/ C & C1 c= A1 & C2 c= A2 & C c= A1 /\ A2 & C1 is closed & C2 is closed & C is open ) ) )
assume A1: A1 \/ A2 = the carrier of X ; ::_thesis: ( A1,A2 are_weakly_separated iff ex C1, C2, C being Subset of X st
( A1 \/ A2 = (C1 \/ C2) \/ C & C1 c= A1 & C2 c= A2 & C c= A1 /\ A2 & C1 is closed & C2 is closed & C is open ) )
thus ( A1,A2 are_weakly_separated implies ex C1, C2, C being Subset of X st
( A1 \/ A2 = (C1 \/ C2) \/ C & C1 c= A1 & C2 c= A2 & C c= A1 /\ A2 & C1 is closed & C2 is closed & C is open ) ) ::_thesis: ( ex C1, C2, C being Subset of X st
( A1 \/ A2 = (C1 \/ C2) \/ C & C1 c= A1 & C2 c= A2 & C c= A1 /\ A2 & C1 is closed & C2 is closed & C is open ) implies A1,A2 are_weakly_separated )
proof
assume A1,A2 are_weakly_separated ; ::_thesis: ex C1, C2, C being Subset of X st
( A1 \/ A2 = (C1 \/ C2) \/ C & C1 c= A1 & C2 c= A2 & C c= A1 /\ A2 & C1 is closed & C2 is closed & C is open )
then consider C1, C2, C being Subset of X such that
A2: ( C1 /\ (A1 \/ A2) c= A1 & C2 /\ (A1 \/ A2) c= A2 & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (C1 \/ C2) \/ C & C1 is closed & C2 is closed & C is open ) by Th54;
take C1 ; ::_thesis: ex C2, C being Subset of X st
( A1 \/ A2 = (C1 \/ C2) \/ C & C1 c= A1 & C2 c= A2 & C c= A1 /\ A2 & C1 is closed & C2 is closed & C is open )
take C2 ; ::_thesis: ex C being Subset of X st
( A1 \/ A2 = (C1 \/ C2) \/ C & C1 c= A1 & C2 c= A2 & C c= A1 /\ A2 & C1 is closed & C2 is closed & C is open )
take C ; ::_thesis: ( A1 \/ A2 = (C1 \/ C2) \/ C & C1 c= A1 & C2 c= A2 & C c= A1 /\ A2 & C1 is closed & C2 is closed & C is open )
thus ( A1 \/ A2 = (C1 \/ C2) \/ C & C1 c= A1 & C2 c= A2 & C c= A1 /\ A2 & C1 is closed & C2 is closed & C is open ) by A1, A2, XBOOLE_1:28; ::_thesis: verum
end;
given C1, C2, C being Subset of X such that A3: ( A1 \/ A2 = (C1 \/ C2) \/ C & C1 c= A1 & C2 c= A2 & C c= A1 /\ A2 & C1 is closed & C2 is closed & C is open ) ; ::_thesis: A1,A2 are_weakly_separated
ex C1, C2, C being Subset of X st
( C1 /\ (A1 \/ A2) c= A1 & C2 /\ (A1 \/ A2) c= A2 & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (C1 \/ C2) \/ C & C1 is closed & C2 is closed & C is open )
proof
take C1 ; ::_thesis: ex C2, C being Subset of X st
( C1 /\ (A1 \/ A2) c= A1 & C2 /\ (A1 \/ A2) c= A2 & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (C1 \/ C2) \/ C & C1 is closed & C2 is closed & C is open )
take C2 ; ::_thesis: ex C being Subset of X st
( C1 /\ (A1 \/ A2) c= A1 & C2 /\ (A1 \/ A2) c= A2 & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (C1 \/ C2) \/ C & C1 is closed & C2 is closed & C is open )
take C ; ::_thesis: ( C1 /\ (A1 \/ A2) c= A1 & C2 /\ (A1 \/ A2) c= A2 & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (C1 \/ C2) \/ C & C1 is closed & C2 is closed & C is open )
thus ( C1 /\ (A1 \/ A2) c= A1 & C2 /\ (A1 \/ A2) c= A2 & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (C1 \/ C2) \/ C & C1 is closed & C2 is closed & C is open ) by A1, A3, XBOOLE_1:28; ::_thesis: verum
end;
hence A1,A2 are_weakly_separated by Th54; ::_thesis: verum
end;
theorem Th57: :: TSEP_1:57
for X being non empty TopSpace
for A1, A2 being Subset of X st A1 \/ A2 = the carrier of X & A1,A2 are_weakly_separated & not A1 c= A2 & not A2 c= A1 holds
ex C1, C2 being non empty Subset of X st
( C1 is closed & C2 is closed & C1 c= A1 & C2 c= A2 & ( A1 \/ A2 = C1 \/ C2 or ex C being non empty Subset of X st
( A1 \/ A2 = (C1 \/ C2) \/ C & C c= A1 /\ A2 & C is open ) ) )
proof
let X be non empty TopSpace; ::_thesis: for A1, A2 being Subset of X st A1 \/ A2 = the carrier of X & A1,A2 are_weakly_separated & not A1 c= A2 & not A2 c= A1 holds
ex C1, C2 being non empty Subset of X st
( C1 is closed & C2 is closed & C1 c= A1 & C2 c= A2 & ( A1 \/ A2 = C1 \/ C2 or ex C being non empty Subset of X st
( A1 \/ A2 = (C1 \/ C2) \/ C & C c= A1 /\ A2 & C is open ) ) )
let A1, A2 be Subset of X; ::_thesis: ( A1 \/ A2 = the carrier of X & A1,A2 are_weakly_separated & not A1 c= A2 & not A2 c= A1 implies ex C1, C2 being non empty Subset of X st
( C1 is closed & C2 is closed & C1 c= A1 & C2 c= A2 & ( A1 \/ A2 = C1 \/ C2 or ex C being non empty Subset of X st
( A1 \/ A2 = (C1 \/ C2) \/ C & C c= A1 /\ A2 & C is open ) ) ) )
assume A1: A1 \/ A2 = the carrier of X ; ::_thesis: ( not A1,A2 are_weakly_separated or A1 c= A2 or A2 c= A1 or ex C1, C2 being non empty Subset of X st
( C1 is closed & C2 is closed & C1 c= A1 & C2 c= A2 & ( A1 \/ A2 = C1 \/ C2 or ex C being non empty Subset of X st
( A1 \/ A2 = (C1 \/ C2) \/ C & C c= A1 /\ A2 & C is open ) ) ) )
assume ( A1,A2 are_weakly_separated & not A1 c= A2 & not A2 c= A1 ) ; ::_thesis: ex C1, C2 being non empty Subset of X st
( C1 is closed & C2 is closed & C1 c= A1 & C2 c= A2 & ( A1 \/ A2 = C1 \/ C2 or ex C being non empty Subset of X st
( A1 \/ A2 = (C1 \/ C2) \/ C & C c= A1 /\ A2 & C is open ) ) )
then consider C1, C2 being non empty Subset of X such that
A2: ( C1 is closed & C2 is closed & C1 /\ (A1 \/ A2) c= A1 & C2 /\ (A1 \/ A2) c= A2 ) and
A3: ( A1 \/ A2 c= C1 \/ C2 or ex C being non empty Subset of X st
( C is open & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (C1 \/ C2) \/ C ) ) by Th55;
take C1 ; ::_thesis: ex C2 being non empty Subset of X st
( C1 is closed & C2 is closed & C1 c= A1 & C2 c= A2 & ( A1 \/ A2 = C1 \/ C2 or ex C being non empty Subset of X st
( A1 \/ A2 = (C1 \/ C2) \/ C & C c= A1 /\ A2 & C is open ) ) )
take C2 ; ::_thesis: ( C1 is closed & C2 is closed & C1 c= A1 & C2 c= A2 & ( A1 \/ A2 = C1 \/ C2 or ex C being non empty Subset of X st
( A1 \/ A2 = (C1 \/ C2) \/ C & C c= A1 /\ A2 & C is open ) ) )
now__::_thesis:_(_not_A1_\/_A2_=_C1_\/_C2_implies_ex_C_being_non_empty_Subset_of_X_st_
(_A1_\/_A2_=_(C1_\/_C2)_\/_C_&_C_c=_A1_/\_A2_&_C_is_open_)_)
assume not A1 \/ A2 = C1 \/ C2 ; ::_thesis: ex C being non empty Subset of X st
( A1 \/ A2 = (C1 \/ C2) \/ C & C c= A1 /\ A2 & C is open )
then consider C being non empty Subset of X such that
A4: ( C is open & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (C1 \/ C2) \/ C ) by A1, A3, XBOOLE_0:def_10;
thus ex C being non empty Subset of X st
( A1 \/ A2 = (C1 \/ C2) \/ C & C c= A1 /\ A2 & C is open ) ::_thesis: verum
proof
take C ; ::_thesis: ( A1 \/ A2 = (C1 \/ C2) \/ C & C c= A1 /\ A2 & C is open )
thus ( A1 \/ A2 = (C1 \/ C2) \/ C & C c= A1 /\ A2 & C is open ) by A1, A4, XBOOLE_1:28; ::_thesis: verum
end;
end;
hence ( C1 is closed & C2 is closed & C1 c= A1 & C2 c= A2 & ( A1 \/ A2 = C1 \/ C2 or ex C being non empty Subset of X st
( A1 \/ A2 = (C1 \/ C2) \/ C & C c= A1 /\ A2 & C is open ) ) ) by A1, A2, XBOOLE_1:28; ::_thesis: verum
end;
theorem Th58: :: TSEP_1:58
for X being non empty TopSpace
for A1, A2 being Subset of X holds
( A1,A2 are_weakly_separated iff ex C1, C2, C being Subset of X st
( C1 /\ (A1 \/ A2) c= A1 & C2 /\ (A1 \/ A2) c= A2 & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (C1 \/ C2) \/ C & C1 is open & C2 is open & C is closed ) )
proof
let X be non empty TopSpace; ::_thesis: for A1, A2 being Subset of X holds
( A1,A2 are_weakly_separated iff ex C1, C2, C being Subset of X st
( C1 /\ (A1 \/ A2) c= A1 & C2 /\ (A1 \/ A2) c= A2 & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (C1 \/ C2) \/ C & C1 is open & C2 is open & C is closed ) )
let A1, A2 be Subset of X; ::_thesis: ( A1,A2 are_weakly_separated iff ex C1, C2, C being Subset of X st
( C1 /\ (A1 \/ A2) c= A1 & C2 /\ (A1 \/ A2) c= A2 & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (C1 \/ C2) \/ C & C1 is open & C2 is open & C is closed ) )
set B1 = A1 \ A2;
set B2 = A2 \ A1;
A1: (A1 \/ A2) ` misses A1 \/ A2 by XBOOLE_1:79;
thus ( A1,A2 are_weakly_separated implies ex C1, C2, C being Subset of X st
( C1 /\ (A1 \/ A2) c= A1 & C2 /\ (A1 \/ A2) c= A2 & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (C1 \/ C2) \/ C & C1 is open & C2 is open & C is closed ) ) ::_thesis: ( ex C1, C2, C being Subset of X st
( C1 /\ (A1 \/ A2) c= A1 & C2 /\ (A1 \/ A2) c= A2 & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (C1 \/ C2) \/ C & C1 is open & C2 is open & C is closed ) implies A1,A2 are_weakly_separated )
proof
assume A1,A2 are_weakly_separated ; ::_thesis: ex C1, C2, C being Subset of X st
( C1 /\ (A1 \/ A2) c= A1 & C2 /\ (A1 \/ A2) c= A2 & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (C1 \/ C2) \/ C & C1 is open & C2 is open & C is closed )
then A1 \ A2,A2 \ A1 are_separated by Def5;
then consider C1, C2 being Subset of X such that
A2: ( A1 \ A2 c= C1 & A2 \ A1 c= C2 ) and
A3: C1 misses A2 \ A1 and
A4: C2 misses A1 \ A2 and
A5: ( C1 is open & C2 is open ) by Th44;
C1 /\ (A2 \ A1) = {} by A3, XBOOLE_0:def_7;
then (C1 /\ A2) \ (C1 /\ A1) = {} by XBOOLE_1:50;
then A6: C1 /\ A2 c= C1 /\ A1 by XBOOLE_1:37;
take C1 ; ::_thesis: ex C2, C being Subset of X st
( C1 /\ (A1 \/ A2) c= A1 & C2 /\ (A1 \/ A2) c= A2 & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (C1 \/ C2) \/ C & C1 is open & C2 is open & C is closed )
take C2 ; ::_thesis: ex C being Subset of X st
( C1 /\ (A1 \/ A2) c= A1 & C2 /\ (A1 \/ A2) c= A2 & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (C1 \/ C2) \/ C & C1 is open & C2 is open & C is closed )
take C = (C1 \/ C2) ` ; ::_thesis: ( C1 /\ (A1 \/ A2) c= A1 & C2 /\ (A1 \/ A2) c= A2 & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (C1 \/ C2) \/ C & C1 is open & C2 is open & C is closed )
(A1 \ A2) \/ (A2 \ A1) c= C1 \/ C2 by A2, XBOOLE_1:13;
then C c= ((A1 \ A2) \/ (A2 \ A1)) ` by SUBSET_1:12;
then C c= (A1 \+\ A2) ` by XBOOLE_0:def_6;
then C c= ((A1 \/ A2) \ (A1 /\ A2)) ` by XBOOLE_1:101;
then C c= ((A1 \/ A2) `) \/ (A1 /\ A2) by SUBSET_1:14;
then C /\ (A1 \/ A2) c= (((A1 \/ A2) `) \/ (A1 /\ A2)) /\ (A1 \/ A2) by XBOOLE_1:26;
then C /\ (A1 \/ A2) c= (((A1 \/ A2) `) /\ (A1 \/ A2)) \/ ((A1 /\ A2) /\ (A1 \/ A2)) by XBOOLE_1:23;
then A7: C /\ (A1 \/ A2) c= ({} X) \/ ((A1 /\ A2) /\ (A1 \/ A2)) by A1, XBOOLE_0:def_7;
C2 /\ (A1 \ A2) = {} by A4, XBOOLE_0:def_7;
then (C2 /\ A1) \ (C2 /\ A2) = {} by XBOOLE_1:50;
then A8: C2 /\ A1 c= C2 /\ A2 by XBOOLE_1:37;
C2 /\ (A1 \/ A2) = (C2 /\ A1) \/ (C2 /\ A2) by XBOOLE_1:23;
then A9: C2 /\ (A1 \/ A2) = C2 /\ A2 by A8, XBOOLE_1:12;
C1 /\ (A1 \/ A2) = (C1 /\ A1) \/ (C1 /\ A2) by XBOOLE_1:23;
then A10: C1 /\ (A1 \/ A2) = C1 /\ A1 by A6, XBOOLE_1:12;
( [#] X = (C `) \/ C & (A1 /\ A2) /\ (A1 \/ A2) c= A1 /\ A2 ) by PRE_TOPC:2, XBOOLE_1:17;
hence ( C1 /\ (A1 \/ A2) c= A1 & C2 /\ (A1 \/ A2) c= A2 & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (C1 \/ C2) \/ C & C1 is open & C2 is open & C is closed ) by A5, A10, A9, A7, XBOOLE_1:1, XBOOLE_1:17; ::_thesis: verum
end;
given C1, C2, C being Subset of X such that A11: C1 /\ (A1 \/ A2) c= A1 and
A12: C2 /\ (A1 \/ A2) c= A2 and
A13: C /\ (A1 \/ A2) c= A1 /\ A2 and
A14: the carrier of X = (C1 \/ C2) \/ C and
A15: ( C1 is open & C2 is open ) and
C is closed ; ::_thesis: A1,A2 are_weakly_separated
ex C1, C2 being Subset of X st
( A1 \ A2 c= C1 & A2 \ A1 c= C2 & C1 /\ C2 misses (A1 \ A2) \/ (A2 \ A1) & C1 is open & C2 is open )
proof
(C1 /\ (A1 \/ A2)) /\ (C2 /\ (A1 \/ A2)) c= A1 /\ A2 by A11, A12, XBOOLE_1:27;
then (C1 /\ ((A1 \/ A2) /\ C2)) /\ (A1 \/ A2) c= A1 /\ A2 by XBOOLE_1:16;
then ((C1 /\ C2) /\ (A1 \/ A2)) /\ (A1 \/ A2) c= A1 /\ A2 by XBOOLE_1:16;
then (C1 /\ C2) /\ ((A1 \/ A2) /\ (A1 \/ A2)) c= A1 /\ A2 by XBOOLE_1:16;
then ((C1 /\ C2) /\ (A1 \/ A2)) \ (A1 /\ A2) = {} by XBOOLE_1:37;
then (C1 /\ C2) /\ ((A1 \/ A2) \ (A1 /\ A2)) = {} by XBOOLE_1:49;
then A16: (C1 /\ C2) /\ ((A1 \ A2) \/ (A2 \ A1)) = {} by XBOOLE_1:55;
A1 /\ A2 c= A2 by XBOOLE_1:17;
then C /\ (A1 \/ A2) c= A2 by A13, XBOOLE_1:1;
then (C2 /\ (A1 \/ A2)) \/ (C /\ (A1 \/ A2)) c= A2 by A12, XBOOLE_1:8;
then A17: (C2 \/ C) /\ (A1 \/ A2) c= A2 by XBOOLE_1:23;
A1 c= A1 \/ A2 by XBOOLE_1:7;
then A1 \ A2 c= (A1 \/ A2) \ ((C2 \/ C) /\ (A1 \/ A2)) by A17, XBOOLE_1:35;
then A18: A1 \ A2 c= (A1 \/ A2) \ (C2 \/ C) by XBOOLE_1:47;
A1 /\ A2 c= A1 by XBOOLE_1:17;
then C /\ (A1 \/ A2) c= A1 by A13, XBOOLE_1:1;
then (C /\ (A1 \/ A2)) \/ (C1 /\ (A1 \/ A2)) c= A1 by A11, XBOOLE_1:8;
then A19: (C \/ C1) /\ (A1 \/ A2) c= A1 by XBOOLE_1:23;
A2 c= A1 \/ A2 by XBOOLE_1:7;
then A2 \ A1 c= (A1 \/ A2) \ ((C \/ C1) /\ (A1 \/ A2)) by A19, XBOOLE_1:35;
then A20: A2 \ A1 c= (A1 \/ A2) \ (C \/ C1) by XBOOLE_1:47;
take C1 ; ::_thesis: ex C2 being Subset of X st
( A1 \ A2 c= C1 & A2 \ A1 c= C2 & C1 /\ C2 misses (A1 \ A2) \/ (A2 \ A1) & C1 is open & C2 is open )
take C2 ; ::_thesis: ( A1 \ A2 c= C1 & A2 \ A1 c= C2 & C1 /\ C2 misses (A1 \ A2) \/ (A2 \ A1) & C1 is open & C2 is open )
A21: A1 \/ A2 c= [#] X ;
then A1 \/ A2 c= (C2 \/ C) \/ C1 by A14, XBOOLE_1:4;
then A22: (A1 \/ A2) \ (C2 \/ C) c= C1 by XBOOLE_1:43;
A1 \/ A2 c= (C \/ C1) \/ C2 by A14, A21, XBOOLE_1:4;
then (A1 \/ A2) \ (C \/ C1) c= C2 by XBOOLE_1:43;
hence ( A1 \ A2 c= C1 & A2 \ A1 c= C2 & C1 /\ C2 misses (A1 \ A2) \/ (A2 \ A1) & C1 is open & C2 is open ) by A15, A20, A18, A22, A16, XBOOLE_0:def_7, XBOOLE_1:1; ::_thesis: verum
end;
then A1 \ A2,A2 \ A1 are_separated by Th45;
hence A1,A2 are_weakly_separated by Def5; ::_thesis: verum
end;
theorem Th59: :: TSEP_1:59
for X being non empty TopSpace
for A1, A2 being Subset of X st A1,A2 are_weakly_separated & not A1 c= A2 & not A2 c= A1 holds
ex C1, C2 being non empty Subset of X st
( C1 is open & C2 is open & C1 /\ (A1 \/ A2) c= A1 & C2 /\ (A1 \/ A2) c= A2 & ( A1 \/ A2 c= C1 \/ C2 or ex C being non empty Subset of X st
( C is closed & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (C1 \/ C2) \/ C ) ) )
proof
let X be non empty TopSpace; ::_thesis: for A1, A2 being Subset of X st A1,A2 are_weakly_separated & not A1 c= A2 & not A2 c= A1 holds
ex C1, C2 being non empty Subset of X st
( C1 is open & C2 is open & C1 /\ (A1 \/ A2) c= A1 & C2 /\ (A1 \/ A2) c= A2 & ( A1 \/ A2 c= C1 \/ C2 or ex C being non empty Subset of X st
( C is closed & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (C1 \/ C2) \/ C ) ) )
let A1, A2 be Subset of X; ::_thesis: ( A1,A2 are_weakly_separated & not A1 c= A2 & not A2 c= A1 implies ex C1, C2 being non empty Subset of X st
( C1 is open & C2 is open & C1 /\ (A1 \/ A2) c= A1 & C2 /\ (A1 \/ A2) c= A2 & ( A1 \/ A2 c= C1 \/ C2 or ex C being non empty Subset of X st
( C is closed & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (C1 \/ C2) \/ C ) ) ) )
assume that
A1: A1,A2 are_weakly_separated and
A2: not A1 c= A2 and
A3: not A2 c= A1 ; ::_thesis: ex C1, C2 being non empty Subset of X st
( C1 is open & C2 is open & C1 /\ (A1 \/ A2) c= A1 & C2 /\ (A1 \/ A2) c= A2 & ( A1 \/ A2 c= C1 \/ C2 or ex C being non empty Subset of X st
( C is closed & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (C1 \/ C2) \/ C ) ) )
set B1 = A1 \ A2;
set B2 = A2 \ A1;
A4: A1 \ A2 <> {} by A2, XBOOLE_1:37;
A5: A2 \ A1 <> {} by A3, XBOOLE_1:37;
A6: A1 c= A1 \/ A2 by XBOOLE_1:7;
A7: A2 c= A1 \/ A2 by XBOOLE_1:7;
consider C1, C2, C being Subset of X such that
A8: C1 /\ (A1 \/ A2) c= A1 and
A9: C2 /\ (A1 \/ A2) c= A2 and
A10: C /\ (A1 \/ A2) c= A1 /\ A2 and
A11: the carrier of X = (C1 \/ C2) \/ C and
A12: ( C1 is open & C2 is open ) and
A13: C is closed by A1, Th58;
A1 /\ A2 c= A1 by XBOOLE_1:17;
then C /\ (A1 \/ A2) c= A1 by A10, XBOOLE_1:1;
then (C /\ (A1 \/ A2)) \/ (C1 /\ (A1 \/ A2)) c= A1 by A8, XBOOLE_1:8;
then (C \/ C1) /\ (A1 \/ A2) c= A1 by XBOOLE_1:23;
then A2 \ A1 c= (A1 \/ A2) \ ((C \/ C1) /\ (A1 \/ A2)) by A7, XBOOLE_1:35;
then A14: A2 \ A1 c= (A1 \/ A2) \ (C \/ C1) by XBOOLE_1:47;
A1 /\ A2 c= A2 by XBOOLE_1:17;
then C /\ (A1 \/ A2) c= A2 by A10, XBOOLE_1:1;
then (C2 /\ (A1 \/ A2)) \/ (C /\ (A1 \/ A2)) c= A2 by A9, XBOOLE_1:8;
then (C2 \/ C) /\ (A1 \/ A2) c= A2 by XBOOLE_1:23;
then A1 \ A2 c= (A1 \/ A2) \ ((C2 \/ C) /\ (A1 \/ A2)) by A6, XBOOLE_1:35;
then A15: A1 \ A2 c= (A1 \/ A2) \ (C2 \/ C) by XBOOLE_1:47;
A16: A1 \/ A2 c= [#] X ;
then A1 \/ A2 c= (C \/ C1) \/ C2 by A11, XBOOLE_1:4;
then (A1 \/ A2) \ (C \/ C1) c= C2 by XBOOLE_1:43;
then reconsider D2 = C2 as non empty Subset of X by A14, A5, XBOOLE_1:1, XBOOLE_1:3;
A1 \/ A2 c= (C2 \/ C) \/ C1 by A11, A16, XBOOLE_1:4;
then (A1 \/ A2) \ (C2 \/ C) c= C1 by XBOOLE_1:43;
then reconsider D1 = C1 as non empty Subset of X by A15, A4, XBOOLE_1:1, XBOOLE_1:3;
take D1 ; ::_thesis: ex C2 being non empty Subset of X st
( D1 is open & C2 is open & D1 /\ (A1 \/ A2) c= A1 & C2 /\ (A1 \/ A2) c= A2 & ( A1 \/ A2 c= D1 \/ C2 or ex C being non empty Subset of X st
( C is closed & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (D1 \/ C2) \/ C ) ) )
take D2 ; ::_thesis: ( D1 is open & D2 is open & D1 /\ (A1 \/ A2) c= A1 & D2 /\ (A1 \/ A2) c= A2 & ( A1 \/ A2 c= D1 \/ D2 or ex C being non empty Subset of X st
( C is closed & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (D1 \/ D2) \/ C ) ) )
now__::_thesis:_(_not_A1_\/_A2_c=_C1_\/_C2_implies_ex_C_being_non_empty_Subset_of_X_st_
(_the_carrier_of_X_=_(C1_\/_C2)_\/_C_&_C_/\_(A1_\/_A2)_c=_A1_/\_A2_&_C_is_closed_)_)
assume A17: not A1 \/ A2 c= C1 \/ C2 ; ::_thesis: ex C being non empty Subset of X st
( the carrier of X = (C1 \/ C2) \/ C & C /\ (A1 \/ A2) c= A1 /\ A2 & C is closed )
thus ex C being non empty Subset of X st
( the carrier of X = (C1 \/ C2) \/ C & C /\ (A1 \/ A2) c= A1 /\ A2 & C is closed ) ::_thesis: verum
proof
reconsider C0 = C as non empty Subset of X by A11, A17;
take C0 ; ::_thesis: ( the carrier of X = (C1 \/ C2) \/ C0 & C0 /\ (A1 \/ A2) c= A1 /\ A2 & C0 is closed )
thus ( the carrier of X = (C1 \/ C2) \/ C0 & C0 /\ (A1 \/ A2) c= A1 /\ A2 & C0 is closed ) by A10, A11, A13; ::_thesis: verum
end;
end;
hence ( D1 is open & D2 is open & D1 /\ (A1 \/ A2) c= A1 & D2 /\ (A1 \/ A2) c= A2 & ( A1 \/ A2 c= D1 \/ D2 or ex C being non empty Subset of X st
( C is closed & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (D1 \/ D2) \/ C ) ) ) by A8, A9, A12; ::_thesis: verum
end;
theorem Th60: :: TSEP_1:60
for X being non empty TopSpace
for A1, A2 being Subset of X st A1 \/ A2 = the carrier of X holds
( A1,A2 are_weakly_separated iff ex C1, C2, C being Subset of X st
( A1 \/ A2 = (C1 \/ C2) \/ C & C1 c= A1 & C2 c= A2 & C c= A1 /\ A2 & C1 is open & C2 is open & C is closed ) )
proof
let X be non empty TopSpace; ::_thesis: for A1, A2 being Subset of X st A1 \/ A2 = the carrier of X holds
( A1,A2 are_weakly_separated iff ex C1, C2, C being Subset of X st
( A1 \/ A2 = (C1 \/ C2) \/ C & C1 c= A1 & C2 c= A2 & C c= A1 /\ A2 & C1 is open & C2 is open & C is closed ) )
let A1, A2 be Subset of X; ::_thesis: ( A1 \/ A2 = the carrier of X implies ( A1,A2 are_weakly_separated iff ex C1, C2, C being Subset of X st
( A1 \/ A2 = (C1 \/ C2) \/ C & C1 c= A1 & C2 c= A2 & C c= A1 /\ A2 & C1 is open & C2 is open & C is closed ) ) )
assume A1: A1 \/ A2 = the carrier of X ; ::_thesis: ( A1,A2 are_weakly_separated iff ex C1, C2, C being Subset of X st
( A1 \/ A2 = (C1 \/ C2) \/ C & C1 c= A1 & C2 c= A2 & C c= A1 /\ A2 & C1 is open & C2 is open & C is closed ) )
thus ( A1,A2 are_weakly_separated implies ex C1, C2, C being Subset of X st
( A1 \/ A2 = (C1 \/ C2) \/ C & C1 c= A1 & C2 c= A2 & C c= A1 /\ A2 & C1 is open & C2 is open & C is closed ) ) ::_thesis: ( ex C1, C2, C being Subset of X st
( A1 \/ A2 = (C1 \/ C2) \/ C & C1 c= A1 & C2 c= A2 & C c= A1 /\ A2 & C1 is open & C2 is open & C is closed ) implies A1,A2 are_weakly_separated )
proof
assume A1,A2 are_weakly_separated ; ::_thesis: ex C1, C2, C being Subset of X st
( A1 \/ A2 = (C1 \/ C2) \/ C & C1 c= A1 & C2 c= A2 & C c= A1 /\ A2 & C1 is open & C2 is open & C is closed )
then consider C1, C2, C being Subset of X such that
A2: ( C1 /\ (A1 \/ A2) c= A1 & C2 /\ (A1 \/ A2) c= A2 & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (C1 \/ C2) \/ C & C1 is open & C2 is open & C is closed ) by Th58;
take C1 ; ::_thesis: ex C2, C being Subset of X st
( A1 \/ A2 = (C1 \/ C2) \/ C & C1 c= A1 & C2 c= A2 & C c= A1 /\ A2 & C1 is open & C2 is open & C is closed )
take C2 ; ::_thesis: ex C being Subset of X st
( A1 \/ A2 = (C1 \/ C2) \/ C & C1 c= A1 & C2 c= A2 & C c= A1 /\ A2 & C1 is open & C2 is open & C is closed )
take C ; ::_thesis: ( A1 \/ A2 = (C1 \/ C2) \/ C & C1 c= A1 & C2 c= A2 & C c= A1 /\ A2 & C1 is open & C2 is open & C is closed )
thus ( A1 \/ A2 = (C1 \/ C2) \/ C & C1 c= A1 & C2 c= A2 & C c= A1 /\ A2 & C1 is open & C2 is open & C is closed ) by A1, A2, XBOOLE_1:28; ::_thesis: verum
end;
given C1, C2, C being Subset of X such that A3: ( A1 \/ A2 = (C1 \/ C2) \/ C & C1 c= A1 & C2 c= A2 & C c= A1 /\ A2 & C1 is open & C2 is open & C is closed ) ; ::_thesis: A1,A2 are_weakly_separated
ex C1, C2, C being Subset of X st
( C1 /\ (A1 \/ A2) c= A1 & C2 /\ (A1 \/ A2) c= A2 & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (C1 \/ C2) \/ C & C1 is open & C2 is open & C is closed )
proof
take C1 ; ::_thesis: ex C2, C being Subset of X st
( C1 /\ (A1 \/ A2) c= A1 & C2 /\ (A1 \/ A2) c= A2 & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (C1 \/ C2) \/ C & C1 is open & C2 is open & C is closed )
take C2 ; ::_thesis: ex C being Subset of X st
( C1 /\ (A1 \/ A2) c= A1 & C2 /\ (A1 \/ A2) c= A2 & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (C1 \/ C2) \/ C & C1 is open & C2 is open & C is closed )
take C ; ::_thesis: ( C1 /\ (A1 \/ A2) c= A1 & C2 /\ (A1 \/ A2) c= A2 & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (C1 \/ C2) \/ C & C1 is open & C2 is open & C is closed )
thus ( C1 /\ (A1 \/ A2) c= A1 & C2 /\ (A1 \/ A2) c= A2 & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (C1 \/ C2) \/ C & C1 is open & C2 is open & C is closed ) by A1, A3, XBOOLE_1:28; ::_thesis: verum
end;
hence A1,A2 are_weakly_separated by Th58; ::_thesis: verum
end;
theorem Th61: :: TSEP_1:61
for X being non empty TopSpace
for A1, A2 being Subset of X st A1 \/ A2 = the carrier of X & A1,A2 are_weakly_separated & not A1 c= A2 & not A2 c= A1 holds
ex C1, C2 being non empty Subset of X st
( C1 is open & C2 is open & C1 c= A1 & C2 c= A2 & ( A1 \/ A2 = C1 \/ C2 or ex C being non empty Subset of X st
( A1 \/ A2 = (C1 \/ C2) \/ C & C c= A1 /\ A2 & C is closed ) ) )
proof
let X be non empty TopSpace; ::_thesis: for A1, A2 being Subset of X st A1 \/ A2 = the carrier of X & A1,A2 are_weakly_separated & not A1 c= A2 & not A2 c= A1 holds
ex C1, C2 being non empty Subset of X st
( C1 is open & C2 is open & C1 c= A1 & C2 c= A2 & ( A1 \/ A2 = C1 \/ C2 or ex C being non empty Subset of X st
( A1 \/ A2 = (C1 \/ C2) \/ C & C c= A1 /\ A2 & C is closed ) ) )
let A1, A2 be Subset of X; ::_thesis: ( A1 \/ A2 = the carrier of X & A1,A2 are_weakly_separated & not A1 c= A2 & not A2 c= A1 implies ex C1, C2 being non empty Subset of X st
( C1 is open & C2 is open & C1 c= A1 & C2 c= A2 & ( A1 \/ A2 = C1 \/ C2 or ex C being non empty Subset of X st
( A1 \/ A2 = (C1 \/ C2) \/ C & C c= A1 /\ A2 & C is closed ) ) ) )
assume A1: A1 \/ A2 = the carrier of X ; ::_thesis: ( not A1,A2 are_weakly_separated or A1 c= A2 or A2 c= A1 or ex C1, C2 being non empty Subset of X st
( C1 is open & C2 is open & C1 c= A1 & C2 c= A2 & ( A1 \/ A2 = C1 \/ C2 or ex C being non empty Subset of X st
( A1 \/ A2 = (C1 \/ C2) \/ C & C c= A1 /\ A2 & C is closed ) ) ) )
assume ( A1,A2 are_weakly_separated & not A1 c= A2 & not A2 c= A1 ) ; ::_thesis: ex C1, C2 being non empty Subset of X st
( C1 is open & C2 is open & C1 c= A1 & C2 c= A2 & ( A1 \/ A2 = C1 \/ C2 or ex C being non empty Subset of X st
( A1 \/ A2 = (C1 \/ C2) \/ C & C c= A1 /\ A2 & C is closed ) ) )
then consider C1, C2 being non empty Subset of X such that
A2: ( C1 is open & C2 is open & C1 /\ (A1 \/ A2) c= A1 & C2 /\ (A1 \/ A2) c= A2 ) and
A3: ( A1 \/ A2 c= C1 \/ C2 or ex C being non empty Subset of X st
( C is closed & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (C1 \/ C2) \/ C ) ) by Th59;
take C1 ; ::_thesis: ex C2 being non empty Subset of X st
( C1 is open & C2 is open & C1 c= A1 & C2 c= A2 & ( A1 \/ A2 = C1 \/ C2 or ex C being non empty Subset of X st
( A1 \/ A2 = (C1 \/ C2) \/ C & C c= A1 /\ A2 & C is closed ) ) )
take C2 ; ::_thesis: ( C1 is open & C2 is open & C1 c= A1 & C2 c= A2 & ( A1 \/ A2 = C1 \/ C2 or ex C being non empty Subset of X st
( A1 \/ A2 = (C1 \/ C2) \/ C & C c= A1 /\ A2 & C is closed ) ) )
now__::_thesis:_(_not_A1_\/_A2_=_C1_\/_C2_implies_ex_C_being_non_empty_Subset_of_X_st_
(_A1_\/_A2_=_(C1_\/_C2)_\/_C_&_C_c=_A1_/\_A2_&_C_is_closed_)_)
assume not A1 \/ A2 = C1 \/ C2 ; ::_thesis: ex C being non empty Subset of X st
( A1 \/ A2 = (C1 \/ C2) \/ C & C c= A1 /\ A2 & C is closed )
then consider C being non empty Subset of X such that
A4: ( C is closed & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (C1 \/ C2) \/ C ) by A1, A3, XBOOLE_0:def_10;
thus ex C being non empty Subset of X st
( A1 \/ A2 = (C1 \/ C2) \/ C & C c= A1 /\ A2 & C is closed ) ::_thesis: verum
proof
take C ; ::_thesis: ( A1 \/ A2 = (C1 \/ C2) \/ C & C c= A1 /\ A2 & C is closed )
thus ( A1 \/ A2 = (C1 \/ C2) \/ C & C c= A1 /\ A2 & C is closed ) by A1, A4, XBOOLE_1:28; ::_thesis: verum
end;
end;
hence ( C1 is open & C2 is open & C1 c= A1 & C2 c= A2 & ( A1 \/ A2 = C1 \/ C2 or ex C being non empty Subset of X st
( A1 \/ A2 = (C1 \/ C2) \/ C & C c= A1 /\ A2 & C is closed ) ) ) by A1, A2, XBOOLE_1:28; ::_thesis: verum
end;
theorem Th62: :: TSEP_1:62
for X being non empty TopSpace
for A1, A2 being Subset of X holds
( A1,A2 are_separated iff ex B1, B2 being Subset of X st
( B1,B2 are_weakly_separated & A1 c= B1 & A2 c= B2 & B1 /\ B2 misses A1 \/ A2 ) )
proof
let X be non empty TopSpace; ::_thesis: for A1, A2 being Subset of X holds
( A1,A2 are_separated iff ex B1, B2 being Subset of X st
( B1,B2 are_weakly_separated & A1 c= B1 & A2 c= B2 & B1 /\ B2 misses A1 \/ A2 ) )
let A1, A2 be Subset of X; ::_thesis: ( A1,A2 are_separated iff ex B1, B2 being Subset of X st
( B1,B2 are_weakly_separated & A1 c= B1 & A2 c= B2 & B1 /\ B2 misses A1 \/ A2 ) )
thus ( A1,A2 are_separated implies ex B1, B2 being Subset of X st
( B1,B2 are_weakly_separated & A1 c= B1 & A2 c= B2 & B1 /\ B2 misses A1 \/ A2 ) ) ::_thesis: ( ex B1, B2 being Subset of X st
( B1,B2 are_weakly_separated & A1 c= B1 & A2 c= B2 & B1 /\ B2 misses A1 \/ A2 ) implies A1,A2 are_separated )
proof
assume A1,A2 are_separated ; ::_thesis: ex B1, B2 being Subset of X st
( B1,B2 are_weakly_separated & A1 c= B1 & A2 c= B2 & B1 /\ B2 misses A1 \/ A2 )
then consider B1, B2 being Subset of X such that
A1: ( A1 c= B1 & A2 c= B2 & B1 /\ B2 misses A1 \/ A2 & B1 is open & B2 is open ) by Th45;
take B1 ; ::_thesis: ex B2 being Subset of X st
( B1,B2 are_weakly_separated & A1 c= B1 & A2 c= B2 & B1 /\ B2 misses A1 \/ A2 )
take B2 ; ::_thesis: ( B1,B2 are_weakly_separated & A1 c= B1 & A2 c= B2 & B1 /\ B2 misses A1 \/ A2 )
thus ( B1,B2 are_weakly_separated & A1 c= B1 & A2 c= B2 & B1 /\ B2 misses A1 \/ A2 ) by A1, Th49; ::_thesis: verum
end;
given B1, B2 being Subset of X such that A2: B1,B2 are_weakly_separated and
A3: A1 c= B1 and
A4: A2 c= B2 and
A5: B1 /\ B2 misses A1 \/ A2 ; ::_thesis: A1,A2 are_separated
B1 /\ B2 misses A1 by A5, XBOOLE_1:7, XBOOLE_1:63;
then A6: (B1 /\ B2) /\ A1 = {} by XBOOLE_0:def_7;
B1 /\ B2 misses A2 by A5, XBOOLE_1:7, XBOOLE_1:63;
then A7: (B1 /\ B2) /\ A2 = {} by XBOOLE_0:def_7;
( B1 /\ A2 c= A2 & B1 /\ A2 c= B1 /\ B2 ) by A4, XBOOLE_1:17, XBOOLE_1:26;
then A8: B1 /\ A2 = {} by A7, XBOOLE_1:3, XBOOLE_1:19;
A2 \ B1 c= B2 \ B1 by A4, XBOOLE_1:33;
then A9: A2 \ (B1 /\ A2) c= B2 \ B1 by XBOOLE_1:47;
( A1 /\ B2 c= A1 & A1 /\ B2 c= B1 /\ B2 ) by A3, XBOOLE_1:17, XBOOLE_1:26;
then A10: A1 /\ B2 = {} by A6, XBOOLE_1:3, XBOOLE_1:19;
A1 \ B2 c= B1 \ B2 by A3, XBOOLE_1:33;
then A11: A1 \ (A1 /\ B2) c= B1 \ B2 by XBOOLE_1:47;
B1 \ B2,B2 \ B1 are_separated by A2, Def5;
hence A1,A2 are_separated by A10, A8, A11, A9, CONNSP_1:7; ::_thesis: verum
end;
begin
definition
let X be TopStruct ;
let X1, X2 be SubSpace of X;
predX1,X2 are_separated means :Def6: :: TSEP_1:def 6
for A1, A2 being Subset of X st A1 = the carrier of X1 & A2 = the carrier of X2 holds
A1,A2 are_separated ;
symmetry
for X1, X2 being SubSpace of X st ( for A1, A2 being Subset of X st A1 = the carrier of X1 & A2 = the carrier of X2 holds
A1,A2 are_separated ) holds
for A1, A2 being Subset of X st A1 = the carrier of X2 & A2 = the carrier of X1 holds
A1,A2 are_separated ;
end;
:: deftheorem Def6 defines are_separated TSEP_1:def_6_:_
for X being TopStruct
for X1, X2 being SubSpace of X holds
( X1,X2 are_separated iff for A1, A2 being Subset of X st A1 = the carrier of X1 & A2 = the carrier of X2 holds
A1,A2 are_separated );
notation
let X be TopStruct ;
let X1, X2 be SubSpace of X;
antonym X1,X2 are_not_separated for X1,X2 are_separated ;
end;
theorem :: TSEP_1:63
for X being non empty TopSpace
for X1, X2 being non empty SubSpace of X st X1,X2 are_separated holds
X1 misses X2
proof
let X be non empty TopSpace; ::_thesis: for X1, X2 being non empty SubSpace of X st X1,X2 are_separated holds
X1 misses X2
let X1, X2 be non empty SubSpace of X; ::_thesis: ( X1,X2 are_separated implies X1 misses X2 )
reconsider A2 = the carrier of X2 as Subset of X by Th1;
reconsider A1 = the carrier of X1 as Subset of X by Th1;
assume X1,X2 are_separated ; ::_thesis: X1 misses X2
then A1,A2 are_separated by Def6;
then A1 misses A2 by CONNSP_1:1;
hence X1 misses X2 by Def3; ::_thesis: verum
end;
theorem :: TSEP_1:64
for X being non empty TopSpace
for X1, X2 being non empty closed SubSpace of X holds
( X1 misses X2 iff X1,X2 are_separated )
proof
let X be non empty TopSpace; ::_thesis: for X1, X2 being non empty closed SubSpace of X holds
( X1 misses X2 iff X1,X2 are_separated )
let X1, X2 be non empty closed SubSpace of X; ::_thesis: ( X1 misses X2 iff X1,X2 are_separated )
reconsider A2 = the carrier of X2 as Subset of X by Th1;
reconsider A1 = the carrier of X1 as Subset of X by Th1;
A1: ( A1 is closed & A2 is closed ) by Th11;
thus ( X1 misses X2 implies X1,X2 are_separated ) ::_thesis: ( X1,X2 are_separated implies X1 misses X2 )
proof
assume X1 misses X2 ; ::_thesis: X1,X2 are_separated
then A1 misses A2 by Def3;
then for A1, A2 being Subset of X st A1 = the carrier of X1 & A2 = the carrier of X2 holds
A1,A2 are_separated by A1, Th34;
hence X1,X2 are_separated by Def6; ::_thesis: verum
end;
assume X1,X2 are_separated ; ::_thesis: X1 misses X2
then A1,A2 are_separated by Def6;
then A1 misses A2 by CONNSP_1:1;
hence X1 misses X2 by Def3; ::_thesis: verum
end;
theorem :: TSEP_1:65
for X being non empty TopSpace
for X1, X2 being non empty SubSpace of X st X = X1 union X2 & X1,X2 are_separated holds
X1 is closed SubSpace of X
proof
let X be non empty TopSpace; ::_thesis: for X1, X2 being non empty SubSpace of X st X = X1 union X2 & X1,X2 are_separated holds
X1 is closed SubSpace of X
let X1, X2 be non empty SubSpace of X; ::_thesis: ( X = X1 union X2 & X1,X2 are_separated implies X1 is closed SubSpace of X )
reconsider A2 = the carrier of X2 as Subset of X by Th1;
reconsider A1 = the carrier of X1 as Subset of X by Th1;
assume X = X1 union X2 ; ::_thesis: ( not X1,X2 are_separated or X1 is closed SubSpace of X )
then A1: A1 \/ A2 = [#] X by Def2;
assume X1,X2 are_separated ; ::_thesis: X1 is closed SubSpace of X
then A1,A2 are_separated by Def6;
then A1 is closed by A1, CONNSP_1:4;
hence X1 is closed SubSpace of X by Th11; ::_thesis: verum
end;
theorem :: TSEP_1:66
for X being non empty TopSpace
for X1, X2 being non empty SubSpace of X st X1 union X2 is closed SubSpace of X & X1,X2 are_separated holds
X1 is closed SubSpace of X
proof
let X be non empty TopSpace; ::_thesis: for X1, X2 being non empty SubSpace of X st X1 union X2 is closed SubSpace of X & X1,X2 are_separated holds
X1 is closed SubSpace of X
let X1, X2 be non empty SubSpace of X; ::_thesis: ( X1 union X2 is closed SubSpace of X & X1,X2 are_separated implies X1 is closed SubSpace of X )
reconsider A2 = the carrier of X2 as Subset of X by Th1;
reconsider A1 = the carrier of X1 as Subset of X by Th1;
assume A1: X1 union X2 is closed SubSpace of X ; ::_thesis: ( not X1,X2 are_separated or X1 is closed SubSpace of X )
assume X1,X2 are_separated ; ::_thesis: X1 is closed SubSpace of X
then A2: A1,A2 are_separated by Def6;
A1 \/ A2 = the carrier of (X1 union X2) by Def2;
then A1 \/ A2 is closed by A1, Th11;
then A1 is closed by A2, Th35;
hence X1 is closed SubSpace of X by Th11; ::_thesis: verum
end;
theorem :: TSEP_1:67
for X being non empty TopSpace
for X1, X2 being non empty open SubSpace of X holds
( X1 misses X2 iff X1,X2 are_separated )
proof
let X be non empty TopSpace; ::_thesis: for X1, X2 being non empty open SubSpace of X holds
( X1 misses X2 iff X1,X2 are_separated )
let X1, X2 be non empty open SubSpace of X; ::_thesis: ( X1 misses X2 iff X1,X2 are_separated )
reconsider A2 = the carrier of X2 as Subset of X by Th1;
reconsider A1 = the carrier of X1 as Subset of X by Th1;
A1: ( A1 is open & A2 is open ) by Th16;
thus ( X1 misses X2 implies X1,X2 are_separated ) ::_thesis: ( X1,X2 are_separated implies X1 misses X2 )
proof
assume X1 misses X2 ; ::_thesis: X1,X2 are_separated
then A1 misses A2 by Def3;
then for A1, A2 being Subset of X st A1 = the carrier of X1 & A2 = the carrier of X2 holds
A1,A2 are_separated by A1, Th37;
hence X1,X2 are_separated by Def6; ::_thesis: verum
end;
assume X1,X2 are_separated ; ::_thesis: X1 misses X2
then A1,A2 are_separated by Def6;
then A1 misses A2 by CONNSP_1:1;
hence X1 misses X2 by Def3; ::_thesis: verum
end;
theorem :: TSEP_1:68
for X being non empty TopSpace
for X1, X2 being non empty SubSpace of X st X = X1 union X2 & X1,X2 are_separated holds
X1 is open SubSpace of X
proof
let X be non empty TopSpace; ::_thesis: for X1, X2 being non empty SubSpace of X st X = X1 union X2 & X1,X2 are_separated holds
X1 is open SubSpace of X
let X1, X2 be non empty SubSpace of X; ::_thesis: ( X = X1 union X2 & X1,X2 are_separated implies X1 is open SubSpace of X )
reconsider A2 = the carrier of X2 as Subset of X by Th1;
reconsider A1 = the carrier of X1 as Subset of X by Th1;
assume X = X1 union X2 ; ::_thesis: ( not X1,X2 are_separated or X1 is open SubSpace of X )
then A1: A1 \/ A2 = [#] X by Def2;
assume X1,X2 are_separated ; ::_thesis: X1 is open SubSpace of X
then A1,A2 are_separated by Def6;
then A1 is open by A1, CONNSP_1:4;
hence X1 is open SubSpace of X by Th16; ::_thesis: verum
end;
theorem :: TSEP_1:69
for X being non empty TopSpace
for X1, X2 being non empty SubSpace of X st X1 union X2 is open SubSpace of X & X1,X2 are_separated holds
X1 is open SubSpace of X
proof
let X be non empty TopSpace; ::_thesis: for X1, X2 being non empty SubSpace of X st X1 union X2 is open SubSpace of X & X1,X2 are_separated holds
X1 is open SubSpace of X
let X1, X2 be non empty SubSpace of X; ::_thesis: ( X1 union X2 is open SubSpace of X & X1,X2 are_separated implies X1 is open SubSpace of X )
reconsider A2 = the carrier of X2 as Subset of X by Th1;
reconsider A1 = the carrier of X1 as Subset of X by Th1;
assume A1: X1 union X2 is open SubSpace of X ; ::_thesis: ( not X1,X2 are_separated or X1 is open SubSpace of X )
assume X1,X2 are_separated ; ::_thesis: X1 is open SubSpace of X
then A2: A1,A2 are_separated by Def6;
A1 \/ A2 = the carrier of (X1 union X2) by Def2;
then A1 \/ A2 is open by A1, Th16;
then A1 is open by A2, Th38;
hence X1 is open SubSpace of X by Th16; ::_thesis: verum
end;
theorem :: TSEP_1:70
for X being non empty TopSpace
for Y, X1, X2 being non empty SubSpace of X st X1 meets Y & X2 meets Y & X1,X2 are_separated holds
( X1 meet Y,X2 meet Y are_separated & Y meet X1,Y meet X2 are_separated )
proof
let X be non empty TopSpace; ::_thesis: for Y, X1, X2 being non empty SubSpace of X st X1 meets Y & X2 meets Y & X1,X2 are_separated holds
( X1 meet Y,X2 meet Y are_separated & Y meet X1,Y meet X2 are_separated )
let Y, X1, X2 be non empty SubSpace of X; ::_thesis: ( X1 meets Y & X2 meets Y & X1,X2 are_separated implies ( X1 meet Y,X2 meet Y are_separated & Y meet X1,Y meet X2 are_separated ) )
assume that
A1: X1 meets Y and
A2: X2 meets Y ; ::_thesis: ( not X1,X2 are_separated or ( X1 meet Y,X2 meet Y are_separated & Y meet X1,Y meet X2 are_separated ) )
reconsider A2 = the carrier of X2 as Subset of X by Th1;
reconsider A1 = the carrier of X1 as Subset of X by Th1;
reconsider C = the carrier of Y as Subset of X by Th1;
assume X1,X2 are_separated ; ::_thesis: ( X1 meet Y,X2 meet Y are_separated & Y meet X1,Y meet X2 are_separated )
then A3: A1,A2 are_separated by Def6;
now__::_thesis:_for_D1,_D2_being_Subset_of_X_st_D1_=_the_carrier_of_(X1_meet_Y)_&_D2_=_the_carrier_of_(X2_meet_Y)_holds_
D1,D2_are_separated
let D1, D2 be Subset of X; ::_thesis: ( D1 = the carrier of (X1 meet Y) & D2 = the carrier of (X2 meet Y) implies D1,D2 are_separated )
assume ( D1 = the carrier of (X1 meet Y) & D2 = the carrier of (X2 meet Y) ) ; ::_thesis: D1,D2 are_separated
then ( A1 /\ C = D1 & A2 /\ C = D2 ) by A1, A2, Def4;
hence D1,D2 are_separated by A3, Th39; ::_thesis: verum
end;
hence X1 meet Y,X2 meet Y are_separated by Def6; ::_thesis: Y meet X1,Y meet X2 are_separated
then X1 meet Y,Y meet X2 are_separated by A2, Th26;
hence Y meet X1,Y meet X2 are_separated by A1, Th26; ::_thesis: verum
end;
theorem Th71: :: TSEP_1:71
for X being non empty TopSpace
for X1, X2 being non empty SubSpace of X
for Y1, Y2 being SubSpace of X st Y1 is SubSpace of X1 & Y2 is SubSpace of X2 & X1,X2 are_separated holds
Y1,Y2 are_separated
proof
let X be non empty TopSpace; ::_thesis: for X1, X2 being non empty SubSpace of X
for Y1, Y2 being SubSpace of X st Y1 is SubSpace of X1 & Y2 is SubSpace of X2 & X1,X2 are_separated holds
Y1,Y2 are_separated
let X1, X2 be non empty SubSpace of X; ::_thesis: for Y1, Y2 being SubSpace of X st Y1 is SubSpace of X1 & Y2 is SubSpace of X2 & X1,X2 are_separated holds
Y1,Y2 are_separated
reconsider A2 = the carrier of X2 as Subset of X by Th1;
reconsider A1 = the carrier of X1 as Subset of X by Th1;
let Y1, Y2 be SubSpace of X; ::_thesis: ( Y1 is SubSpace of X1 & Y2 is SubSpace of X2 & X1,X2 are_separated implies Y1,Y2 are_separated )
assume A1: ( Y1 is SubSpace of X1 & Y2 is SubSpace of X2 ) ; ::_thesis: ( not X1,X2 are_separated or Y1,Y2 are_separated )
assume A2: X1,X2 are_separated ; ::_thesis: Y1,Y2 are_separated
now__::_thesis:_for_B1,_B2_being_Subset_of_X_st_B1_=_the_carrier_of_Y1_&_B2_=_the_carrier_of_Y2_holds_
B1,B2_are_separated
let B1, B2 be Subset of X; ::_thesis: ( B1 = the carrier of Y1 & B2 = the carrier of Y2 implies B1,B2 are_separated )
assume ( B1 = the carrier of Y1 & B2 = the carrier of Y2 ) ; ::_thesis: B1,B2 are_separated
then A3: ( B1 c= A1 & B2 c= A2 ) by A1, Th4;
A1,A2 are_separated by A2, Def6;
hence B1,B2 are_separated by A3, CONNSP_1:7; ::_thesis: verum
end;
hence Y1,Y2 are_separated by Def6; ::_thesis: verum
end;
theorem :: TSEP_1:72
for X being non empty TopSpace
for X1, X2, Y being non empty SubSpace of X st X1 meets X2 & X1,Y are_separated holds
X1 meet X2,Y are_separated
proof
let X be non empty TopSpace; ::_thesis: for X1, X2, Y being non empty SubSpace of X st X1 meets X2 & X1,Y are_separated holds
X1 meet X2,Y are_separated
let X1, X2, Y be non empty SubSpace of X; ::_thesis: ( X1 meets X2 & X1,Y are_separated implies X1 meet X2,Y are_separated )
assume X1 meets X2 ; ::_thesis: ( not X1,Y are_separated or X1 meet X2,Y are_separated )
then A1: X1 meet X2 is SubSpace of X1 by Th27;
Y is SubSpace of Y by Th2;
hence ( not X1,Y are_separated or X1 meet X2,Y are_separated ) by A1, Th71; ::_thesis: verum
end;
theorem :: TSEP_1:73
for X being non empty TopSpace
for X1, X2, Y being non empty SubSpace of X holds
( ( X1,Y are_separated & X2,Y are_separated ) iff X1 union X2,Y are_separated )
proof
let X be non empty TopSpace; ::_thesis: for X1, X2, Y being non empty SubSpace of X holds
( ( X1,Y are_separated & X2,Y are_separated ) iff X1 union X2,Y are_separated )
let X1, X2 be non empty SubSpace of X; ::_thesis: for Y being non empty SubSpace of X holds
( ( X1,Y are_separated & X2,Y are_separated ) iff X1 union X2,Y are_separated )
reconsider A2 = the carrier of X2 as Subset of X by Th1;
reconsider A1 = the carrier of X1 as Subset of X by Th1;
let Y be non empty SubSpace of X; ::_thesis: ( ( X1,Y are_separated & X2,Y are_separated ) iff X1 union X2,Y are_separated )
reconsider C = the carrier of Y as Subset of X by Th1;
A1: Y is SubSpace of Y by Th2;
thus ( X1,Y are_separated & X2,Y are_separated implies X1 union X2,Y are_separated ) ::_thesis: ( X1 union X2,Y are_separated implies ( X1,Y are_separated & X2,Y are_separated ) )
proof
assume ( X1,Y are_separated & X2,Y are_separated ) ; ::_thesis: X1 union X2,Y are_separated
then A2: ( A1,C are_separated & A2,C are_separated ) by Def6;
now__::_thesis:_for_D,_C_being_Subset_of_X_st_D_=_the_carrier_of_(X1_union_X2)_&_C_=_the_carrier_of_Y_holds_
D,C_are_separated
let D, C be Subset of X; ::_thesis: ( D = the carrier of (X1 union X2) & C = the carrier of Y implies D,C are_separated )
assume that
A3: D = the carrier of (X1 union X2) and
A4: C = the carrier of Y ; ::_thesis: D,C are_separated
A1 \/ A2 = D by A3, Def2;
hence D,C are_separated by A2, A4, Th41; ::_thesis: verum
end;
hence X1 union X2,Y are_separated by Def6; ::_thesis: verum
end;
assume A5: X1 union X2,Y are_separated ; ::_thesis: ( X1,Y are_separated & X2,Y are_separated )
( X1 is SubSpace of X1 union X2 & X2 is SubSpace of X1 union X2 ) by Th22;
hence ( X1,Y are_separated & X2,Y are_separated ) by A5, A1, Th71; ::_thesis: verum
end;
theorem :: TSEP_1:74
for X being non empty TopSpace
for X1, X2 being non empty SubSpace of X holds
( X1,X2 are_separated iff ex Y1, Y2 being non empty closed SubSpace of X st
( X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & Y1 misses X2 & Y2 misses X1 ) )
proof
let X be non empty TopSpace; ::_thesis: for X1, X2 being non empty SubSpace of X holds
( X1,X2 are_separated iff ex Y1, Y2 being non empty closed SubSpace of X st
( X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & Y1 misses X2 & Y2 misses X1 ) )
let X1, X2 be non empty SubSpace of X; ::_thesis: ( X1,X2 are_separated iff ex Y1, Y2 being non empty closed SubSpace of X st
( X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & Y1 misses X2 & Y2 misses X1 ) )
reconsider A2 = the carrier of X2 as Subset of X by Th1;
reconsider A1 = the carrier of X1 as Subset of X by Th1;
thus ( X1,X2 are_separated implies ex Y1, Y2 being non empty closed SubSpace of X st
( X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & Y1 misses X2 & Y2 misses X1 ) ) ::_thesis: ( ex Y1, Y2 being non empty closed SubSpace of X st
( X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & Y1 misses X2 & Y2 misses X1 ) implies X1,X2 are_separated )
proof
assume X1,X2 are_separated ; ::_thesis: ex Y1, Y2 being non empty closed SubSpace of X st
( X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & Y1 misses X2 & Y2 misses X1 )
then A1,A2 are_separated by Def6;
then consider C1, C2 being Subset of X such that
A1: A1 c= C1 and
A2: A2 c= C2 and
A3: ( C1 misses A2 & C2 misses A1 ) and
A4: C1 is closed and
A5: C2 is closed by Th42;
not C1 is empty by A1, XBOOLE_1:3;
then consider Y1 being non empty strict closed SubSpace of X such that
A6: C1 = the carrier of Y1 by A4, Th15;
not C2 is empty by A2, XBOOLE_1:3;
then consider Y2 being non empty strict closed SubSpace of X such that
A7: C2 = the carrier of Y2 by A5, Th15;
take Y1 ; ::_thesis: ex Y2 being non empty closed SubSpace of X st
( X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & Y1 misses X2 & Y2 misses X1 )
take Y2 ; ::_thesis: ( X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & Y1 misses X2 & Y2 misses X1 )
thus ( X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & Y1 misses X2 & Y2 misses X1 ) by A1, A2, A3, A6, A7, Def3, Th4; ::_thesis: verum
end;
given Y1, Y2 being non empty closed SubSpace of X such that A8: ( X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & Y1 misses X2 & Y2 misses X1 ) ; ::_thesis: X1,X2 are_separated
now__::_thesis:_for_A1,_A2_being_Subset_of_X_st_A1_=_the_carrier_of_X1_&_A2_=_the_carrier_of_X2_holds_
A1,A2_are_separated
let A1, A2 be Subset of X; ::_thesis: ( A1 = the carrier of X1 & A2 = the carrier of X2 implies A1,A2 are_separated )
assume A9: ( A1 = the carrier of X1 & A2 = the carrier of X2 ) ; ::_thesis: A1,A2 are_separated
ex C1, C2 being Subset of X st
( A1 c= C1 & A2 c= C2 & C1 misses A2 & C2 misses A1 & C1 is closed & C2 is closed )
proof
reconsider C2 = the carrier of Y2 as Subset of X by Th1;
reconsider C1 = the carrier of Y1 as Subset of X by Th1;
take C1 ; ::_thesis: ex C2 being Subset of X st
( A1 c= C1 & A2 c= C2 & C1 misses A2 & C2 misses A1 & C1 is closed & C2 is closed )
take C2 ; ::_thesis: ( A1 c= C1 & A2 c= C2 & C1 misses A2 & C2 misses A1 & C1 is closed & C2 is closed )
thus ( A1 c= C1 & A2 c= C2 & C1 misses A2 & C2 misses A1 & C1 is closed & C2 is closed ) by A8, A9, Def3, Th4, Th11; ::_thesis: verum
end;
hence A1,A2 are_separated by Th42; ::_thesis: verum
end;
hence X1,X2 are_separated by Def6; ::_thesis: verum
end;
theorem :: TSEP_1:75
for X being non empty TopSpace
for X1, X2 being non empty SubSpace of X holds
( X1,X2 are_separated iff ex Y1, Y2 being non empty closed SubSpace of X st
( X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & ( Y1 misses Y2 or Y1 meet Y2 misses X1 union X2 ) ) )
proof
let X be non empty TopSpace; ::_thesis: for X1, X2 being non empty SubSpace of X holds
( X1,X2 are_separated iff ex Y1, Y2 being non empty closed SubSpace of X st
( X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & ( Y1 misses Y2 or Y1 meet Y2 misses X1 union X2 ) ) )
let X1, X2 be non empty SubSpace of X; ::_thesis: ( X1,X2 are_separated iff ex Y1, Y2 being non empty closed SubSpace of X st
( X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & ( Y1 misses Y2 or Y1 meet Y2 misses X1 union X2 ) ) )
reconsider A2 = the carrier of X2 as Subset of X by Th1;
reconsider A1 = the carrier of X1 as Subset of X by Th1;
thus ( X1,X2 are_separated implies ex Y1, Y2 being non empty closed SubSpace of X st
( X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & ( Y1 misses Y2 or Y1 meet Y2 misses X1 union X2 ) ) ) ::_thesis: ( ex Y1, Y2 being non empty closed SubSpace of X st
( X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & ( Y1 misses Y2 or Y1 meet Y2 misses X1 union X2 ) ) implies X1,X2 are_separated )
proof
assume X1,X2 are_separated ; ::_thesis: ex Y1, Y2 being non empty closed SubSpace of X st
( X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & ( Y1 misses Y2 or Y1 meet Y2 misses X1 union X2 ) )
then A1,A2 are_separated by Def6;
then consider C1, C2 being Subset of X such that
A1: A1 c= C1 and
A2: A2 c= C2 and
A3: C1 /\ C2 misses A1 \/ A2 and
A4: C1 is closed and
A5: C2 is closed by Th43;
not C1 is empty by A1, XBOOLE_1:3;
then consider Y1 being non empty strict closed SubSpace of X such that
A6: C1 = the carrier of Y1 by A4, Th15;
not C2 is empty by A2, XBOOLE_1:3;
then consider Y2 being non empty strict closed SubSpace of X such that
A7: C2 = the carrier of Y2 by A5, Th15;
take Y1 ; ::_thesis: ex Y2 being non empty closed SubSpace of X st
( X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & ( Y1 misses Y2 or Y1 meet Y2 misses X1 union X2 ) )
take Y2 ; ::_thesis: ( X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & ( Y1 misses Y2 or Y1 meet Y2 misses X1 union X2 ) )
now__::_thesis:_(_not_Y1_misses_Y2_implies_Y1_meet_Y2_misses_X1_union_X2_)
assume not Y1 misses Y2 ; ::_thesis: Y1 meet Y2 misses X1 union X2
then A8: the carrier of (Y1 meet Y2) = C1 /\ C2 by A6, A7, Def4;
the carrier of (X1 union X2) = A1 \/ A2 by Def2;
hence Y1 meet Y2 misses X1 union X2 by A3, A8, Def3; ::_thesis: verum
end;
hence ( X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & ( Y1 misses Y2 or Y1 meet Y2 misses X1 union X2 ) ) by A1, A2, A6, A7, Th4; ::_thesis: verum
end;
given Y1, Y2 being non empty closed SubSpace of X such that A9: ( X1 is SubSpace of Y1 & X2 is SubSpace of Y2 ) and
A10: ( Y1 misses Y2 or Y1 meet Y2 misses X1 union X2 ) ; ::_thesis: X1,X2 are_separated
now__::_thesis:_for_A1,_A2_being_Subset_of_X_st_A1_=_the_carrier_of_X1_&_A2_=_the_carrier_of_X2_holds_
A1,A2_are_separated
let A1, A2 be Subset of X; ::_thesis: ( A1 = the carrier of X1 & A2 = the carrier of X2 implies A1,A2 are_separated )
assume A11: ( A1 = the carrier of X1 & A2 = the carrier of X2 ) ; ::_thesis: A1,A2 are_separated
ex C1, C2 being Subset of X st
( A1 c= C1 & A2 c= C2 & C1 /\ C2 misses A1 \/ A2 & C1 is closed & C2 is closed )
proof
reconsider C2 = the carrier of Y2 as Subset of X by Th1;
reconsider C1 = the carrier of Y1 as Subset of X by Th1;
take C1 ; ::_thesis: ex C2 being Subset of X st
( A1 c= C1 & A2 c= C2 & C1 /\ C2 misses A1 \/ A2 & C1 is closed & C2 is closed )
take C2 ; ::_thesis: ( A1 c= C1 & A2 c= C2 & C1 /\ C2 misses A1 \/ A2 & C1 is closed & C2 is closed )
now__::_thesis:_C1_/\_C2_misses_A1_\/_A2
percases ( Y1 misses Y2 or not Y1 misses Y2 ) ;
suppose Y1 misses Y2 ; ::_thesis: C1 /\ C2 misses A1 \/ A2
then C1 misses C2 by Def3;
then C1 /\ C2 = {} by XBOOLE_0:def_7;
hence C1 /\ C2 misses A1 \/ A2 by XBOOLE_1:65; ::_thesis: verum
end;
supposeA12: not Y1 misses Y2 ; ::_thesis: C1 /\ C2 misses A1 \/ A2
A13: the carrier of (X1 union X2) = A1 \/ A2 by A11, Def2;
the carrier of (Y1 meet Y2) = C1 /\ C2 by A12, Def4;
hence C1 /\ C2 misses A1 \/ A2 by A10, A12, A13, Def3; ::_thesis: verum
end;
end;
end;
hence ( A1 c= C1 & A2 c= C2 & C1 /\ C2 misses A1 \/ A2 & C1 is closed & C2 is closed ) by A9, A11, Th4, Th11; ::_thesis: verum
end;
hence A1,A2 are_separated by Th43; ::_thesis: verum
end;
hence X1,X2 are_separated by Def6; ::_thesis: verum
end;
theorem :: TSEP_1:76
for X being non empty TopSpace
for X1, X2 being non empty SubSpace of X holds
( X1,X2 are_separated iff ex Y1, Y2 being non empty open SubSpace of X st
( X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & Y1 misses X2 & Y2 misses X1 ) )
proof
let X be non empty TopSpace; ::_thesis: for X1, X2 being non empty SubSpace of X holds
( X1,X2 are_separated iff ex Y1, Y2 being non empty open SubSpace of X st
( X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & Y1 misses X2 & Y2 misses X1 ) )
let X1, X2 be non empty SubSpace of X; ::_thesis: ( X1,X2 are_separated iff ex Y1, Y2 being non empty open SubSpace of X st
( X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & Y1 misses X2 & Y2 misses X1 ) )
reconsider A2 = the carrier of X2 as Subset of X by Th1;
reconsider A1 = the carrier of X1 as Subset of X by Th1;
thus ( X1,X2 are_separated implies ex Y1, Y2 being non empty open SubSpace of X st
( X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & Y1 misses X2 & Y2 misses X1 ) ) ::_thesis: ( ex Y1, Y2 being non empty open SubSpace of X st
( X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & Y1 misses X2 & Y2 misses X1 ) implies X1,X2 are_separated )
proof
assume X1,X2 are_separated ; ::_thesis: ex Y1, Y2 being non empty open SubSpace of X st
( X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & Y1 misses X2 & Y2 misses X1 )
then A1,A2 are_separated by Def6;
then consider C1, C2 being Subset of X such that
A1: A1 c= C1 and
A2: A2 c= C2 and
A3: ( C1 misses A2 & C2 misses A1 ) and
A4: C1 is open and
A5: C2 is open by Th44;
not C1 is empty by A1, XBOOLE_1:3;
then consider Y1 being non empty strict open SubSpace of X such that
A6: C1 = the carrier of Y1 by A4, Th20;
not C2 is empty by A2, XBOOLE_1:3;
then consider Y2 being non empty strict open SubSpace of X such that
A7: C2 = the carrier of Y2 by A5, Th20;
take Y1 ; ::_thesis: ex Y2 being non empty open SubSpace of X st
( X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & Y1 misses X2 & Y2 misses X1 )
take Y2 ; ::_thesis: ( X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & Y1 misses X2 & Y2 misses X1 )
thus ( X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & Y1 misses X2 & Y2 misses X1 ) by A1, A2, A3, A6, A7, Def3, Th4; ::_thesis: verum
end;
given Y1, Y2 being non empty open SubSpace of X such that A8: ( X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & Y1 misses X2 & Y2 misses X1 ) ; ::_thesis: X1,X2 are_separated
now__::_thesis:_for_A1,_A2_being_Subset_of_X_st_A1_=_the_carrier_of_X1_&_A2_=_the_carrier_of_X2_holds_
A1,A2_are_separated
let A1, A2 be Subset of X; ::_thesis: ( A1 = the carrier of X1 & A2 = the carrier of X2 implies A1,A2 are_separated )
assume A9: ( A1 = the carrier of X1 & A2 = the carrier of X2 ) ; ::_thesis: A1,A2 are_separated
ex C1, C2 being Subset of X st
( A1 c= C1 & A2 c= C2 & C1 misses A2 & C2 misses A1 & C1 is open & C2 is open )
proof
reconsider C2 = the carrier of Y2 as Subset of X by Th1;
reconsider C1 = the carrier of Y1 as Subset of X by Th1;
take C1 ; ::_thesis: ex C2 being Subset of X st
( A1 c= C1 & A2 c= C2 & C1 misses A2 & C2 misses A1 & C1 is open & C2 is open )
take C2 ; ::_thesis: ( A1 c= C1 & A2 c= C2 & C1 misses A2 & C2 misses A1 & C1 is open & C2 is open )
thus ( A1 c= C1 & A2 c= C2 & C1 misses A2 & C2 misses A1 & C1 is open & C2 is open ) by A8, A9, Def3, Th4, Th16; ::_thesis: verum
end;
hence A1,A2 are_separated by Th44; ::_thesis: verum
end;
hence X1,X2 are_separated by Def6; ::_thesis: verum
end;
theorem Th77: :: TSEP_1:77
for X being non empty TopSpace
for X1, X2 being non empty SubSpace of X holds
( X1,X2 are_separated iff ex Y1, Y2 being non empty open SubSpace of X st
( X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & ( Y1 misses Y2 or Y1 meet Y2 misses X1 union X2 ) ) )
proof
let X be non empty TopSpace; ::_thesis: for X1, X2 being non empty SubSpace of X holds
( X1,X2 are_separated iff ex Y1, Y2 being non empty open SubSpace of X st
( X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & ( Y1 misses Y2 or Y1 meet Y2 misses X1 union X2 ) ) )
let X1, X2 be non empty SubSpace of X; ::_thesis: ( X1,X2 are_separated iff ex Y1, Y2 being non empty open SubSpace of X st
( X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & ( Y1 misses Y2 or Y1 meet Y2 misses X1 union X2 ) ) )
reconsider A2 = the carrier of X2 as Subset of X by Th1;
reconsider A1 = the carrier of X1 as Subset of X by Th1;
thus ( X1,X2 are_separated implies ex Y1, Y2 being non empty open SubSpace of X st
( X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & ( Y1 misses Y2 or Y1 meet Y2 misses X1 union X2 ) ) ) ::_thesis: ( ex Y1, Y2 being non empty open SubSpace of X st
( X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & ( Y1 misses Y2 or Y1 meet Y2 misses X1 union X2 ) ) implies X1,X2 are_separated )
proof
assume X1,X2 are_separated ; ::_thesis: ex Y1, Y2 being non empty open SubSpace of X st
( X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & ( Y1 misses Y2 or Y1 meet Y2 misses X1 union X2 ) )
then A1,A2 are_separated by Def6;
then consider C1, C2 being Subset of X such that
A1: A1 c= C1 and
A2: A2 c= C2 and
A3: C1 /\ C2 misses A1 \/ A2 and
A4: C1 is open and
A5: C2 is open by Th45;
not C1 is empty by A1, XBOOLE_1:3;
then consider Y1 being non empty strict open SubSpace of X such that
A6: C1 = the carrier of Y1 by A4, Th20;
not C2 is empty by A2, XBOOLE_1:3;
then consider Y2 being non empty strict open SubSpace of X such that
A7: C2 = the carrier of Y2 by A5, Th20;
take Y1 ; ::_thesis: ex Y2 being non empty open SubSpace of X st
( X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & ( Y1 misses Y2 or Y1 meet Y2 misses X1 union X2 ) )
take Y2 ; ::_thesis: ( X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & ( Y1 misses Y2 or Y1 meet Y2 misses X1 union X2 ) )
now__::_thesis:_(_not_Y1_misses_Y2_implies_Y1_meet_Y2_misses_X1_union_X2_)
assume not Y1 misses Y2 ; ::_thesis: Y1 meet Y2 misses X1 union X2
then A8: the carrier of (Y1 meet Y2) = C1 /\ C2 by A6, A7, Def4;
the carrier of (X1 union X2) = A1 \/ A2 by Def2;
hence Y1 meet Y2 misses X1 union X2 by A3, A8, Def3; ::_thesis: verum
end;
hence ( X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & ( Y1 misses Y2 or Y1 meet Y2 misses X1 union X2 ) ) by A1, A2, A6, A7, Th4; ::_thesis: verum
end;
given Y1, Y2 being non empty open SubSpace of X such that A9: ( X1 is SubSpace of Y1 & X2 is SubSpace of Y2 ) and
A10: ( Y1 misses Y2 or Y1 meet Y2 misses X1 union X2 ) ; ::_thesis: X1,X2 are_separated
now__::_thesis:_for_A1,_A2_being_Subset_of_X_st_A1_=_the_carrier_of_X1_&_A2_=_the_carrier_of_X2_holds_
A1,A2_are_separated
let A1, A2 be Subset of X; ::_thesis: ( A1 = the carrier of X1 & A2 = the carrier of X2 implies A1,A2 are_separated )
assume A11: ( A1 = the carrier of X1 & A2 = the carrier of X2 ) ; ::_thesis: A1,A2 are_separated
ex C1, C2 being Subset of X st
( A1 c= C1 & A2 c= C2 & C1 /\ C2 misses A1 \/ A2 & C1 is open & C2 is open )
proof
reconsider C2 = the carrier of Y2 as Subset of X by Th1;
reconsider C1 = the carrier of Y1 as Subset of X by Th1;
take C1 ; ::_thesis: ex C2 being Subset of X st
( A1 c= C1 & A2 c= C2 & C1 /\ C2 misses A1 \/ A2 & C1 is open & C2 is open )
take C2 ; ::_thesis: ( A1 c= C1 & A2 c= C2 & C1 /\ C2 misses A1 \/ A2 & C1 is open & C2 is open )
now__::_thesis:_C1_/\_C2_misses_A1_\/_A2
percases ( Y1 misses Y2 or not Y1 misses Y2 ) ;
suppose Y1 misses Y2 ; ::_thesis: C1 /\ C2 misses A1 \/ A2
then C1 misses C2 by Def3;
then C1 /\ C2 = {} by XBOOLE_0:def_7;
hence C1 /\ C2 misses A1 \/ A2 by XBOOLE_1:65; ::_thesis: verum
end;
supposeA12: not Y1 misses Y2 ; ::_thesis: C1 /\ C2 misses A1 \/ A2
A13: the carrier of (X1 union X2) = A1 \/ A2 by A11, Def2;
the carrier of (Y1 meet Y2) = C1 /\ C2 by A12, Def4;
hence C1 /\ C2 misses A1 \/ A2 by A10, A12, A13, Def3; ::_thesis: verum
end;
end;
end;
hence ( A1 c= C1 & A2 c= C2 & C1 /\ C2 misses A1 \/ A2 & C1 is open & C2 is open ) by A9, A11, Th4, Th16; ::_thesis: verum
end;
hence A1,A2 are_separated by Th45; ::_thesis: verum
end;
hence X1,X2 are_separated by Def6; ::_thesis: verum
end;
definition
let X be TopStruct ;
let X1, X2 be SubSpace of X;
predX1,X2 are_weakly_separated means :Def7: :: TSEP_1:def 7
for A1, A2 being Subset of X st A1 = the carrier of X1 & A2 = the carrier of X2 holds
A1,A2 are_weakly_separated ;
symmetry
for X1, X2 being SubSpace of X st ( for A1, A2 being Subset of X st A1 = the carrier of X1 & A2 = the carrier of X2 holds
A1,A2 are_weakly_separated ) holds
for A1, A2 being Subset of X st A1 = the carrier of X2 & A2 = the carrier of X1 holds
A1,A2 are_weakly_separated ;
end;
:: deftheorem Def7 defines are_weakly_separated TSEP_1:def_7_:_
for X being TopStruct
for X1, X2 being SubSpace of X holds
( X1,X2 are_weakly_separated iff for A1, A2 being Subset of X st A1 = the carrier of X1 & A2 = the carrier of X2 holds
A1,A2 are_weakly_separated );
notation
let X be TopStruct ;
let X1, X2 be SubSpace of X;
antonym X1,X2 are_not_weakly_separated for X1,X2 are_weakly_separated ;
end;
theorem Th78: :: TSEP_1:78
for X being non empty TopSpace
for X1, X2 being non empty SubSpace of X holds
( ( X1 misses X2 & X1,X2 are_weakly_separated ) iff X1,X2 are_separated )
proof
let X be non empty TopSpace; ::_thesis: for X1, X2 being non empty SubSpace of X holds
( ( X1 misses X2 & X1,X2 are_weakly_separated ) iff X1,X2 are_separated )
let X1, X2 be non empty SubSpace of X; ::_thesis: ( ( X1 misses X2 & X1,X2 are_weakly_separated ) iff X1,X2 are_separated )
reconsider A2 = the carrier of X2 as Subset of X by Th1;
reconsider A1 = the carrier of X1 as Subset of X by Th1;
thus ( X1 misses X2 & X1,X2 are_weakly_separated implies X1,X2 are_separated ) ::_thesis: ( X1,X2 are_separated implies ( X1 misses X2 & X1,X2 are_weakly_separated ) )
proof
assume ( X1 misses X2 & X1,X2 are_weakly_separated ) ; ::_thesis: X1,X2 are_separated
then ( A1 misses A2 & A1,A2 are_weakly_separated ) by Def3, Def7;
then for A1, A2 being Subset of X st A1 = the carrier of X1 & A2 = the carrier of X2 holds
A1,A2 are_separated by Th46;
hence X1,X2 are_separated by Def6; ::_thesis: verum
end;
assume X1,X2 are_separated ; ::_thesis: ( X1 misses X2 & X1,X2 are_weakly_separated )
then A1: A1,A2 are_separated by Def6;
then A1 misses A2 by Th46;
hence X1 misses X2 by Def3; ::_thesis: X1,X2 are_weakly_separated
for A1, A2 being Subset of X st A1 = the carrier of X1 & A2 = the carrier of X2 holds
A1,A2 are_weakly_separated by A1, Th46;
hence X1,X2 are_weakly_separated by Def7; ::_thesis: verum
end;
theorem Th79: :: TSEP_1:79
for X being non empty TopSpace
for X1, X2 being non empty SubSpace of X st X1 is SubSpace of X2 holds
X1,X2 are_weakly_separated
proof
let X be non empty TopSpace; ::_thesis: for X1, X2 being non empty SubSpace of X st X1 is SubSpace of X2 holds
X1,X2 are_weakly_separated
let X1, X2 be non empty SubSpace of X; ::_thesis: ( X1 is SubSpace of X2 implies X1,X2 are_weakly_separated )
A1: now__::_thesis:_(_X1_is_SubSpace_of_X2_&_X1_is_SubSpace_of_X2_implies_X1,X2_are_weakly_separated_)
assume A2: X1 is SubSpace of X2 ; ::_thesis: ( X1 is SubSpace of X2 implies X1,X2 are_weakly_separated )
now__::_thesis:_for_A1,_A2_being_Subset_of_X_st_A1_=_the_carrier_of_X1_&_A2_=_the_carrier_of_X2_holds_
A1,A2_are_weakly_separated
let A1, A2 be Subset of X; ::_thesis: ( A1 = the carrier of X1 & A2 = the carrier of X2 implies A1,A2 are_weakly_separated )
assume ( A1 = the carrier of X1 & A2 = the carrier of X2 ) ; ::_thesis: A1,A2 are_weakly_separated
then A1 c= A2 by A2, Th4;
hence A1,A2 are_weakly_separated by Th47; ::_thesis: verum
end;
hence ( X1 is SubSpace of X2 implies X1,X2 are_weakly_separated ) by Def7; ::_thesis: verum
end;
assume X1 is SubSpace of X2 ; ::_thesis: X1,X2 are_weakly_separated
hence X1,X2 are_weakly_separated by A1; ::_thesis: verum
end;
theorem Th80: :: TSEP_1:80
for X being non empty TopSpace
for X1, X2 being closed SubSpace of X holds X1,X2 are_weakly_separated
proof
let X be non empty TopSpace; ::_thesis: for X1, X2 being closed SubSpace of X holds X1,X2 are_weakly_separated
let X1, X2 be closed SubSpace of X; ::_thesis: X1,X2 are_weakly_separated
now__::_thesis:_for_A1,_A2_being_Subset_of_X_st_A1_=_the_carrier_of_X1_&_A2_=_the_carrier_of_X2_holds_
A1,A2_are_weakly_separated
let A1, A2 be Subset of X; ::_thesis: ( A1 = the carrier of X1 & A2 = the carrier of X2 implies A1,A2 are_weakly_separated )
reconsider B1 = A1, B2 = A2 as Subset of X ;
assume ( A1 = the carrier of X1 & A2 = the carrier of X2 ) ; ::_thesis: A1,A2 are_weakly_separated
then ( B1 is closed & B2 is closed ) by Th11;
hence A1,A2 are_weakly_separated by Th48; ::_thesis: verum
end;
hence X1,X2 are_weakly_separated by Def7; ::_thesis: verum
end;
theorem Th81: :: TSEP_1:81
for X being non empty TopSpace
for X1, X2 being open SubSpace of X holds X1,X2 are_weakly_separated
proof
let X be non empty TopSpace; ::_thesis: for X1, X2 being open SubSpace of X holds X1,X2 are_weakly_separated
let X1, X2 be open SubSpace of X; ::_thesis: X1,X2 are_weakly_separated
now__::_thesis:_for_A1,_A2_being_Subset_of_X_st_A1_=_the_carrier_of_X1_&_A2_=_the_carrier_of_X2_holds_
A1,A2_are_weakly_separated
let A1, A2 be Subset of X; ::_thesis: ( A1 = the carrier of X1 & A2 = the carrier of X2 implies A1,A2 are_weakly_separated )
reconsider B1 = A1, B2 = A2 as Subset of X ;
assume ( A1 = the carrier of X1 & A2 = the carrier of X2 ) ; ::_thesis: A1,A2 are_weakly_separated
then ( B1 is open & B2 is open ) by Th16;
hence A1,A2 are_weakly_separated by Th49; ::_thesis: verum
end;
hence X1,X2 are_weakly_separated by Def7; ::_thesis: verum
end;
theorem :: TSEP_1:82
for X being non empty TopSpace
for X1, X2, Y being non empty SubSpace of X st X1,X2 are_weakly_separated holds
X1 union Y,X2 union Y are_weakly_separated
proof
let X be non empty TopSpace; ::_thesis: for X1, X2, Y being non empty SubSpace of X st X1,X2 are_weakly_separated holds
X1 union Y,X2 union Y are_weakly_separated
let X1, X2 be non empty SubSpace of X; ::_thesis: for Y being non empty SubSpace of X st X1,X2 are_weakly_separated holds
X1 union Y,X2 union Y are_weakly_separated
reconsider A2 = the carrier of X2 as Subset of X by Th1;
reconsider A1 = the carrier of X1 as Subset of X by Th1;
let Y be non empty SubSpace of X; ::_thesis: ( X1,X2 are_weakly_separated implies X1 union Y,X2 union Y are_weakly_separated )
reconsider C = the carrier of Y as Subset of X by Th1;
assume X1,X2 are_weakly_separated ; ::_thesis: X1 union Y,X2 union Y are_weakly_separated
then A1: A1,A2 are_weakly_separated by Def7;
now__::_thesis:_for_D1,_D2_being_Subset_of_X_st_D1_=_the_carrier_of_(X1_union_Y)_&_D2_=_the_carrier_of_(X2_union_Y)_holds_
D1,D2_are_weakly_separated
let D1, D2 be Subset of X; ::_thesis: ( D1 = the carrier of (X1 union Y) & D2 = the carrier of (X2 union Y) implies D1,D2 are_weakly_separated )
assume ( D1 = the carrier of (X1 union Y) & D2 = the carrier of (X2 union Y) ) ; ::_thesis: D1,D2 are_weakly_separated
then ( A1 \/ C = D1 & A2 \/ C = D2 ) by Def2;
hence D1,D2 are_weakly_separated by A1, Th50; ::_thesis: verum
end;
hence X1 union Y,X2 union Y are_weakly_separated by Def7; ::_thesis: verum
end;
theorem :: TSEP_1:83
for X being non empty TopSpace
for X2, X1, Y1, Y2 being non empty SubSpace of X st Y1 is SubSpace of X2 & Y2 is SubSpace of X1 & X1,X2 are_weakly_separated holds
( X1 union Y1,X2 union Y2 are_weakly_separated & Y1 union X1,Y2 union X2 are_weakly_separated )
proof
let X be non empty TopSpace; ::_thesis: for X2, X1, Y1, Y2 being non empty SubSpace of X st Y1 is SubSpace of X2 & Y2 is SubSpace of X1 & X1,X2 are_weakly_separated holds
( X1 union Y1,X2 union Y2 are_weakly_separated & Y1 union X1,Y2 union X2 are_weakly_separated )
let X2, X1 be non empty SubSpace of X; ::_thesis: for Y1, Y2 being non empty SubSpace of X st Y1 is SubSpace of X2 & Y2 is SubSpace of X1 & X1,X2 are_weakly_separated holds
( X1 union Y1,X2 union Y2 are_weakly_separated & Y1 union X1,Y2 union X2 are_weakly_separated )
reconsider A2 = the carrier of X2 as Subset of X by Th1;
reconsider A1 = the carrier of X1 as Subset of X by Th1;
let Y1, Y2 be non empty SubSpace of X; ::_thesis: ( Y1 is SubSpace of X2 & Y2 is SubSpace of X1 & X1,X2 are_weakly_separated implies ( X1 union Y1,X2 union Y2 are_weakly_separated & Y1 union X1,Y2 union X2 are_weakly_separated ) )
assume A1: ( Y1 is SubSpace of X2 & Y2 is SubSpace of X1 ) ; ::_thesis: ( not X1,X2 are_weakly_separated or ( X1 union Y1,X2 union Y2 are_weakly_separated & Y1 union X1,Y2 union X2 are_weakly_separated ) )
reconsider B2 = the carrier of Y2 as Subset of X by Th1;
reconsider B1 = the carrier of Y1 as Subset of X by Th1;
assume X1,X2 are_weakly_separated ; ::_thesis: ( X1 union Y1,X2 union Y2 are_weakly_separated & Y1 union X1,Y2 union X2 are_weakly_separated )
then A2: A1,A2 are_weakly_separated by Def7;
A3: ( B1 c= A2 & B2 c= A1 ) by A1, Th4;
A4: now__::_thesis:_for_D1,_D2_being_Subset_of_X_st_D1_=_the_carrier_of_(X1_union_Y1)_&_D2_=_the_carrier_of_(X2_union_Y2)_holds_
D1,D2_are_weakly_separated
let D1, D2 be Subset of X; ::_thesis: ( D1 = the carrier of (X1 union Y1) & D2 = the carrier of (X2 union Y2) implies D1,D2 are_weakly_separated )
assume ( D1 = the carrier of (X1 union Y1) & D2 = the carrier of (X2 union Y2) ) ; ::_thesis: D1,D2 are_weakly_separated
then ( A1 \/ B1 = D1 & A2 \/ B2 = D2 ) by Def2;
hence D1,D2 are_weakly_separated by A3, A2, Th51; ::_thesis: verum
end;
hence X1 union Y1,X2 union Y2 are_weakly_separated by Def7; ::_thesis: Y1 union X1,Y2 union X2 are_weakly_separated
thus Y1 union X1,Y2 union X2 are_weakly_separated by A4, Def7; ::_thesis: verum
end;
theorem :: TSEP_1:84
for X being non empty TopSpace
for Y, X1, X2 being non empty SubSpace of X st X1 meets X2 holds
( ( X1,Y are_weakly_separated & X2,Y are_weakly_separated implies X1 meet X2,Y are_weakly_separated ) & ( Y,X1 are_weakly_separated & Y,X2 are_weakly_separated implies Y,X1 meet X2 are_weakly_separated ) )
proof
let X be non empty TopSpace; ::_thesis: for Y, X1, X2 being non empty SubSpace of X st X1 meets X2 holds
( ( X1,Y are_weakly_separated & X2,Y are_weakly_separated implies X1 meet X2,Y are_weakly_separated ) & ( Y,X1 are_weakly_separated & Y,X2 are_weakly_separated implies Y,X1 meet X2 are_weakly_separated ) )
let Y, X1, X2 be non empty SubSpace of X; ::_thesis: ( X1 meets X2 implies ( ( X1,Y are_weakly_separated & X2,Y are_weakly_separated implies X1 meet X2,Y are_weakly_separated ) & ( Y,X1 are_weakly_separated & Y,X2 are_weakly_separated implies Y,X1 meet X2 are_weakly_separated ) ) )
assume A1: X1 meets X2 ; ::_thesis: ( ( X1,Y are_weakly_separated & X2,Y are_weakly_separated implies X1 meet X2,Y are_weakly_separated ) & ( Y,X1 are_weakly_separated & Y,X2 are_weakly_separated implies Y,X1 meet X2 are_weakly_separated ) )
reconsider A2 = the carrier of X2 as Subset of X by Th1;
reconsider A1 = the carrier of X1 as Subset of X by Th1;
reconsider C = the carrier of Y as Subset of X by Th1;
thus ( X1,Y are_weakly_separated & X2,Y are_weakly_separated implies X1 meet X2,Y are_weakly_separated ) ::_thesis: ( Y,X1 are_weakly_separated & Y,X2 are_weakly_separated implies Y,X1 meet X2 are_weakly_separated )
proof
assume ( X1,Y are_weakly_separated & X2,Y are_weakly_separated ) ; ::_thesis: X1 meet X2,Y are_weakly_separated
then A2: ( A1,C are_weakly_separated & A2,C are_weakly_separated ) by Def7;
now__::_thesis:_for_D,_C_being_Subset_of_X_st_D_=_the_carrier_of_(X1_meet_X2)_&_C_=_the_carrier_of_Y_holds_
D,C_are_weakly_separated
let D, C be Subset of X; ::_thesis: ( D = the carrier of (X1 meet X2) & C = the carrier of Y implies D,C are_weakly_separated )
assume that
A3: D = the carrier of (X1 meet X2) and
A4: C = the carrier of Y ; ::_thesis: D,C are_weakly_separated
A1 /\ A2 = D by A1, A3, Def4;
hence D,C are_weakly_separated by A2, A4, Th52; ::_thesis: verum
end;
hence X1 meet X2,Y are_weakly_separated by Def7; ::_thesis: verum
end;
hence ( Y,X1 are_weakly_separated & Y,X2 are_weakly_separated implies Y,X1 meet X2 are_weakly_separated ) ; ::_thesis: verum
end;
theorem :: TSEP_1:85
for X being non empty TopSpace
for X1, X2, Y being non empty SubSpace of X holds
( ( X1,Y are_weakly_separated & X2,Y are_weakly_separated implies X1 union X2,Y are_weakly_separated ) & ( Y,X1 are_weakly_separated & Y,X2 are_weakly_separated implies Y,X1 union X2 are_weakly_separated ) )
proof
let X be non empty TopSpace; ::_thesis: for X1, X2, Y being non empty SubSpace of X holds
( ( X1,Y are_weakly_separated & X2,Y are_weakly_separated implies X1 union X2,Y are_weakly_separated ) & ( Y,X1 are_weakly_separated & Y,X2 are_weakly_separated implies Y,X1 union X2 are_weakly_separated ) )
let X1, X2 be non empty SubSpace of X; ::_thesis: for Y being non empty SubSpace of X holds
( ( X1,Y are_weakly_separated & X2,Y are_weakly_separated implies X1 union X2,Y are_weakly_separated ) & ( Y,X1 are_weakly_separated & Y,X2 are_weakly_separated implies Y,X1 union X2 are_weakly_separated ) )
reconsider A2 = the carrier of X2 as Subset of X by Th1;
reconsider A1 = the carrier of X1 as Subset of X by Th1;
let Y be non empty SubSpace of X; ::_thesis: ( ( X1,Y are_weakly_separated & X2,Y are_weakly_separated implies X1 union X2,Y are_weakly_separated ) & ( Y,X1 are_weakly_separated & Y,X2 are_weakly_separated implies Y,X1 union X2 are_weakly_separated ) )
reconsider C = the carrier of Y as Subset of X by Th1;
thus ( X1,Y are_weakly_separated & X2,Y are_weakly_separated implies X1 union X2,Y are_weakly_separated ) ::_thesis: ( Y,X1 are_weakly_separated & Y,X2 are_weakly_separated implies Y,X1 union X2 are_weakly_separated )
proof
assume ( X1,Y are_weakly_separated & X2,Y are_weakly_separated ) ; ::_thesis: X1 union X2,Y are_weakly_separated
then A1: ( A1,C are_weakly_separated & A2,C are_weakly_separated ) by Def7;
now__::_thesis:_for_D,_C_being_Subset_of_X_st_D_=_the_carrier_of_(X1_union_X2)_&_C_=_the_carrier_of_Y_holds_
D,C_are_weakly_separated
let D, C be Subset of X; ::_thesis: ( D = the carrier of (X1 union X2) & C = the carrier of Y implies D,C are_weakly_separated )
assume that
A2: D = the carrier of (X1 union X2) and
A3: C = the carrier of Y ; ::_thesis: D,C are_weakly_separated
A1 \/ A2 = D by A2, Def2;
hence D,C are_weakly_separated by A1, A3, Th53; ::_thesis: verum
end;
hence X1 union X2,Y are_weakly_separated by Def7; ::_thesis: verum
end;
hence ( Y,X1 are_weakly_separated & Y,X2 are_weakly_separated implies Y,X1 union X2 are_weakly_separated ) ; ::_thesis: verum
end;
theorem :: TSEP_1:86
for X being non empty TopSpace
for X1, X2 being non empty SubSpace of X st X1 meets X2 holds
( X1,X2 are_weakly_separated iff ( X1 is SubSpace of X2 or X2 is SubSpace of X1 or ex Y1, Y2 being non empty closed SubSpace of X st
( Y1 meet (X1 union X2) is SubSpace of X1 & Y2 meet (X1 union X2) is SubSpace of X2 & ( X1 union X2 is SubSpace of Y1 union Y2 or ex Y being non empty open SubSpace of X st
( TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union Y & Y meet (X1 union X2) is SubSpace of X1 meet X2 ) ) ) ) )
proof
let X be non empty TopSpace; ::_thesis: for X1, X2 being non empty SubSpace of X st X1 meets X2 holds
( X1,X2 are_weakly_separated iff ( X1 is SubSpace of X2 or X2 is SubSpace of X1 or ex Y1, Y2 being non empty closed SubSpace of X st
( Y1 meet (X1 union X2) is SubSpace of X1 & Y2 meet (X1 union X2) is SubSpace of X2 & ( X1 union X2 is SubSpace of Y1 union Y2 or ex Y being non empty open SubSpace of X st
( TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union Y & Y meet (X1 union X2) is SubSpace of X1 meet X2 ) ) ) ) )
let X1, X2 be non empty SubSpace of X; ::_thesis: ( X1 meets X2 implies ( X1,X2 are_weakly_separated iff ( X1 is SubSpace of X2 or X2 is SubSpace of X1 or ex Y1, Y2 being non empty closed SubSpace of X st
( Y1 meet (X1 union X2) is SubSpace of X1 & Y2 meet (X1 union X2) is SubSpace of X2 & ( X1 union X2 is SubSpace of Y1 union Y2 or ex Y being non empty open SubSpace of X st
( TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union Y & Y meet (X1 union X2) is SubSpace of X1 meet X2 ) ) ) ) ) )
reconsider A2 = the carrier of X2 as Subset of X by Th1;
reconsider A1 = the carrier of X1 as Subset of X by Th1;
assume A1: X1 meets X2 ; ::_thesis: ( X1,X2 are_weakly_separated iff ( X1 is SubSpace of X2 or X2 is SubSpace of X1 or ex Y1, Y2 being non empty closed SubSpace of X st
( Y1 meet (X1 union X2) is SubSpace of X1 & Y2 meet (X1 union X2) is SubSpace of X2 & ( X1 union X2 is SubSpace of Y1 union Y2 or ex Y being non empty open SubSpace of X st
( TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union Y & Y meet (X1 union X2) is SubSpace of X1 meet X2 ) ) ) ) )
A2: [#] X = the carrier of X ;
A3: now__::_thesis:_(_(_X1_is_SubSpace_of_X2_or_X2_is_SubSpace_of_X1_or_ex_Y1,_Y2_being_non_empty_closed_SubSpace_of_X_st_
(_Y1_meet_(X1_union_X2)_is_SubSpace_of_X1_&_Y2_meet_(X1_union_X2)_is_SubSpace_of_X2_&_(_X1_union_X2_is_SubSpace_of_Y1_union_Y2_or_ex_Y_being_non_empty_open_SubSpace_of_X_st_
(_TopStruct(#_the_carrier_of_X,_the_topology_of_X_#)_=_(Y1_union_Y2)_union_Y_&_Y_meet_(X1_union_X2)_is_SubSpace_of_X1_meet_X2_)_)_)_)_implies_X1,X2_are_weakly_separated_)
assume A4: ( X1 is SubSpace of X2 or X2 is SubSpace of X1 or ex Y1, Y2 being non empty closed SubSpace of X st
( Y1 meet (X1 union X2) is SubSpace of X1 & Y2 meet (X1 union X2) is SubSpace of X2 & ( X1 union X2 is SubSpace of Y1 union Y2 or ex Y being non empty open SubSpace of X st
( TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union Y & Y meet (X1 union X2) is SubSpace of X1 meet X2 ) ) ) ) ; ::_thesis: X1,X2 are_weakly_separated
now__::_thesis:_(_X1_is_not_SubSpace_of_X2_&_X2_is_not_SubSpace_of_X1_implies_X1,X2_are_weakly_separated_)
assume that
A5: X1 is not SubSpace of X2 and
A6: X2 is not SubSpace of X1 ; ::_thesis: X1,X2 are_weakly_separated
consider Y1, Y2 being non empty closed SubSpace of X such that
A7: Y1 meet (X1 union X2) is SubSpace of X1 and
A8: Y2 meet (X1 union X2) is SubSpace of X2 and
A9: ( X1 union X2 is SubSpace of Y1 union Y2 or ex Y being non empty open SubSpace of X st
( TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union Y & Y meet (X1 union X2) is SubSpace of X1 meet X2 ) ) by A4, A5, A6;
reconsider C2 = the carrier of Y2 as Subset of X by Th1;
reconsider C1 = the carrier of Y1 as Subset of X by Th1;
A10: the carrier of (X1 union X2) = A1 \/ A2 by Def2;
A11: the carrier of (Y1 union Y2) = C1 \/ C2 by Def2;
now__::_thesis:_not_Y1_misses_X1_union_X2
assume Y1 misses X1 union X2 ; ::_thesis: contradiction
then A12: C1 misses A1 \/ A2 by A10, Def3;
A13: now__::_thesis:_A1_\/_A2_c=_A2
percases ( X1 union X2 is SubSpace of Y1 union Y2 or not X1 union X2 is SubSpace of Y1 union Y2 ) ;
suppose X1 union X2 is SubSpace of Y1 union Y2 ; ::_thesis: A1 \/ A2 c= A2
then A1 \/ A2 c= C1 \/ C2 by A10, A11, Th4;
then A14: A1 \/ A2 = (C1 \/ C2) /\ (A1 \/ A2) by XBOOLE_1:28
.= (C1 /\ (A1 \/ A2)) \/ (C2 /\ (A1 \/ A2)) by XBOOLE_1:23
.= {} \/ (C2 /\ (A1 \/ A2)) by A12, XBOOLE_0:def_7
.= C2 /\ (A1 \/ A2) ;
then C2 meets A1 \/ A2 by XBOOLE_0:def_7;
then Y2 meets X1 union X2 by A10, Def3;
then the carrier of (Y2 meet (X1 union X2)) = C2 /\ (A1 \/ A2) by A10, Def4;
hence A1 \/ A2 c= A2 by A8, A14, Th4; ::_thesis: verum
end;
suppose X1 union X2 is not SubSpace of Y1 union Y2 ; ::_thesis: A1 \/ A2 c= A2
then consider Y being non empty open SubSpace of X such that
A15: TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union Y and
A16: Y meet (X1 union X2) is SubSpace of X1 meet X2 by A9;
reconsider C = the carrier of Y as Subset of X by Th1;
the carrier of X = (C1 \/ C2) \/ C by A11, A15, Def2;
then A17: A1 \/ A2 = ((C1 \/ C2) \/ C) /\ (A1 \/ A2) by XBOOLE_1:28
.= (C1 \/ (C2 \/ C)) /\ (A1 \/ A2) by XBOOLE_1:4
.= (C1 /\ (A1 \/ A2)) \/ ((C2 \/ C) /\ (A1 \/ A2)) by XBOOLE_1:23
.= {} \/ ((C2 \/ C) /\ (A1 \/ A2)) by A12, XBOOLE_0:def_7
.= (C2 /\ (A1 \/ A2)) \/ (C /\ (A1 \/ A2)) by XBOOLE_1:23 ;
A18: now__::_thesis:_(_C_/\_(A1_\/_A2)_<>_{}_implies_A1_\/_A2_c=_A2_)
assume C /\ (A1 \/ A2) <> {} ; ::_thesis: A1 \/ A2 c= A2
then C meets A1 \/ A2 by XBOOLE_0:def_7;
then Y meets X1 union X2 by A10, Def3;
then A19: the carrier of (Y meet (X1 union X2)) = C /\ (A1 \/ A2) by A10, Def4;
the carrier of (X1 meet X2) = A1 /\ A2 by A1, Def4;
then A20: C /\ (A1 \/ A2) c= A1 /\ A2 by A16, A19, Th4;
A21: A1 /\ A2 c= A2 by XBOOLE_1:17;
then A22: C /\ (A1 \/ A2) c= A2 by A20, XBOOLE_1:1;
now__::_thesis:_A1_\/_A2_c=_A2
percases ( C2 /\ (A1 \/ A2) = {} or C2 /\ (A1 \/ A2) <> {} ) ;
suppose C2 /\ (A1 \/ A2) = {} ; ::_thesis: A1 \/ A2 c= A2
hence A1 \/ A2 c= A2 by A17, A20, A21, XBOOLE_1:1; ::_thesis: verum
end;
suppose C2 /\ (A1 \/ A2) <> {} ; ::_thesis: A1 \/ A2 c= A2
then C2 meets A1 \/ A2 by XBOOLE_0:def_7;
then Y2 meets X1 union X2 by A10, Def3;
then the carrier of (Y2 meet (X1 union X2)) = C2 /\ (A1 \/ A2) by A10, Def4;
then C2 /\ (A1 \/ A2) c= A2 by A8, Th4;
hence A1 \/ A2 c= A2 by A17, A22, XBOOLE_1:8; ::_thesis: verum
end;
end;
end;
hence A1 \/ A2 c= A2 ; ::_thesis: verum
end;
now__::_thesis:_(_C2_/\_(A1_\/_A2)_<>_{}_implies_A1_\/_A2_c=_A2_)
assume C2 /\ (A1 \/ A2) <> {} ; ::_thesis: A1 \/ A2 c= A2
then C2 meets A1 \/ A2 by XBOOLE_0:def_7;
then Y2 meets X1 union X2 by A10, Def3;
then A23: the carrier of (Y2 meet (X1 union X2)) = C2 /\ (A1 \/ A2) by A10, Def4;
then A24: C2 /\ (A1 \/ A2) c= A2 by A8, Th4;
now__::_thesis:_A1_\/_A2_c=_A2
percases ( C /\ (A1 \/ A2) = {} or C /\ (A1 \/ A2) <> {} ) ;
suppose C /\ (A1 \/ A2) = {} ; ::_thesis: A1 \/ A2 c= A2
hence A1 \/ A2 c= A2 by A8, A17, A23, Th4; ::_thesis: verum
end;
suppose C /\ (A1 \/ A2) <> {} ; ::_thesis: A1 \/ A2 c= A2
then C meets A1 \/ A2 by XBOOLE_0:def_7;
then Y meets X1 union X2 by A10, Def3;
then A25: the carrier of (Y meet (X1 union X2)) = C /\ (A1 \/ A2) by A10, Def4;
the carrier of (X1 meet X2) = A1 /\ A2 by A1, Def4;
then C /\ (A1 \/ A2) c= A1 /\ A2 by A16, A25, Th4;
then A1 \/ A2 c= A2 \/ (A1 /\ A2) by A17, A24, XBOOLE_1:13;
hence A1 \/ A2 c= A2 by XBOOLE_1:12, XBOOLE_1:17; ::_thesis: verum
end;
end;
end;
hence A1 \/ A2 c= A2 ; ::_thesis: verum
end;
hence A1 \/ A2 c= A2 by A17, A18; ::_thesis: verum
end;
end;
end;
A1 c= A1 \/ A2 by XBOOLE_1:7;
then A1 c= A2 by A13, XBOOLE_1:1;
hence contradiction by A5, Th4; ::_thesis: verum
end;
then the carrier of (Y1 meet (X1 union X2)) = C1 /\ (A1 \/ A2) by A10, Def4;
then A26: C1 /\ (A1 \/ A2) c= A1 by A7, Th4;
now__::_thesis:_Y2_meets_X1_union_X2
assume not Y2 meets X1 union X2 ; ::_thesis: contradiction
then A27: C2 misses A1 \/ A2 by A10, Def3;
A28: now__::_thesis:_A1_\/_A2_c=_A1
percases ( X1 union X2 is SubSpace of Y1 union Y2 or not X1 union X2 is SubSpace of Y1 union Y2 ) ;
suppose X1 union X2 is SubSpace of Y1 union Y2 ; ::_thesis: A1 \/ A2 c= A1
then A1 \/ A2 c= C1 \/ C2 by A10, A11, Th4;
then A29: A1 \/ A2 = (C1 \/ C2) /\ (A1 \/ A2) by XBOOLE_1:28
.= (C1 /\ (A1 \/ A2)) \/ (C2 /\ (A1 \/ A2)) by XBOOLE_1:23
.= (C1 /\ (A1 \/ A2)) \/ {} by A27, XBOOLE_0:def_7
.= C1 /\ (A1 \/ A2) ;
then C1 meets A1 \/ A2 by XBOOLE_0:def_7;
then Y1 meets X1 union X2 by A10, Def3;
then the carrier of (Y1 meet (X1 union X2)) = C1 /\ (A1 \/ A2) by A10, Def4;
hence A1 \/ A2 c= A1 by A7, A29, Th4; ::_thesis: verum
end;
suppose X1 union X2 is not SubSpace of Y1 union Y2 ; ::_thesis: A1 \/ A2 c= A1
then consider Y being non empty open SubSpace of X such that
A30: TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union Y and
A31: Y meet (X1 union X2) is SubSpace of X1 meet X2 by A9;
reconsider C = the carrier of Y as Subset of X by Th1;
the carrier of X = (C1 \/ C2) \/ C by A11, A30, Def2;
then A32: A1 \/ A2 = ((C2 \/ C1) \/ C) /\ (A1 \/ A2) by XBOOLE_1:28
.= (C2 \/ (C1 \/ C)) /\ (A1 \/ A2) by XBOOLE_1:4
.= (C2 /\ (A1 \/ A2)) \/ ((C1 \/ C) /\ (A1 \/ A2)) by XBOOLE_1:23
.= {} \/ ((C1 \/ C) /\ (A1 \/ A2)) by A27, XBOOLE_0:def_7
.= (C1 /\ (A1 \/ A2)) \/ (C /\ (A1 \/ A2)) by XBOOLE_1:23 ;
A33: now__::_thesis:_(_C_/\_(A1_\/_A2)_<>_{}_implies_A1_\/_A2_c=_A1_)
assume C /\ (A1 \/ A2) <> {} ; ::_thesis: A1 \/ A2 c= A1
then C meets A1 \/ A2 by XBOOLE_0:def_7;
then Y meets X1 union X2 by A10, Def3;
then A34: the carrier of (Y meet (X1 union X2)) = C /\ (A1 \/ A2) by A10, Def4;
the carrier of (X1 meet X2) = A1 /\ A2 by A1, Def4;
then A35: C /\ (A1 \/ A2) c= A1 /\ A2 by A31, A34, Th4;
A36: A1 /\ A2 c= A1 by XBOOLE_1:17;
then A37: C /\ (A1 \/ A2) c= A1 by A35, XBOOLE_1:1;
now__::_thesis:_A1_\/_A2_c=_A1
percases ( C1 /\ (A1 \/ A2) = {} or C1 /\ (A1 \/ A2) <> {} ) ;
suppose C1 /\ (A1 \/ A2) = {} ; ::_thesis: A1 \/ A2 c= A1
hence A1 \/ A2 c= A1 by A32, A35, A36, XBOOLE_1:1; ::_thesis: verum
end;
suppose C1 /\ (A1 \/ A2) <> {} ; ::_thesis: A1 \/ A2 c= A1
then C1 meets A1 \/ A2 by XBOOLE_0:def_7;
then Y1 meets X1 union X2 by A10, Def3;
then the carrier of (Y1 meet (X1 union X2)) = C1 /\ (A1 \/ A2) by A10, Def4;
then C1 /\ (A1 \/ A2) c= A1 by A7, Th4;
hence A1 \/ A2 c= A1 by A32, A37, XBOOLE_1:8; ::_thesis: verum
end;
end;
end;
hence A1 \/ A2 c= A1 ; ::_thesis: verum
end;
now__::_thesis:_(_C1_/\_(A1_\/_A2)_<>_{}_implies_A1_\/_A2_c=_A1_)
assume C1 /\ (A1 \/ A2) <> {} ; ::_thesis: A1 \/ A2 c= A1
then C1 meets A1 \/ A2 by XBOOLE_0:def_7;
then Y1 meets X1 union X2 by A10, Def3;
then A38: the carrier of (Y1 meet (X1 union X2)) = C1 /\ (A1 \/ A2) by A10, Def4;
then A39: C1 /\ (A1 \/ A2) c= A1 by A7, Th4;
now__::_thesis:_A1_\/_A2_c=_A1
percases ( C /\ (A1 \/ A2) = {} or C /\ (A1 \/ A2) <> {} ) ;
suppose C /\ (A1 \/ A2) = {} ; ::_thesis: A1 \/ A2 c= A1
hence A1 \/ A2 c= A1 by A7, A32, A38, Th4; ::_thesis: verum
end;
suppose C /\ (A1 \/ A2) <> {} ; ::_thesis: A1 \/ A2 c= A1
then C meets A1 \/ A2 by XBOOLE_0:def_7;
then Y meets X1 union X2 by A10, Def3;
then A40: the carrier of (Y meet (X1 union X2)) = C /\ (A1 \/ A2) by A10, Def4;
the carrier of (X1 meet X2) = A1 /\ A2 by A1, Def4;
then C /\ (A1 \/ A2) c= A1 /\ A2 by A31, A40, Th4;
then A1 \/ A2 c= A1 \/ (A1 /\ A2) by A32, A39, XBOOLE_1:13;
hence A1 \/ A2 c= A1 by XBOOLE_1:12, XBOOLE_1:17; ::_thesis: verum
end;
end;
end;
hence A1 \/ A2 c= A1 ; ::_thesis: verum
end;
hence A1 \/ A2 c= A1 by A32, A33; ::_thesis: verum
end;
end;
end;
A2 c= A1 \/ A2 by XBOOLE_1:7;
then A2 c= A1 by A28, XBOOLE_1:1;
hence contradiction by A6, Th4; ::_thesis: verum
end;
then the carrier of (Y2 meet (X1 union X2)) = C2 /\ (A1 \/ A2) by A10, Def4;
then A41: C2 /\ (A1 \/ A2) c= A2 by A8, Th4;
A42: ( C1 is closed & C2 is closed ) by Th11;
now__::_thesis:_ex_C_being_Subset_of_X_st_
(_the_carrier_of_X_=_(C1_\/_C2)_\/_C_&_C_/\_(A1_\/_A2)_c=_A1_/\_A2_&_C_is_open_)
percases ( A1 \/ A2 c= C1 \/ C2 or not A1 \/ A2 c= C1 \/ C2 ) ;
supposeA43: A1 \/ A2 c= C1 \/ C2 ; ::_thesis: ex C being Subset of X st
( the carrier of X = (C1 \/ C2) \/ C & C /\ (A1 \/ A2) c= A1 /\ A2 & C is open )
thus ex C being Subset of X st
( the carrier of X = (C1 \/ C2) \/ C & C /\ (A1 \/ A2) c= A1 /\ A2 & C is open ) ::_thesis: verum
proof
take C = (C1 \/ C2) ` ; ::_thesis: ( the carrier of X = (C1 \/ C2) \/ C & C /\ (A1 \/ A2) c= A1 /\ A2 & C is open )
C misses A1 \/ A2 by A43, SUBSET_1:24;
then C /\ (A1 \/ A2) = {} by XBOOLE_0:def_7;
hence ( the carrier of X = (C1 \/ C2) \/ C & C /\ (A1 \/ A2) c= A1 /\ A2 & C is open ) by A2, A42, PRE_TOPC:2, XBOOLE_1:2; ::_thesis: verum
end;
end;
supposeA44: not A1 \/ A2 c= C1 \/ C2 ; ::_thesis: ex C being Subset of X st
( the carrier of X = (C1 \/ C2) \/ C & C /\ (A1 \/ A2) c= A1 /\ A2 & C is open )
thus ex C being Subset of X st
( the carrier of X = (C1 \/ C2) \/ C & C /\ (A1 \/ A2) c= A1 /\ A2 & C is open ) ::_thesis: verum
proof
consider Y being non empty open SubSpace of X such that
A45: TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union Y and
A46: Y meet (X1 union X2) is SubSpace of X1 meet X2 by A9, A10, A11, A44, Th4;
reconsider C = the carrier of Y as Subset of X by Th1;
A47: the carrier of X = the carrier of (Y1 union Y2) \/ C by A45, Def2
.= (C1 \/ C2) \/ C by Def2 ;
now__::_thesis:_Y_meets_X1_union_X2
assume not Y meets X1 union X2 ; ::_thesis: contradiction
then A48: C misses A1 \/ A2 by A10, Def3;
the carrier of X = (C1 \/ C2) \/ C by A11, A45, Def2;
then A1 \/ A2 = ((C1 \/ C2) \/ C) /\ (A1 \/ A2) by XBOOLE_1:28
.= ((C1 \/ C2) /\ (A1 \/ A2)) \/ (C /\ (A1 \/ A2)) by XBOOLE_1:23
.= ((C1 \/ C2) /\ (A1 \/ A2)) \/ {} by A48, XBOOLE_0:def_7
.= (C1 \/ C2) /\ (A1 \/ A2) ;
hence contradiction by A44, XBOOLE_1:17; ::_thesis: verum
end;
then A49: the carrier of (Y meet (X1 union X2)) = C /\ (A1 \/ A2) by A10, Def4;
A50: C is open by Th16;
the carrier of (X1 meet X2) = A1 /\ A2 by A1, Def4;
then C /\ (A1 \/ A2) c= A1 /\ A2 by A46, A49, Th4;
hence ex C being Subset of X st
( the carrier of X = (C1 \/ C2) \/ C & C /\ (A1 \/ A2) c= A1 /\ A2 & C is open ) by A50, A47; ::_thesis: verum
end;
end;
end;
end;
then for A1, A2 being Subset of X st A1 = the carrier of X1 & A2 = the carrier of X2 holds
A1,A2 are_weakly_separated by A42, A26, A41, Th54;
hence X1,X2 are_weakly_separated by Def7; ::_thesis: verum
end;
hence X1,X2 are_weakly_separated by Th79; ::_thesis: verum
end;
A51: X is SubSpace of X by Th2;
now__::_thesis:_(_not_X1,X2_are_weakly_separated_or_X1_is_SubSpace_of_X2_or_X2_is_SubSpace_of_X1_or_ex_Y1,_Y2_being_non_empty_closed_SubSpace_of_X_st_
(_Y1_meet_(X1_union_X2)_is_SubSpace_of_X1_&_Y2_meet_(X1_union_X2)_is_SubSpace_of_X2_&_(_X1_union_X2_is_SubSpace_of_Y1_union_Y2_or_ex_Y_being_non_empty_open_SubSpace_of_X_st_
(_TopStruct(#_the_carrier_of_X,_the_topology_of_X_#)_=_(Y1_union_Y2)_union_Y_&_Y_meet_(X1_union_X2)_is_SubSpace_of_X1_meet_X2_)_)_)_)
assume X1,X2 are_weakly_separated ; ::_thesis: ( X1 is SubSpace of X2 or X2 is SubSpace of X1 or ex Y1, Y2 being non empty closed SubSpace of X st
( Y1 meet (X1 union X2) is SubSpace of X1 & Y2 meet (X1 union X2) is SubSpace of X2 & ( X1 union X2 is SubSpace of Y1 union Y2 or ex Y being non empty open SubSpace of X st
( TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union Y & Y meet (X1 union X2) is SubSpace of X1 meet X2 ) ) ) )
then A52: A1,A2 are_weakly_separated by Def7;
now__::_thesis:_(_X1_is_not_SubSpace_of_X2_&_X2_is_not_SubSpace_of_X1_implies_ex_Y1,_Y2_being_non_empty_closed_SubSpace_of_X_st_
(_Y1_meet_(X1_union_X2)_is_SubSpace_of_X1_&_Y2_meet_(X1_union_X2)_is_SubSpace_of_X2_&_(_X1_union_X2_is_SubSpace_of_Y1_union_Y2_or_ex_Y_being_non_empty_open_SubSpace_of_X_st_
(_TopStruct(#_the_carrier_of_X,_the_topology_of_X_#)_=_(Y1_union_Y2)_union_Y_&_Y_meet_(X1_union_X2)_is_SubSpace_of_X1_meet_X2_)_)_)_)
assume that
A53: X1 is not SubSpace of X2 and
A54: X2 is not SubSpace of X1 ; ::_thesis: ex Y1, Y2 being non empty closed SubSpace of X st
( Y1 meet (X1 union X2) is SubSpace of X1 & Y2 meet (X1 union X2) is SubSpace of X2 & ( X1 union X2 is SubSpace of Y1 union Y2 or ex Y being non empty open SubSpace of X st
( TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union Y & Y meet (X1 union X2) is SubSpace of X1 meet X2 ) ) )
A55: not A2 c= A1 by A54, Th4;
A56: not A1 c= A2 by A53, Th4;
then consider C1, C2 being non empty Subset of X such that
A57: C1 is closed and
A58: C2 is closed and
A59: C1 /\ (A1 \/ A2) c= A1 and
A60: C2 /\ (A1 \/ A2) c= A2 and
A61: ( A1 \/ A2 c= C1 \/ C2 or ex C being non empty Subset of X st
( C is open & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (C1 \/ C2) \/ C ) ) by A52, A55, Th55;
A62: now__::_thesis:_not_C2_misses_A1_\/_A2
assume C2 misses A1 \/ A2 ; ::_thesis: contradiction
then A63: C2 /\ (A1 \/ A2) = {} by XBOOLE_0:def_7;
now__::_thesis:_contradiction
percases ( A1 \/ A2 c= C1 \/ C2 or not A1 \/ A2 c= C1 \/ C2 ) ;
supposeA64: A1 \/ A2 c= C1 \/ C2 ; ::_thesis: contradiction
A65: A2 c= A1 \/ A2 by XBOOLE_1:7;
A1 \/ A2 = (C1 \/ C2) /\ (A1 \/ A2) by A64, XBOOLE_1:28
.= (C1 /\ (A1 \/ A2)) \/ {} by A63, XBOOLE_1:23
.= C1 /\ (A1 \/ A2) ;
hence contradiction by A55, A59, A65, XBOOLE_1:1; ::_thesis: verum
end;
suppose not A1 \/ A2 c= C1 \/ C2 ; ::_thesis: contradiction
then consider C being non empty Subset of X such that
C is open and
A66: C /\ (A1 \/ A2) c= A1 /\ A2 and
A67: the carrier of X = (C1 \/ C2) \/ C by A61;
A1 \/ A2 = ((C2 \/ C1) \/ C) /\ (A1 \/ A2) by A67, XBOOLE_1:28
.= (C2 \/ (C1 \/ C)) /\ (A1 \/ A2) by XBOOLE_1:4
.= {} \/ ((C1 \/ C) /\ (A1 \/ A2)) by A63, XBOOLE_1:23
.= (C1 /\ (A1 \/ A2)) \/ (C /\ (A1 \/ A2)) by XBOOLE_1:23 ;
then A1 \/ A2 c= A1 \/ (A1 /\ A2) by A59, A66, XBOOLE_1:13;
then A68: A1 \/ A2 c= A1 by XBOOLE_1:12, XBOOLE_1:17;
A2 c= A1 \/ A2 by XBOOLE_1:7;
hence contradiction by A55, A68, XBOOLE_1:1; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
A69: now__::_thesis:_not_C1_misses_A1_\/_A2
assume C1 misses A1 \/ A2 ; ::_thesis: contradiction
then A70: C1 /\ (A1 \/ A2) = {} by XBOOLE_0:def_7;
now__::_thesis:_contradiction
percases ( A1 \/ A2 c= C1 \/ C2 or not A1 \/ A2 c= C1 \/ C2 ) ;
supposeA71: A1 \/ A2 c= C1 \/ C2 ; ::_thesis: contradiction
A72: A1 c= A1 \/ A2 by XBOOLE_1:7;
A1 \/ A2 = (C1 \/ C2) /\ (A1 \/ A2) by A71, XBOOLE_1:28
.= {} \/ (C2 /\ (A1 \/ A2)) by A70, XBOOLE_1:23
.= C2 /\ (A1 \/ A2) ;
hence contradiction by A56, A60, A72, XBOOLE_1:1; ::_thesis: verum
end;
suppose not A1 \/ A2 c= C1 \/ C2 ; ::_thesis: contradiction
then consider C being non empty Subset of X such that
C is open and
A73: C /\ (A1 \/ A2) c= A1 /\ A2 and
A74: the carrier of X = (C1 \/ C2) \/ C by A61;
A1 \/ A2 = ((C1 \/ C2) \/ C) /\ (A1 \/ A2) by A74, XBOOLE_1:28
.= (C1 \/ (C2 \/ C)) /\ (A1 \/ A2) by XBOOLE_1:4
.= {} \/ ((C2 \/ C) /\ (A1 \/ A2)) by A70, XBOOLE_1:23
.= (C2 /\ (A1 \/ A2)) \/ (C /\ (A1 \/ A2)) by XBOOLE_1:23 ;
then A1 \/ A2 c= A2 \/ (A1 /\ A2) by A60, A73, XBOOLE_1:13;
then A75: A1 \/ A2 c= A2 by XBOOLE_1:12, XBOOLE_1:17;
A1 c= A1 \/ A2 by XBOOLE_1:7;
hence contradiction by A56, A75, XBOOLE_1:1; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
thus ex Y1, Y2 being non empty closed SubSpace of X st
( Y1 meet (X1 union X2) is SubSpace of X1 & Y2 meet (X1 union X2) is SubSpace of X2 & ( X1 union X2 is SubSpace of Y1 union Y2 or ex Y being non empty open SubSpace of X st
( TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union Y & Y meet (X1 union X2) is SubSpace of X1 meet X2 ) ) ) ::_thesis: verum
proof
consider Y2 being non empty strict closed SubSpace of X such that
A76: C2 = the carrier of Y2 by A58, Th15;
A77: the carrier of (X1 union X2) = A1 \/ A2 by Def2;
then Y2 meets X1 union X2 by A62, A76, Def3;
then A78: the carrier of (Y2 meet (X1 union X2)) = C2 /\ (A1 \/ A2) by A76, A77, Def4;
consider Y1 being non empty strict closed SubSpace of X such that
A79: C1 = the carrier of Y1 by A57, Th15;
A80: the carrier of (Y1 union Y2) = C1 \/ C2 by A79, A76, Def2;
A81: now__::_thesis:_(_X1_union_X2_is_not_SubSpace_of_Y1_union_Y2_implies_ex_Y_being_non_empty_open_SubSpace_of_X_st_
(_TopStruct(#_the_carrier_of_X,_the_topology_of_X_#)_=_(Y1_union_Y2)_union_Y_&_Y_meet_(X1_union_X2)_is_SubSpace_of_X1_meet_X2_)_)
assume A82: X1 union X2 is not SubSpace of Y1 union Y2 ; ::_thesis: ex Y being non empty open SubSpace of X st
( TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union Y & Y meet (X1 union X2) is SubSpace of X1 meet X2 )
then consider C being non empty Subset of X such that
A83: C is open and
A84: C /\ (A1 \/ A2) c= A1 /\ A2 and
A85: the carrier of X = (C1 \/ C2) \/ C by A61, A77, A80, Th4;
A86: not A1 \/ A2 c= C1 \/ C2 by A77, A80, A82, Th4;
thus ex Y being non empty open SubSpace of X st
( TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union Y & Y meet (X1 union X2) is SubSpace of X1 meet X2 ) ::_thesis: verum
proof
consider Y being non empty strict open SubSpace of X such that
A87: C = the carrier of Y by A83, Th20;
now__::_thesis:_not_C_misses_A1_\/_A2
assume C misses A1 \/ A2 ; ::_thesis: contradiction
then A88: C /\ (A1 \/ A2) = {} by XBOOLE_0:def_7;
A1 \/ A2 = ((C1 \/ C2) \/ C) /\ (A1 \/ A2) by A85, XBOOLE_1:28
.= ((C1 \/ C2) /\ (A1 \/ A2)) \/ {} by A88, XBOOLE_1:23
.= (C1 \/ C2) /\ (A1 \/ A2) ;
hence contradiction by A86, XBOOLE_1:17; ::_thesis: verum
end;
then Y meets X1 union X2 by A77, A87, Def3;
then A89: the carrier of (Y meet (X1 union X2)) = C /\ (A1 \/ A2) by A77, A87, Def4;
take Y ; ::_thesis: ( TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union Y & Y meet (X1 union X2) is SubSpace of X1 meet X2 )
A90: the carrier of (X1 meet X2) = A1 /\ A2 by A1, Def4;
the carrier of X = the carrier of (Y1 union Y2) \/ C by A79, A76, A85, Def2
.= the carrier of ((Y1 union Y2) union Y) by A87, Def2 ;
hence ( TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union Y & Y meet (X1 union X2) is SubSpace of X1 meet X2 ) by A51, A84, A89, A90, Th4, Th5; ::_thesis: verum
end;
end;
take Y1 ; ::_thesis: ex Y2 being non empty closed SubSpace of X st
( Y1 meet (X1 union X2) is SubSpace of X1 & Y2 meet (X1 union X2) is SubSpace of X2 & ( X1 union X2 is SubSpace of Y1 union Y2 or ex Y being non empty open SubSpace of X st
( TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union Y & Y meet (X1 union X2) is SubSpace of X1 meet X2 ) ) )
take Y2 ; ::_thesis: ( Y1 meet (X1 union X2) is SubSpace of X1 & Y2 meet (X1 union X2) is SubSpace of X2 & ( X1 union X2 is SubSpace of Y1 union Y2 or ex Y being non empty open SubSpace of X st
( TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union Y & Y meet (X1 union X2) is SubSpace of X1 meet X2 ) ) )
Y1 meets X1 union X2 by A69, A79, A77, Def3;
then the carrier of (Y1 meet (X1 union X2)) = C1 /\ (A1 \/ A2) by A79, A77, Def4;
hence ( Y1 meet (X1 union X2) is SubSpace of X1 & Y2 meet (X1 union X2) is SubSpace of X2 & ( X1 union X2 is SubSpace of Y1 union Y2 or ex Y being non empty open SubSpace of X st
( TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union Y & Y meet (X1 union X2) is SubSpace of X1 meet X2 ) ) ) by A59, A60, A78, A81, Th4; ::_thesis: verum
end;
end;
hence ( X1 is SubSpace of X2 or X2 is SubSpace of X1 or ex Y1, Y2 being non empty closed SubSpace of X st
( Y1 meet (X1 union X2) is SubSpace of X1 & Y2 meet (X1 union X2) is SubSpace of X2 & ( X1 union X2 is SubSpace of Y1 union Y2 or ex Y being non empty open SubSpace of X st
( TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union Y & Y meet (X1 union X2) is SubSpace of X1 meet X2 ) ) ) ) ; ::_thesis: verum
end;
hence ( X1,X2 are_weakly_separated iff ( X1 is SubSpace of X2 or X2 is SubSpace of X1 or ex Y1, Y2 being non empty closed SubSpace of X st
( Y1 meet (X1 union X2) is SubSpace of X1 & Y2 meet (X1 union X2) is SubSpace of X2 & ( X1 union X2 is SubSpace of Y1 union Y2 or ex Y being non empty open SubSpace of X st
( TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union Y & Y meet (X1 union X2) is SubSpace of X1 meet X2 ) ) ) ) ) by A3; ::_thesis: verum
end;
theorem :: TSEP_1:87
for X being non empty TopSpace
for X1, X2 being non empty SubSpace of X st X = X1 union X2 & X1 meets X2 holds
( X1,X2 are_weakly_separated iff ( X1 is SubSpace of X2 or X2 is SubSpace of X1 or ex Y1, Y2 being non empty closed SubSpace of X st
( Y1 is SubSpace of X1 & Y2 is SubSpace of X2 & ( X = Y1 union Y2 or ex Y being non empty open SubSpace of X st
( X = (Y1 union Y2) union Y & Y is SubSpace of X1 meet X2 ) ) ) ) )
proof
let X be non empty TopSpace; ::_thesis: for X1, X2 being non empty SubSpace of X st X = X1 union X2 & X1 meets X2 holds
( X1,X2 are_weakly_separated iff ( X1 is SubSpace of X2 or X2 is SubSpace of X1 or ex Y1, Y2 being non empty closed SubSpace of X st
( Y1 is SubSpace of X1 & Y2 is SubSpace of X2 & ( X = Y1 union Y2 or ex Y being non empty open SubSpace of X st
( X = (Y1 union Y2) union Y & Y is SubSpace of X1 meet X2 ) ) ) ) )
let X1, X2 be non empty SubSpace of X; ::_thesis: ( X = X1 union X2 & X1 meets X2 implies ( X1,X2 are_weakly_separated iff ( X1 is SubSpace of X2 or X2 is SubSpace of X1 or ex Y1, Y2 being non empty closed SubSpace of X st
( Y1 is SubSpace of X1 & Y2 is SubSpace of X2 & ( X = Y1 union Y2 or ex Y being non empty open SubSpace of X st
( X = (Y1 union Y2) union Y & Y is SubSpace of X1 meet X2 ) ) ) ) ) )
reconsider A2 = the carrier of X2 as Subset of X by Th1;
reconsider A1 = the carrier of X1 as Subset of X by Th1;
assume A1: X = X1 union X2 ; ::_thesis: ( not X1 meets X2 or ( X1,X2 are_weakly_separated iff ( X1 is SubSpace of X2 or X2 is SubSpace of X1 or ex Y1, Y2 being non empty closed SubSpace of X st
( Y1 is SubSpace of X1 & Y2 is SubSpace of X2 & ( X = Y1 union Y2 or ex Y being non empty open SubSpace of X st
( X = (Y1 union Y2) union Y & Y is SubSpace of X1 meet X2 ) ) ) ) ) )
then A2: A1 \/ A2 = the carrier of X by Def2;
assume A3: X1 meets X2 ; ::_thesis: ( X1,X2 are_weakly_separated iff ( X1 is SubSpace of X2 or X2 is SubSpace of X1 or ex Y1, Y2 being non empty closed SubSpace of X st
( Y1 is SubSpace of X1 & Y2 is SubSpace of X2 & ( X = Y1 union Y2 or ex Y being non empty open SubSpace of X st
( X = (Y1 union Y2) union Y & Y is SubSpace of X1 meet X2 ) ) ) ) )
A4: now__::_thesis:_(_(_X1_is_SubSpace_of_X2_or_X2_is_SubSpace_of_X1_or_X1_is_SubSpace_of_X2_or_X2_is_SubSpace_of_X1_or_ex_Y1,_Y2_being_non_empty_closed_SubSpace_of_X_st_
(_Y1_is_SubSpace_of_X1_&_Y2_is_SubSpace_of_X2_&_(_X_=_Y1_union_Y2_or_ex_Y_being_non_empty_open_SubSpace_of_X_st_
(_X_=_(Y1_union_Y2)_union_Y_&_Y_is_SubSpace_of_X1_meet_X2_)_)_)_)_implies_X1,X2_are_weakly_separated_)
assume A5: ( X1 is SubSpace of X2 or X2 is SubSpace of X1 or X1 is SubSpace of X2 or X2 is SubSpace of X1 or ex Y1, Y2 being non empty closed SubSpace of X st
( Y1 is SubSpace of X1 & Y2 is SubSpace of X2 & ( X = Y1 union Y2 or ex Y being non empty open SubSpace of X st
( X = (Y1 union Y2) union Y & Y is SubSpace of X1 meet X2 ) ) ) ) ; ::_thesis: X1,X2 are_weakly_separated
now__::_thesis:_(_X1_is_not_SubSpace_of_X2_&_X2_is_not_SubSpace_of_X1_implies_X1,X2_are_weakly_separated_)
assume ( X1 is not SubSpace of X2 & X2 is not SubSpace of X1 ) ; ::_thesis: X1,X2 are_weakly_separated
then consider Y1, Y2 being non empty closed SubSpace of X such that
A6: ( Y1 is SubSpace of X1 & Y2 is SubSpace of X2 ) and
A7: ( X = Y1 union Y2 or ex Y being non empty open SubSpace of X st
( X = (Y1 union Y2) union Y & Y is SubSpace of X1 meet X2 ) ) by A5;
reconsider C2 = the carrier of Y2 as Subset of X by Th1;
reconsider C1 = the carrier of Y1 as Subset of X by Th1;
A8: now__::_thesis:_ex_C_being_Subset_of_X_st_
(_A1_\/_A2_=_(C1_\/_C2)_\/_C_&_C_c=_A1_/\_A2_&_C_is_open_)
percases ( A1 \/ A2 = C1 \/ C2 or A1 \/ A2 <> C1 \/ C2 ) ;
supposeA9: A1 \/ A2 = C1 \/ C2 ; ::_thesis: ex C being Subset of X st
( A1 \/ A2 = (C1 \/ C2) \/ C & C c= A1 /\ A2 & C is open )
thus ex C being Subset of X st
( A1 \/ A2 = (C1 \/ C2) \/ C & C c= A1 /\ A2 & C is open ) ::_thesis: verum
proof
take {} X ; ::_thesis: ( A1 \/ A2 = (C1 \/ C2) \/ ({} X) & {} X c= A1 /\ A2 & {} X is open )
thus ( A1 \/ A2 = (C1 \/ C2) \/ ({} X) & {} X c= A1 /\ A2 & {} X is open ) by A9, XBOOLE_1:2; ::_thesis: verum
end;
end;
supposeA10: A1 \/ A2 <> C1 \/ C2 ; ::_thesis: ex C being Subset of X st
( A1 \/ A2 = (C1 \/ C2) \/ C & C c= A1 /\ A2 & C is open )
thus ex C being Subset of X st
( A1 \/ A2 = (C1 \/ C2) \/ C & C c= A1 /\ A2 & C is open ) ::_thesis: verum
proof
consider Y being non empty open SubSpace of X such that
A11: X = (Y1 union Y2) union Y and
A12: Y is SubSpace of X1 meet X2 by A2, A7, A10, Def2;
reconsider C = the carrier of Y as Subset of X by Th1;
A1 /\ A2 = the carrier of (X1 meet X2) by A3, Def4;
then A13: C c= A1 /\ A2 by A12, Th4;
A14: C is open by Th16;
A1 \/ A2 = the carrier of (Y1 union Y2) \/ C by A2, A11, Def2
.= (C1 \/ C2) \/ C by Def2 ;
hence ex C being Subset of X st
( A1 \/ A2 = (C1 \/ C2) \/ C & C c= A1 /\ A2 & C is open ) by A14, A13; ::_thesis: verum
end;
end;
end;
end;
A15: ( C1 is closed & C2 is closed ) by Th11;
( C1 c= A1 & C2 c= A2 ) by A6, Th4;
then for A1, A2 being Subset of X st A1 = the carrier of X1 & A2 = the carrier of X2 holds
A1,A2 are_weakly_separated by A2, A15, A8, Th56;
hence X1,X2 are_weakly_separated by Def7; ::_thesis: verum
end;
hence X1,X2 are_weakly_separated by Th79; ::_thesis: verum
end;
now__::_thesis:_(_not_X1,X2_are_weakly_separated_or_X1_is_SubSpace_of_X2_or_X2_is_SubSpace_of_X1_or_ex_Y1,_Y2_being_non_empty_closed_SubSpace_of_X_st_
(_Y1_is_SubSpace_of_X1_&_Y2_is_SubSpace_of_X2_&_(_X_=_Y1_union_Y2_or_ex_Y_being_non_empty_open_SubSpace_of_X_st_
(_X_=_(Y1_union_Y2)_union_Y_&_Y_is_SubSpace_of_X1_meet_X2_)_)_)_)
assume X1,X2 are_weakly_separated ; ::_thesis: ( X1 is SubSpace of X2 or X2 is SubSpace of X1 or ex Y1, Y2 being non empty closed SubSpace of X st
( Y1 is SubSpace of X1 & Y2 is SubSpace of X2 & ( X = Y1 union Y2 or ex Y being non empty open SubSpace of X st
( X = (Y1 union Y2) union Y & Y is SubSpace of X1 meet X2 ) ) ) )
then A16: A1,A2 are_weakly_separated by Def7;
now__::_thesis:_(_X1_is_not_SubSpace_of_X2_&_X2_is_not_SubSpace_of_X1_implies_ex_Y1,_Y2_being_non_empty_closed_SubSpace_of_X_st_
(_Y1_is_SubSpace_of_X1_&_Y2_is_SubSpace_of_X2_&_(_X_=_Y1_union_Y2_or_ex_Y_being_non_empty_open_SubSpace_of_X_st_
(_X_=_(Y1_union_Y2)_union_Y_&_Y_is_SubSpace_of_X1_meet_X2_)_)_)_)
assume ( X1 is not SubSpace of X2 & X2 is not SubSpace of X1 ) ; ::_thesis: ex Y1, Y2 being non empty closed SubSpace of X st
( Y1 is SubSpace of X1 & Y2 is SubSpace of X2 & ( X = Y1 union Y2 or ex Y being non empty open SubSpace of X st
( X = (Y1 union Y2) union Y & Y is SubSpace of X1 meet X2 ) ) )
then ( not A1 c= A2 & not A2 c= A1 ) by Th4;
then consider C1, C2 being non empty Subset of X such that
A17: C1 is closed and
A18: C2 is closed and
A19: ( C1 c= A1 & C2 c= A2 ) and
A20: ( A1 \/ A2 = C1 \/ C2 or ex C being non empty Subset of X st
( A1 \/ A2 = (C1 \/ C2) \/ C & C c= A1 /\ A2 & C is open ) ) by A2, A16, Th57;
thus ex Y1, Y2 being non empty closed SubSpace of X st
( Y1 is SubSpace of X1 & Y2 is SubSpace of X2 & ( X = Y1 union Y2 or ex Y being non empty open SubSpace of X st
( X = (Y1 union Y2) union Y & Y is SubSpace of X1 meet X2 ) ) ) ::_thesis: verum
proof
consider Y2 being non empty strict closed SubSpace of X such that
A21: C2 = the carrier of Y2 by A18, Th15;
consider Y1 being non empty strict closed SubSpace of X such that
A22: C1 = the carrier of Y1 by A17, Th15;
take Y1 ; ::_thesis: ex Y2 being non empty closed SubSpace of X st
( Y1 is SubSpace of X1 & Y2 is SubSpace of X2 & ( X = Y1 union Y2 or ex Y being non empty open SubSpace of X st
( X = (Y1 union Y2) union Y & Y is SubSpace of X1 meet X2 ) ) )
take Y2 ; ::_thesis: ( Y1 is SubSpace of X1 & Y2 is SubSpace of X2 & ( X = Y1 union Y2 or ex Y being non empty open SubSpace of X st
( X = (Y1 union Y2) union Y & Y is SubSpace of X1 meet X2 ) ) )
now__::_thesis:_(_X_<>_Y1_union_Y2_implies_ex_Y_being_non_empty_open_SubSpace_of_X_st_
(_X_=_(Y1_union_Y2)_union_Y_&_Y_is_SubSpace_of_X1_meet_X2_)_)
assume X <> Y1 union Y2 ; ::_thesis: ex Y being non empty open SubSpace of X st
( X = (Y1 union Y2) union Y & Y is SubSpace of X1 meet X2 )
then consider C being non empty Subset of X such that
A23: A1 \/ A2 = (C1 \/ C2) \/ C and
A24: C c= A1 /\ A2 and
A25: C is open by A1, A2, A20, A22, A21, Def2;
thus ex Y being non empty open SubSpace of X st
( X = (Y1 union Y2) union Y & Y is SubSpace of X1 meet X2 ) ::_thesis: verum
proof
A26: C c= the carrier of (X1 meet X2) by A3, A24, Def4;
consider Y being non empty strict open SubSpace of X such that
A27: C = the carrier of Y by A25, Th20;
take Y ; ::_thesis: ( X = (Y1 union Y2) union Y & Y is SubSpace of X1 meet X2 )
the carrier of X = the carrier of (Y1 union Y2) \/ C by A2, A22, A21, A23, Def2
.= the carrier of ((Y1 union Y2) union Y) by A27, Def2 ;
hence ( X = (Y1 union Y2) union Y & Y is SubSpace of X1 meet X2 ) by A1, A27, A26, Th4, Th5; ::_thesis: verum
end;
end;
hence ( Y1 is SubSpace of X1 & Y2 is SubSpace of X2 & ( X = Y1 union Y2 or ex Y being non empty open SubSpace of X st
( X = (Y1 union Y2) union Y & Y is SubSpace of X1 meet X2 ) ) ) by A19, A22, A21, Th4; ::_thesis: verum
end;
end;
hence ( X1 is SubSpace of X2 or X2 is SubSpace of X1 or ex Y1, Y2 being non empty closed SubSpace of X st
( Y1 is SubSpace of X1 & Y2 is SubSpace of X2 & ( X = Y1 union Y2 or ex Y being non empty open SubSpace of X st
( X = (Y1 union Y2) union Y & Y is SubSpace of X1 meet X2 ) ) ) ) ; ::_thesis: verum
end;
hence ( X1,X2 are_weakly_separated iff ( X1 is SubSpace of X2 or X2 is SubSpace of X1 or ex Y1, Y2 being non empty closed SubSpace of X st
( Y1 is SubSpace of X1 & Y2 is SubSpace of X2 & ( X = Y1 union Y2 or ex Y being non empty open SubSpace of X st
( X = (Y1 union Y2) union Y & Y is SubSpace of X1 meet X2 ) ) ) ) ) by A4; ::_thesis: verum
end;
theorem :: TSEP_1:88
for X being non empty TopSpace
for X1, X2 being non empty SubSpace of X st X = X1 union X2 & X1 misses X2 holds
( X1,X2 are_weakly_separated iff ( X1 is closed SubSpace of X & X2 is closed SubSpace of X ) )
proof
let X be non empty TopSpace; ::_thesis: for X1, X2 being non empty SubSpace of X st X = X1 union X2 & X1 misses X2 holds
( X1,X2 are_weakly_separated iff ( X1 is closed SubSpace of X & X2 is closed SubSpace of X ) )
let X1, X2 be non empty SubSpace of X; ::_thesis: ( X = X1 union X2 & X1 misses X2 implies ( X1,X2 are_weakly_separated iff ( X1 is closed SubSpace of X & X2 is closed SubSpace of X ) ) )
assume A1: X = X1 union X2 ; ::_thesis: ( not X1 misses X2 or ( X1,X2 are_weakly_separated iff ( X1 is closed SubSpace of X & X2 is closed SubSpace of X ) ) )
assume A2: X1 misses X2 ; ::_thesis: ( X1,X2 are_weakly_separated iff ( X1 is closed SubSpace of X & X2 is closed SubSpace of X ) )
thus ( X1,X2 are_weakly_separated implies ( X1 is closed SubSpace of X & X2 is closed SubSpace of X ) ) ::_thesis: ( X1 is closed SubSpace of X & X2 is closed SubSpace of X implies X1,X2 are_weakly_separated )
proof
reconsider A2 = the carrier of X2 as Subset of X by Th1;
reconsider A1 = the carrier of X1 as Subset of X by Th1;
assume X1,X2 are_weakly_separated ; ::_thesis: ( X1 is closed SubSpace of X & X2 is closed SubSpace of X )
then X1,X2 are_separated by A2, Th78;
then A3: A1,A2 are_separated by Def6;
A1 \/ A2 = [#] X by A1, Def2;
then ( A1 is closed & A2 is closed ) by A3, CONNSP_1:4;
hence ( X1 is closed SubSpace of X & X2 is closed SubSpace of X ) by Th11; ::_thesis: verum
end;
thus ( X1 is closed SubSpace of X & X2 is closed SubSpace of X implies X1,X2 are_weakly_separated ) by Th80; ::_thesis: verum
end;
theorem :: TSEP_1:89
for X being non empty TopSpace
for X1, X2 being non empty SubSpace of X st X1 meets X2 holds
( X1,X2 are_weakly_separated iff ( X1 is SubSpace of X2 or X2 is SubSpace of X1 or ex Y1, Y2 being non empty open SubSpace of X st
( Y1 meet (X1 union X2) is SubSpace of X1 & Y2 meet (X1 union X2) is SubSpace of X2 & ( X1 union X2 is SubSpace of Y1 union Y2 or ex Y being non empty closed SubSpace of X st
( TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union Y & Y meet (X1 union X2) is SubSpace of X1 meet X2 ) ) ) ) )
proof
let X be non empty TopSpace; ::_thesis: for X1, X2 being non empty SubSpace of X st X1 meets X2 holds
( X1,X2 are_weakly_separated iff ( X1 is SubSpace of X2 or X2 is SubSpace of X1 or ex Y1, Y2 being non empty open SubSpace of X st
( Y1 meet (X1 union X2) is SubSpace of X1 & Y2 meet (X1 union X2) is SubSpace of X2 & ( X1 union X2 is SubSpace of Y1 union Y2 or ex Y being non empty closed SubSpace of X st
( TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union Y & Y meet (X1 union X2) is SubSpace of X1 meet X2 ) ) ) ) )
let X1, X2 be non empty SubSpace of X; ::_thesis: ( X1 meets X2 implies ( X1,X2 are_weakly_separated iff ( X1 is SubSpace of X2 or X2 is SubSpace of X1 or ex Y1, Y2 being non empty open SubSpace of X st
( Y1 meet (X1 union X2) is SubSpace of X1 & Y2 meet (X1 union X2) is SubSpace of X2 & ( X1 union X2 is SubSpace of Y1 union Y2 or ex Y being non empty closed SubSpace of X st
( TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union Y & Y meet (X1 union X2) is SubSpace of X1 meet X2 ) ) ) ) ) )
reconsider A2 = the carrier of X2 as Subset of X by Th1;
reconsider A1 = the carrier of X1 as Subset of X by Th1;
assume A1: X1 meets X2 ; ::_thesis: ( X1,X2 are_weakly_separated iff ( X1 is SubSpace of X2 or X2 is SubSpace of X1 or ex Y1, Y2 being non empty open SubSpace of X st
( Y1 meet (X1 union X2) is SubSpace of X1 & Y2 meet (X1 union X2) is SubSpace of X2 & ( X1 union X2 is SubSpace of Y1 union Y2 or ex Y being non empty closed SubSpace of X st
( TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union Y & Y meet (X1 union X2) is SubSpace of X1 meet X2 ) ) ) ) )
A2: [#] X = the carrier of X ;
A3: now__::_thesis:_(_(_X1_is_SubSpace_of_X2_or_X2_is_SubSpace_of_X1_or_ex_Y1,_Y2_being_non_empty_open_SubSpace_of_X_st_
(_Y1_meet_(X1_union_X2)_is_SubSpace_of_X1_&_Y2_meet_(X1_union_X2)_is_SubSpace_of_X2_&_(_X1_union_X2_is_SubSpace_of_Y1_union_Y2_or_ex_Y_being_non_empty_closed_SubSpace_of_X_st_
(_TopStruct(#_the_carrier_of_X,_the_topology_of_X_#)_=_(Y1_union_Y2)_union_Y_&_Y_meet_(X1_union_X2)_is_SubSpace_of_X1_meet_X2_)_)_)_)_implies_X1,X2_are_weakly_separated_)
assume A4: ( X1 is SubSpace of X2 or X2 is SubSpace of X1 or ex Y1, Y2 being non empty open SubSpace of X st
( Y1 meet (X1 union X2) is SubSpace of X1 & Y2 meet (X1 union X2) is SubSpace of X2 & ( X1 union X2 is SubSpace of Y1 union Y2 or ex Y being non empty closed SubSpace of X st
( TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union Y & Y meet (X1 union X2) is SubSpace of X1 meet X2 ) ) ) ) ; ::_thesis: X1,X2 are_weakly_separated
now__::_thesis:_(_X1_is_not_SubSpace_of_X2_&_X2_is_not_SubSpace_of_X1_implies_X1,X2_are_weakly_separated_)
assume that
A5: X1 is not SubSpace of X2 and
A6: X2 is not SubSpace of X1 ; ::_thesis: X1,X2 are_weakly_separated
consider Y1, Y2 being non empty open SubSpace of X such that
A7: Y1 meet (X1 union X2) is SubSpace of X1 and
A8: Y2 meet (X1 union X2) is SubSpace of X2 and
A9: ( X1 union X2 is SubSpace of Y1 union Y2 or ex Y being non empty closed SubSpace of X st
( TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union Y & Y meet (X1 union X2) is SubSpace of X1 meet X2 ) ) by A4, A5, A6;
reconsider C2 = the carrier of Y2 as Subset of X by Th1;
reconsider C1 = the carrier of Y1 as Subset of X by Th1;
A10: the carrier of (X1 union X2) = A1 \/ A2 by Def2;
A11: the carrier of (Y1 union Y2) = C1 \/ C2 by Def2;
now__::_thesis:_not_Y1_misses_X1_union_X2
assume Y1 misses X1 union X2 ; ::_thesis: contradiction
then A12: C1 misses A1 \/ A2 by A10, Def3;
A13: now__::_thesis:_A1_\/_A2_c=_A2
percases ( X1 union X2 is SubSpace of Y1 union Y2 or not X1 union X2 is SubSpace of Y1 union Y2 ) ;
suppose X1 union X2 is SubSpace of Y1 union Y2 ; ::_thesis: A1 \/ A2 c= A2
then A1 \/ A2 c= C1 \/ C2 by A10, A11, Th4;
then A14: A1 \/ A2 = (C1 \/ C2) /\ (A1 \/ A2) by XBOOLE_1:28
.= (C1 /\ (A1 \/ A2)) \/ (C2 /\ (A1 \/ A2)) by XBOOLE_1:23
.= {} \/ (C2 /\ (A1 \/ A2)) by A12, XBOOLE_0:def_7
.= C2 /\ (A1 \/ A2) ;
then C2 meets A1 \/ A2 by XBOOLE_0:def_7;
then Y2 meets X1 union X2 by A10, Def3;
then the carrier of (Y2 meet (X1 union X2)) = C2 /\ (A1 \/ A2) by A10, Def4;
hence A1 \/ A2 c= A2 by A8, A14, Th4; ::_thesis: verum
end;
suppose X1 union X2 is not SubSpace of Y1 union Y2 ; ::_thesis: A1 \/ A2 c= A2
then consider Y being non empty closed SubSpace of X such that
A15: TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union Y and
A16: Y meet (X1 union X2) is SubSpace of X1 meet X2 by A9;
reconsider C = the carrier of Y as Subset of X by Th1;
the carrier of X = (C1 \/ C2) \/ C by A11, A15, Def2;
then A17: A1 \/ A2 = ((C1 \/ C2) \/ C) /\ (A1 \/ A2) by XBOOLE_1:28
.= (C1 \/ (C2 \/ C)) /\ (A1 \/ A2) by XBOOLE_1:4
.= (C1 /\ (A1 \/ A2)) \/ ((C2 \/ C) /\ (A1 \/ A2)) by XBOOLE_1:23
.= {} \/ ((C2 \/ C) /\ (A1 \/ A2)) by A12, XBOOLE_0:def_7
.= (C2 /\ (A1 \/ A2)) \/ (C /\ (A1 \/ A2)) by XBOOLE_1:23 ;
A18: now__::_thesis:_(_C_/\_(A1_\/_A2)_<>_{}_implies_A1_\/_A2_c=_A2_)
assume C /\ (A1 \/ A2) <> {} ; ::_thesis: A1 \/ A2 c= A2
then C meets A1 \/ A2 by XBOOLE_0:def_7;
then Y meets X1 union X2 by A10, Def3;
then A19: the carrier of (Y meet (X1 union X2)) = C /\ (A1 \/ A2) by A10, Def4;
the carrier of (X1 meet X2) = A1 /\ A2 by A1, Def4;
then A20: C /\ (A1 \/ A2) c= A1 /\ A2 by A16, A19, Th4;
A21: A1 /\ A2 c= A2 by XBOOLE_1:17;
then A22: C /\ (A1 \/ A2) c= A2 by A20, XBOOLE_1:1;
now__::_thesis:_A1_\/_A2_c=_A2
percases ( C2 /\ (A1 \/ A2) = {} or C2 /\ (A1 \/ A2) <> {} ) ;
suppose C2 /\ (A1 \/ A2) = {} ; ::_thesis: A1 \/ A2 c= A2
hence A1 \/ A2 c= A2 by A17, A20, A21, XBOOLE_1:1; ::_thesis: verum
end;
suppose C2 /\ (A1 \/ A2) <> {} ; ::_thesis: A1 \/ A2 c= A2
then C2 meets A1 \/ A2 by XBOOLE_0:def_7;
then Y2 meets X1 union X2 by A10, Def3;
then the carrier of (Y2 meet (X1 union X2)) = C2 /\ (A1 \/ A2) by A10, Def4;
then C2 /\ (A1 \/ A2) c= A2 by A8, Th4;
hence A1 \/ A2 c= A2 by A17, A22, XBOOLE_1:8; ::_thesis: verum
end;
end;
end;
hence A1 \/ A2 c= A2 ; ::_thesis: verum
end;
now__::_thesis:_(_C2_/\_(A1_\/_A2)_<>_{}_implies_A1_\/_A2_c=_A2_)
assume C2 /\ (A1 \/ A2) <> {} ; ::_thesis: A1 \/ A2 c= A2
then C2 meets A1 \/ A2 by XBOOLE_0:def_7;
then Y2 meets X1 union X2 by A10, Def3;
then A23: the carrier of (Y2 meet (X1 union X2)) = C2 /\ (A1 \/ A2) by A10, Def4;
then A24: C2 /\ (A1 \/ A2) c= A2 by A8, Th4;
now__::_thesis:_A1_\/_A2_c=_A2
percases ( C /\ (A1 \/ A2) = {} or C /\ (A1 \/ A2) <> {} ) ;
suppose C /\ (A1 \/ A2) = {} ; ::_thesis: A1 \/ A2 c= A2
hence A1 \/ A2 c= A2 by A8, A17, A23, Th4; ::_thesis: verum
end;
suppose C /\ (A1 \/ A2) <> {} ; ::_thesis: A1 \/ A2 c= A2
then C meets A1 \/ A2 by XBOOLE_0:def_7;
then Y meets X1 union X2 by A10, Def3;
then A25: the carrier of (Y meet (X1 union X2)) = C /\ (A1 \/ A2) by A10, Def4;
the carrier of (X1 meet X2) = A1 /\ A2 by A1, Def4;
then C /\ (A1 \/ A2) c= A1 /\ A2 by A16, A25, Th4;
then A1 \/ A2 c= A2 \/ (A1 /\ A2) by A17, A24, XBOOLE_1:13;
hence A1 \/ A2 c= A2 by XBOOLE_1:12, XBOOLE_1:17; ::_thesis: verum
end;
end;
end;
hence A1 \/ A2 c= A2 ; ::_thesis: verum
end;
hence A1 \/ A2 c= A2 by A17, A18; ::_thesis: verum
end;
end;
end;
A1 c= A1 \/ A2 by XBOOLE_1:7;
then A1 c= A2 by A13, XBOOLE_1:1;
hence contradiction by A5, Th4; ::_thesis: verum
end;
then the carrier of (Y1 meet (X1 union X2)) = C1 /\ (A1 \/ A2) by A10, Def4;
then A26: C1 /\ (A1 \/ A2) c= A1 by A7, Th4;
now__::_thesis:_Y2_meets_X1_union_X2
assume not Y2 meets X1 union X2 ; ::_thesis: contradiction
then A27: C2 misses A1 \/ A2 by A10, Def3;
A28: now__::_thesis:_A1_\/_A2_c=_A1
percases ( X1 union X2 is SubSpace of Y1 union Y2 or not X1 union X2 is SubSpace of Y1 union Y2 ) ;
suppose X1 union X2 is SubSpace of Y1 union Y2 ; ::_thesis: A1 \/ A2 c= A1
then A1 \/ A2 c= C1 \/ C2 by A10, A11, Th4;
then A29: A1 \/ A2 = (C1 \/ C2) /\ (A1 \/ A2) by XBOOLE_1:28
.= (C1 /\ (A1 \/ A2)) \/ (C2 /\ (A1 \/ A2)) by XBOOLE_1:23
.= (C1 /\ (A1 \/ A2)) \/ {} by A27, XBOOLE_0:def_7
.= C1 /\ (A1 \/ A2) ;
then C1 meets A1 \/ A2 by XBOOLE_0:def_7;
then Y1 meets X1 union X2 by A10, Def3;
then the carrier of (Y1 meet (X1 union X2)) = C1 /\ (A1 \/ A2) by A10, Def4;
hence A1 \/ A2 c= A1 by A7, A29, Th4; ::_thesis: verum
end;
suppose X1 union X2 is not SubSpace of Y1 union Y2 ; ::_thesis: A1 \/ A2 c= A1
then consider Y being non empty closed SubSpace of X such that
A30: TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union Y and
A31: Y meet (X1 union X2) is SubSpace of X1 meet X2 by A9;
reconsider C = the carrier of Y as Subset of X by Th1;
the carrier of X = (C1 \/ C2) \/ C by A11, A30, Def2;
then A32: A1 \/ A2 = ((C2 \/ C1) \/ C) /\ (A1 \/ A2) by XBOOLE_1:28
.= (C2 \/ (C1 \/ C)) /\ (A1 \/ A2) by XBOOLE_1:4
.= (C2 /\ (A1 \/ A2)) \/ ((C1 \/ C) /\ (A1 \/ A2)) by XBOOLE_1:23
.= {} \/ ((C1 \/ C) /\ (A1 \/ A2)) by A27, XBOOLE_0:def_7
.= (C1 /\ (A1 \/ A2)) \/ (C /\ (A1 \/ A2)) by XBOOLE_1:23 ;
A33: now__::_thesis:_(_C_/\_(A1_\/_A2)_<>_{}_implies_A1_\/_A2_c=_A1_)
assume C /\ (A1 \/ A2) <> {} ; ::_thesis: A1 \/ A2 c= A1
then C meets A1 \/ A2 by XBOOLE_0:def_7;
then Y meets X1 union X2 by A10, Def3;
then A34: the carrier of (Y meet (X1 union X2)) = C /\ (A1 \/ A2) by A10, Def4;
the carrier of (X1 meet X2) = A1 /\ A2 by A1, Def4;
then A35: C /\ (A1 \/ A2) c= A1 /\ A2 by A31, A34, Th4;
A36: A1 /\ A2 c= A1 by XBOOLE_1:17;
then A37: C /\ (A1 \/ A2) c= A1 by A35, XBOOLE_1:1;
now__::_thesis:_A1_\/_A2_c=_A1
percases ( C1 /\ (A1 \/ A2) = {} or C1 /\ (A1 \/ A2) <> {} ) ;
suppose C1 /\ (A1 \/ A2) = {} ; ::_thesis: A1 \/ A2 c= A1
hence A1 \/ A2 c= A1 by A32, A35, A36, XBOOLE_1:1; ::_thesis: verum
end;
suppose C1 /\ (A1 \/ A2) <> {} ; ::_thesis: A1 \/ A2 c= A1
then C1 meets A1 \/ A2 by XBOOLE_0:def_7;
then Y1 meets X1 union X2 by A10, Def3;
then the carrier of (Y1 meet (X1 union X2)) = C1 /\ (A1 \/ A2) by A10, Def4;
then C1 /\ (A1 \/ A2) c= A1 by A7, Th4;
hence A1 \/ A2 c= A1 by A32, A37, XBOOLE_1:8; ::_thesis: verum
end;
end;
end;
hence A1 \/ A2 c= A1 ; ::_thesis: verum
end;
now__::_thesis:_(_C1_/\_(A1_\/_A2)_<>_{}_implies_A1_\/_A2_c=_A1_)
assume C1 /\ (A1 \/ A2) <> {} ; ::_thesis: A1 \/ A2 c= A1
then C1 meets A1 \/ A2 by XBOOLE_0:def_7;
then Y1 meets X1 union X2 by A10, Def3;
then A38: the carrier of (Y1 meet (X1 union X2)) = C1 /\ (A1 \/ A2) by A10, Def4;
then A39: C1 /\ (A1 \/ A2) c= A1 by A7, Th4;
now__::_thesis:_A1_\/_A2_c=_A1
percases ( C /\ (A1 \/ A2) = {} or C /\ (A1 \/ A2) <> {} ) ;
suppose C /\ (A1 \/ A2) = {} ; ::_thesis: A1 \/ A2 c= A1
hence A1 \/ A2 c= A1 by A7, A32, A38, Th4; ::_thesis: verum
end;
suppose C /\ (A1 \/ A2) <> {} ; ::_thesis: A1 \/ A2 c= A1
then C meets A1 \/ A2 by XBOOLE_0:def_7;
then Y meets X1 union X2 by A10, Def3;
then A40: the carrier of (Y meet (X1 union X2)) = C /\ (A1 \/ A2) by A10, Def4;
the carrier of (X1 meet X2) = A1 /\ A2 by A1, Def4;
then C /\ (A1 \/ A2) c= A1 /\ A2 by A31, A40, Th4;
then A1 \/ A2 c= A1 \/ (A1 /\ A2) by A32, A39, XBOOLE_1:13;
hence A1 \/ A2 c= A1 by XBOOLE_1:12, XBOOLE_1:17; ::_thesis: verum
end;
end;
end;
hence A1 \/ A2 c= A1 ; ::_thesis: verum
end;
hence A1 \/ A2 c= A1 by A32, A33; ::_thesis: verum
end;
end;
end;
A2 c= A1 \/ A2 by XBOOLE_1:7;
then A2 c= A1 by A28, XBOOLE_1:1;
hence contradiction by A6, Th4; ::_thesis: verum
end;
then the carrier of (Y2 meet (X1 union X2)) = C2 /\ (A1 \/ A2) by A10, Def4;
then A41: C2 /\ (A1 \/ A2) c= A2 by A8, Th4;
A42: ( C1 is open & C2 is open ) by Th16;
now__::_thesis:_ex_C_being_Subset_of_X_st_
(_the_carrier_of_X_=_(C1_\/_C2)_\/_C_&_C_/\_(A1_\/_A2)_c=_A1_/\_A2_&_C_is_closed_)
percases ( A1 \/ A2 c= C1 \/ C2 or not A1 \/ A2 c= C1 \/ C2 ) ;
supposeA43: A1 \/ A2 c= C1 \/ C2 ; ::_thesis: ex C being Subset of X st
( the carrier of X = (C1 \/ C2) \/ C & C /\ (A1 \/ A2) c= A1 /\ A2 & C is closed )
thus ex C being Subset of X st
( the carrier of X = (C1 \/ C2) \/ C & C /\ (A1 \/ A2) c= A1 /\ A2 & C is closed ) ::_thesis: verum
proof
take C = (C1 \/ C2) ` ; ::_thesis: ( the carrier of X = (C1 \/ C2) \/ C & C /\ (A1 \/ A2) c= A1 /\ A2 & C is closed )
C misses A1 \/ A2 by A43, SUBSET_1:24;
then C /\ (A1 \/ A2) = {} by XBOOLE_0:def_7;
hence ( the carrier of X = (C1 \/ C2) \/ C & C /\ (A1 \/ A2) c= A1 /\ A2 & C is closed ) by A2, A42, PRE_TOPC:2, XBOOLE_1:2; ::_thesis: verum
end;
end;
supposeA44: not A1 \/ A2 c= C1 \/ C2 ; ::_thesis: ex C being Subset of X st
( the carrier of X = (C1 \/ C2) \/ C & C /\ (A1 \/ A2) c= A1 /\ A2 & C is closed )
thus ex C being Subset of X st
( the carrier of X = (C1 \/ C2) \/ C & C /\ (A1 \/ A2) c= A1 /\ A2 & C is closed ) ::_thesis: verum
proof
consider Y being non empty closed SubSpace of X such that
A45: TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union Y and
A46: Y meet (X1 union X2) is SubSpace of X1 meet X2 by A9, A10, A11, A44, Th4;
reconsider C = the carrier of Y as Subset of X by Th1;
A47: the carrier of X = the carrier of (Y1 union Y2) \/ C by A45, Def2
.= (C1 \/ C2) \/ C by Def2 ;
now__::_thesis:_Y_meets_X1_union_X2
assume not Y meets X1 union X2 ; ::_thesis: contradiction
then A48: C misses A1 \/ A2 by A10, Def3;
the carrier of X = (C1 \/ C2) \/ C by A11, A45, Def2;
then A1 \/ A2 = ((C1 \/ C2) \/ C) /\ (A1 \/ A2) by XBOOLE_1:28
.= ((C1 \/ C2) /\ (A1 \/ A2)) \/ (C /\ (A1 \/ A2)) by XBOOLE_1:23
.= ((C1 \/ C2) /\ (A1 \/ A2)) \/ {} by A48, XBOOLE_0:def_7
.= (C1 \/ C2) /\ (A1 \/ A2) ;
hence contradiction by A44, XBOOLE_1:17; ::_thesis: verum
end;
then A49: the carrier of (Y meet (X1 union X2)) = C /\ (A1 \/ A2) by A10, Def4;
A50: C is closed by Th11;
the carrier of (X1 meet X2) = A1 /\ A2 by A1, Def4;
then C /\ (A1 \/ A2) c= A1 /\ A2 by A46, A49, Th4;
hence ex C being Subset of X st
( the carrier of X = (C1 \/ C2) \/ C & C /\ (A1 \/ A2) c= A1 /\ A2 & C is closed ) by A50, A47; ::_thesis: verum
end;
end;
end;
end;
then for A1, A2 being Subset of X st A1 = the carrier of X1 & A2 = the carrier of X2 holds
A1,A2 are_weakly_separated by A42, A26, A41, Th58;
hence X1,X2 are_weakly_separated by Def7; ::_thesis: verum
end;
hence X1,X2 are_weakly_separated by Th79; ::_thesis: verum
end;
A51: X is SubSpace of X by Th2;
now__::_thesis:_(_not_X1,X2_are_weakly_separated_or_X1_is_SubSpace_of_X2_or_X2_is_SubSpace_of_X1_or_ex_Y1,_Y2_being_non_empty_open_SubSpace_of_X_st_
(_Y1_meet_(X1_union_X2)_is_SubSpace_of_X1_&_Y2_meet_(X1_union_X2)_is_SubSpace_of_X2_&_(_X1_union_X2_is_SubSpace_of_Y1_union_Y2_or_ex_Y_being_non_empty_closed_SubSpace_of_X_st_
(_TopStruct(#_the_carrier_of_X,_the_topology_of_X_#)_=_(Y1_union_Y2)_union_Y_&_Y_meet_(X1_union_X2)_is_SubSpace_of_X1_meet_X2_)_)_)_)
assume X1,X2 are_weakly_separated ; ::_thesis: ( X1 is SubSpace of X2 or X2 is SubSpace of X1 or ex Y1, Y2 being non empty open SubSpace of X st
( Y1 meet (X1 union X2) is SubSpace of X1 & Y2 meet (X1 union X2) is SubSpace of X2 & ( X1 union X2 is SubSpace of Y1 union Y2 or ex Y being non empty closed SubSpace of X st
( TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union Y & Y meet (X1 union X2) is SubSpace of X1 meet X2 ) ) ) )
then A52: A1,A2 are_weakly_separated by Def7;
now__::_thesis:_(_X1_is_not_SubSpace_of_X2_&_X2_is_not_SubSpace_of_X1_implies_ex_Y1,_Y2_being_non_empty_open_SubSpace_of_X_st_
(_Y1_meet_(X1_union_X2)_is_SubSpace_of_X1_&_Y2_meet_(X1_union_X2)_is_SubSpace_of_X2_&_(_X1_union_X2_is_SubSpace_of_Y1_union_Y2_or_ex_Y_being_non_empty_closed_SubSpace_of_X_st_
(_TopStruct(#_the_carrier_of_X,_the_topology_of_X_#)_=_(Y1_union_Y2)_union_Y_&_Y_meet_(X1_union_X2)_is_SubSpace_of_X1_meet_X2_)_)_)_)
assume that
A53: X1 is not SubSpace of X2 and
A54: X2 is not SubSpace of X1 ; ::_thesis: ex Y1, Y2 being non empty open SubSpace of X st
( Y1 meet (X1 union X2) is SubSpace of X1 & Y2 meet (X1 union X2) is SubSpace of X2 & ( X1 union X2 is SubSpace of Y1 union Y2 or ex Y being non empty closed SubSpace of X st
( TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union Y & Y meet (X1 union X2) is SubSpace of X1 meet X2 ) ) )
A55: not A2 c= A1 by A54, Th4;
A56: not A1 c= A2 by A53, Th4;
then consider C1, C2 being non empty Subset of X such that
A57: C1 is open and
A58: C2 is open and
A59: C1 /\ (A1 \/ A2) c= A1 and
A60: C2 /\ (A1 \/ A2) c= A2 and
A61: ( A1 \/ A2 c= C1 \/ C2 or ex C being non empty Subset of X st
( C is closed & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (C1 \/ C2) \/ C ) ) by A52, A55, Th59;
A62: now__::_thesis:_not_C2_misses_A1_\/_A2
assume C2 misses A1 \/ A2 ; ::_thesis: contradiction
then A63: C2 /\ (A1 \/ A2) = {} by XBOOLE_0:def_7;
now__::_thesis:_contradiction
percases ( A1 \/ A2 c= C1 \/ C2 or not A1 \/ A2 c= C1 \/ C2 ) ;
supposeA64: A1 \/ A2 c= C1 \/ C2 ; ::_thesis: contradiction
A65: A2 c= A1 \/ A2 by XBOOLE_1:7;
A1 \/ A2 = (C1 \/ C2) /\ (A1 \/ A2) by A64, XBOOLE_1:28
.= (C1 /\ (A1 \/ A2)) \/ {} by A63, XBOOLE_1:23
.= C1 /\ (A1 \/ A2) ;
hence contradiction by A55, A59, A65, XBOOLE_1:1; ::_thesis: verum
end;
suppose not A1 \/ A2 c= C1 \/ C2 ; ::_thesis: contradiction
then consider C being non empty Subset of X such that
C is closed and
A66: C /\ (A1 \/ A2) c= A1 /\ A2 and
A67: the carrier of X = (C1 \/ C2) \/ C by A61;
A1 \/ A2 = ((C2 \/ C1) \/ C) /\ (A1 \/ A2) by A67, XBOOLE_1:28
.= (C2 \/ (C1 \/ C)) /\ (A1 \/ A2) by XBOOLE_1:4
.= {} \/ ((C1 \/ C) /\ (A1 \/ A2)) by A63, XBOOLE_1:23
.= (C1 /\ (A1 \/ A2)) \/ (C /\ (A1 \/ A2)) by XBOOLE_1:23 ;
then A1 \/ A2 c= A1 \/ (A1 /\ A2) by A59, A66, XBOOLE_1:13;
then A68: A1 \/ A2 c= A1 by XBOOLE_1:12, XBOOLE_1:17;
A2 c= A1 \/ A2 by XBOOLE_1:7;
hence contradiction by A55, A68, XBOOLE_1:1; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
A69: now__::_thesis:_not_C1_misses_A1_\/_A2
assume C1 misses A1 \/ A2 ; ::_thesis: contradiction
then A70: C1 /\ (A1 \/ A2) = {} by XBOOLE_0:def_7;
now__::_thesis:_contradiction
percases ( A1 \/ A2 c= C1 \/ C2 or not A1 \/ A2 c= C1 \/ C2 ) ;
supposeA71: A1 \/ A2 c= C1 \/ C2 ; ::_thesis: contradiction
A72: A1 c= A1 \/ A2 by XBOOLE_1:7;
A1 \/ A2 = (C1 \/ C2) /\ (A1 \/ A2) by A71, XBOOLE_1:28
.= {} \/ (C2 /\ (A1 \/ A2)) by A70, XBOOLE_1:23
.= C2 /\ (A1 \/ A2) ;
hence contradiction by A56, A60, A72, XBOOLE_1:1; ::_thesis: verum
end;
suppose not A1 \/ A2 c= C1 \/ C2 ; ::_thesis: contradiction
then consider C being non empty Subset of X such that
C is closed and
A73: C /\ (A1 \/ A2) c= A1 /\ A2 and
A74: the carrier of X = (C1 \/ C2) \/ C by A61;
A1 \/ A2 = ((C1 \/ C2) \/ C) /\ (A1 \/ A2) by A74, XBOOLE_1:28
.= (C1 \/ (C2 \/ C)) /\ (A1 \/ A2) by XBOOLE_1:4
.= {} \/ ((C2 \/ C) /\ (A1 \/ A2)) by A70, XBOOLE_1:23
.= (C2 /\ (A1 \/ A2)) \/ (C /\ (A1 \/ A2)) by XBOOLE_1:23 ;
then A1 \/ A2 c= A2 \/ (A1 /\ A2) by A60, A73, XBOOLE_1:13;
then A75: A1 \/ A2 c= A2 by XBOOLE_1:12, XBOOLE_1:17;
A1 c= A1 \/ A2 by XBOOLE_1:7;
hence contradiction by A56, A75, XBOOLE_1:1; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
thus ex Y1, Y2 being non empty open SubSpace of X st
( Y1 meet (X1 union X2) is SubSpace of X1 & Y2 meet (X1 union X2) is SubSpace of X2 & ( X1 union X2 is SubSpace of Y1 union Y2 or ex Y being non empty closed SubSpace of X st
( TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union Y & Y meet (X1 union X2) is SubSpace of X1 meet X2 ) ) ) ::_thesis: verum
proof
consider Y2 being non empty strict open SubSpace of X such that
A76: C2 = the carrier of Y2 by A58, Th20;
A77: the carrier of (X1 union X2) = A1 \/ A2 by Def2;
then Y2 meets X1 union X2 by A62, A76, Def3;
then A78: the carrier of (Y2 meet (X1 union X2)) = C2 /\ (A1 \/ A2) by A76, A77, Def4;
consider Y1 being non empty strict open SubSpace of X such that
A79: C1 = the carrier of Y1 by A57, Th20;
A80: the carrier of (Y1 union Y2) = C1 \/ C2 by A79, A76, Def2;
A81: now__::_thesis:_(_X1_union_X2_is_not_SubSpace_of_Y1_union_Y2_implies_ex_Y_being_non_empty_closed_SubSpace_of_X_st_
(_TopStruct(#_the_carrier_of_X,_the_topology_of_X_#)_=_(Y1_union_Y2)_union_Y_&_Y_meet_(X1_union_X2)_is_SubSpace_of_X1_meet_X2_)_)
assume A82: X1 union X2 is not SubSpace of Y1 union Y2 ; ::_thesis: ex Y being non empty closed SubSpace of X st
( TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union Y & Y meet (X1 union X2) is SubSpace of X1 meet X2 )
then consider C being non empty Subset of X such that
A83: C is closed and
A84: C /\ (A1 \/ A2) c= A1 /\ A2 and
A85: the carrier of X = (C1 \/ C2) \/ C by A61, A77, A80, Th4;
A86: not A1 \/ A2 c= C1 \/ C2 by A77, A80, A82, Th4;
thus ex Y being non empty closed SubSpace of X st
( TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union Y & Y meet (X1 union X2) is SubSpace of X1 meet X2 ) ::_thesis: verum
proof
consider Y being non empty strict closed SubSpace of X such that
A87: C = the carrier of Y by A83, Th15;
now__::_thesis:_not_C_misses_A1_\/_A2
assume C misses A1 \/ A2 ; ::_thesis: contradiction
then A88: C /\ (A1 \/ A2) = {} by XBOOLE_0:def_7;
A1 \/ A2 = ((C1 \/ C2) \/ C) /\ (A1 \/ A2) by A85, XBOOLE_1:28
.= ((C1 \/ C2) /\ (A1 \/ A2)) \/ {} by A88, XBOOLE_1:23
.= (C1 \/ C2) /\ (A1 \/ A2) ;
hence contradiction by A86, XBOOLE_1:17; ::_thesis: verum
end;
then Y meets X1 union X2 by A77, A87, Def3;
then A89: the carrier of (Y meet (X1 union X2)) = C /\ (A1 \/ A2) by A77, A87, Def4;
take Y ; ::_thesis: ( TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union Y & Y meet (X1 union X2) is SubSpace of X1 meet X2 )
A90: the carrier of (X1 meet X2) = A1 /\ A2 by A1, Def4;
the carrier of X = the carrier of (Y1 union Y2) \/ C by A79, A76, A85, Def2
.= the carrier of ((Y1 union Y2) union Y) by A87, Def2 ;
hence ( TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union Y & Y meet (X1 union X2) is SubSpace of X1 meet X2 ) by A51, A84, A89, A90, Th4, Th5; ::_thesis: verum
end;
end;
take Y1 ; ::_thesis: ex Y2 being non empty open SubSpace of X st
( Y1 meet (X1 union X2) is SubSpace of X1 & Y2 meet (X1 union X2) is SubSpace of X2 & ( X1 union X2 is SubSpace of Y1 union Y2 or ex Y being non empty closed SubSpace of X st
( TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union Y & Y meet (X1 union X2) is SubSpace of X1 meet X2 ) ) )
take Y2 ; ::_thesis: ( Y1 meet (X1 union X2) is SubSpace of X1 & Y2 meet (X1 union X2) is SubSpace of X2 & ( X1 union X2 is SubSpace of Y1 union Y2 or ex Y being non empty closed SubSpace of X st
( TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union Y & Y meet (X1 union X2) is SubSpace of X1 meet X2 ) ) )
Y1 meets X1 union X2 by A69, A79, A77, Def3;
then the carrier of (Y1 meet (X1 union X2)) = C1 /\ (A1 \/ A2) by A79, A77, Def4;
hence ( Y1 meet (X1 union X2) is SubSpace of X1 & Y2 meet (X1 union X2) is SubSpace of X2 & ( X1 union X2 is SubSpace of Y1 union Y2 or ex Y being non empty closed SubSpace of X st
( TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union Y & Y meet (X1 union X2) is SubSpace of X1 meet X2 ) ) ) by A59, A60, A78, A81, Th4; ::_thesis: verum
end;
end;
hence ( X1 is SubSpace of X2 or X2 is SubSpace of X1 or ex Y1, Y2 being non empty open SubSpace of X st
( Y1 meet (X1 union X2) is SubSpace of X1 & Y2 meet (X1 union X2) is SubSpace of X2 & ( X1 union X2 is SubSpace of Y1 union Y2 or ex Y being non empty closed SubSpace of X st
( TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union Y & Y meet (X1 union X2) is SubSpace of X1 meet X2 ) ) ) ) ; ::_thesis: verum
end;
hence ( X1,X2 are_weakly_separated iff ( X1 is SubSpace of X2 or X2 is SubSpace of X1 or ex Y1, Y2 being non empty open SubSpace of X st
( Y1 meet (X1 union X2) is SubSpace of X1 & Y2 meet (X1 union X2) is SubSpace of X2 & ( X1 union X2 is SubSpace of Y1 union Y2 or ex Y being non empty closed SubSpace of X st
( TopStruct(# the carrier of X, the topology of X #) = (Y1 union Y2) union Y & Y meet (X1 union X2) is SubSpace of X1 meet X2 ) ) ) ) ) by A3; ::_thesis: verum
end;
theorem :: TSEP_1:90
for X being non empty TopSpace
for X1, X2 being non empty SubSpace of X st X = X1 union X2 & X1 meets X2 holds
( X1,X2 are_weakly_separated iff ( X1 is SubSpace of X2 or X2 is SubSpace of X1 or ex Y1, Y2 being non empty open SubSpace of X st
( Y1 is SubSpace of X1 & Y2 is SubSpace of X2 & ( X = Y1 union Y2 or ex Y being non empty closed SubSpace of X st
( X = (Y1 union Y2) union Y & Y is SubSpace of X1 meet X2 ) ) ) ) )
proof
let X be non empty TopSpace; ::_thesis: for X1, X2 being non empty SubSpace of X st X = X1 union X2 & X1 meets X2 holds
( X1,X2 are_weakly_separated iff ( X1 is SubSpace of X2 or X2 is SubSpace of X1 or ex Y1, Y2 being non empty open SubSpace of X st
( Y1 is SubSpace of X1 & Y2 is SubSpace of X2 & ( X = Y1 union Y2 or ex Y being non empty closed SubSpace of X st
( X = (Y1 union Y2) union Y & Y is SubSpace of X1 meet X2 ) ) ) ) )
let X1, X2 be non empty SubSpace of X; ::_thesis: ( X = X1 union X2 & X1 meets X2 implies ( X1,X2 are_weakly_separated iff ( X1 is SubSpace of X2 or X2 is SubSpace of X1 or ex Y1, Y2 being non empty open SubSpace of X st
( Y1 is SubSpace of X1 & Y2 is SubSpace of X2 & ( X = Y1 union Y2 or ex Y being non empty closed SubSpace of X st
( X = (Y1 union Y2) union Y & Y is SubSpace of X1 meet X2 ) ) ) ) ) )
reconsider A2 = the carrier of X2 as Subset of X by Th1;
reconsider A1 = the carrier of X1 as Subset of X by Th1;
assume A1: X = X1 union X2 ; ::_thesis: ( not X1 meets X2 or ( X1,X2 are_weakly_separated iff ( X1 is SubSpace of X2 or X2 is SubSpace of X1 or ex Y1, Y2 being non empty open SubSpace of X st
( Y1 is SubSpace of X1 & Y2 is SubSpace of X2 & ( X = Y1 union Y2 or ex Y being non empty closed SubSpace of X st
( X = (Y1 union Y2) union Y & Y is SubSpace of X1 meet X2 ) ) ) ) ) )
then A2: A1 \/ A2 = the carrier of X by Def2;
assume A3: X1 meets X2 ; ::_thesis: ( X1,X2 are_weakly_separated iff ( X1 is SubSpace of X2 or X2 is SubSpace of X1 or ex Y1, Y2 being non empty open SubSpace of X st
( Y1 is SubSpace of X1 & Y2 is SubSpace of X2 & ( X = Y1 union Y2 or ex Y being non empty closed SubSpace of X st
( X = (Y1 union Y2) union Y & Y is SubSpace of X1 meet X2 ) ) ) ) )
A4: now__::_thesis:_(_(_X1_is_SubSpace_of_X2_or_X2_is_SubSpace_of_X1_or_ex_Y1,_Y2_being_non_empty_open_SubSpace_of_X_st_
(_Y1_is_SubSpace_of_X1_&_Y2_is_SubSpace_of_X2_&_(_X_=_Y1_union_Y2_or_ex_Y_being_non_empty_closed_SubSpace_of_X_st_
(_X_=_(Y1_union_Y2)_union_Y_&_Y_is_SubSpace_of_X1_meet_X2_)_)_)_)_implies_X1,X2_are_weakly_separated_)
assume A5: ( X1 is SubSpace of X2 or X2 is SubSpace of X1 or ex Y1, Y2 being non empty open SubSpace of X st
( Y1 is SubSpace of X1 & Y2 is SubSpace of X2 & ( X = Y1 union Y2 or ex Y being non empty closed SubSpace of X st
( X = (Y1 union Y2) union Y & Y is SubSpace of X1 meet X2 ) ) ) ) ; ::_thesis: X1,X2 are_weakly_separated
now__::_thesis:_(_X1_is_not_SubSpace_of_X2_&_X2_is_not_SubSpace_of_X1_implies_X1,X2_are_weakly_separated_)
assume ( X1 is not SubSpace of X2 & X2 is not SubSpace of X1 ) ; ::_thesis: X1,X2 are_weakly_separated
then consider Y1, Y2 being non empty open SubSpace of X such that
A6: ( Y1 is SubSpace of X1 & Y2 is SubSpace of X2 ) and
A7: ( X = Y1 union Y2 or ex Y being non empty closed SubSpace of X st
( X = (Y1 union Y2) union Y & Y is SubSpace of X1 meet X2 ) ) by A5;
reconsider C2 = the carrier of Y2 as Subset of X by Th1;
reconsider C1 = the carrier of Y1 as Subset of X by Th1;
A8: now__::_thesis:_ex_C_being_Subset_of_X_st_
(_A1_\/_A2_=_(C1_\/_C2)_\/_C_&_C_c=_A1_/\_A2_&_C_is_closed_)
percases ( A1 \/ A2 = C1 \/ C2 or A1 \/ A2 <> C1 \/ C2 ) ;
supposeA9: A1 \/ A2 = C1 \/ C2 ; ::_thesis: ex C being Subset of X st
( A1 \/ A2 = (C1 \/ C2) \/ C & C c= A1 /\ A2 & C is closed )
thus ex C being Subset of X st
( A1 \/ A2 = (C1 \/ C2) \/ C & C c= A1 /\ A2 & C is closed ) ::_thesis: verum
proof
take {} X ; ::_thesis: ( A1 \/ A2 = (C1 \/ C2) \/ ({} X) & {} X c= A1 /\ A2 & {} X is closed )
thus ( A1 \/ A2 = (C1 \/ C2) \/ ({} X) & {} X c= A1 /\ A2 & {} X is closed ) by A9, XBOOLE_1:2; ::_thesis: verum
end;
end;
supposeA10: A1 \/ A2 <> C1 \/ C2 ; ::_thesis: ex C being Subset of X st
( A1 \/ A2 = (C1 \/ C2) \/ C & C c= A1 /\ A2 & C is closed )
thus ex C being Subset of X st
( A1 \/ A2 = (C1 \/ C2) \/ C & C c= A1 /\ A2 & C is closed ) ::_thesis: verum
proof
consider Y being non empty closed SubSpace of X such that
A11: X = (Y1 union Y2) union Y and
A12: Y is SubSpace of X1 meet X2 by A2, A7, A10, Def2;
reconsider C = the carrier of Y as Subset of X by Th1;
A1 /\ A2 = the carrier of (X1 meet X2) by A3, Def4;
then A13: C c= A1 /\ A2 by A12, Th4;
A14: C is closed by Th11;
A1 \/ A2 = the carrier of (Y1 union Y2) \/ C by A2, A11, Def2
.= (C1 \/ C2) \/ C by Def2 ;
hence ex C being Subset of X st
( A1 \/ A2 = (C1 \/ C2) \/ C & C c= A1 /\ A2 & C is closed ) by A14, A13; ::_thesis: verum
end;
end;
end;
end;
A15: ( C1 is open & C2 is open ) by Th16;
( C1 c= A1 & C2 c= A2 ) by A6, Th4;
then for A1, A2 being Subset of X st A1 = the carrier of X1 & A2 = the carrier of X2 holds
A1,A2 are_weakly_separated by A2, A15, A8, Th60;
hence X1,X2 are_weakly_separated by Def7; ::_thesis: verum
end;
hence X1,X2 are_weakly_separated by Th79; ::_thesis: verum
end;
now__::_thesis:_(_not_X1,X2_are_weakly_separated_or_X1_is_SubSpace_of_X2_or_X2_is_SubSpace_of_X1_or_ex_Y1,_Y2_being_non_empty_open_SubSpace_of_X_st_
(_Y1_is_SubSpace_of_X1_&_Y2_is_SubSpace_of_X2_&_(_X_=_Y1_union_Y2_or_ex_Y_being_non_empty_closed_SubSpace_of_X_st_
(_X_=_(Y1_union_Y2)_union_Y_&_Y_is_SubSpace_of_X1_meet_X2_)_)_)_)
assume X1,X2 are_weakly_separated ; ::_thesis: ( X1 is SubSpace of X2 or X2 is SubSpace of X1 or ex Y1, Y2 being non empty open SubSpace of X st
( Y1 is SubSpace of X1 & Y2 is SubSpace of X2 & ( X = Y1 union Y2 or ex Y being non empty closed SubSpace of X st
( X = (Y1 union Y2) union Y & Y is SubSpace of X1 meet X2 ) ) ) )
then A16: A1,A2 are_weakly_separated by Def7;
now__::_thesis:_(_X1_is_not_SubSpace_of_X2_&_X2_is_not_SubSpace_of_X1_implies_ex_Y1,_Y2_being_non_empty_open_SubSpace_of_X_st_
(_Y1_is_SubSpace_of_X1_&_Y2_is_SubSpace_of_X2_&_(_X_=_Y1_union_Y2_or_ex_Y_being_non_empty_closed_SubSpace_of_X_st_
(_X_=_(Y1_union_Y2)_union_Y_&_Y_is_SubSpace_of_X1_meet_X2_)_)_)_)
assume ( X1 is not SubSpace of X2 & X2 is not SubSpace of X1 ) ; ::_thesis: ex Y1, Y2 being non empty open SubSpace of X st
( Y1 is SubSpace of X1 & Y2 is SubSpace of X2 & ( X = Y1 union Y2 or ex Y being non empty closed SubSpace of X st
( X = (Y1 union Y2) union Y & Y is SubSpace of X1 meet X2 ) ) )
then ( not A1 c= A2 & not A2 c= A1 ) by Th4;
then consider C1, C2 being non empty Subset of X such that
A17: C1 is open and
A18: C2 is open and
A19: ( C1 c= A1 & C2 c= A2 ) and
A20: ( A1 \/ A2 = C1 \/ C2 or ex C being non empty Subset of X st
( A1 \/ A2 = (C1 \/ C2) \/ C & C c= A1 /\ A2 & C is closed ) ) by A2, A16, Th61;
thus ex Y1, Y2 being non empty open SubSpace of X st
( Y1 is SubSpace of X1 & Y2 is SubSpace of X2 & ( X = Y1 union Y2 or ex Y being non empty closed SubSpace of X st
( X = (Y1 union Y2) union Y & Y is SubSpace of X1 meet X2 ) ) ) ::_thesis: verum
proof
consider Y2 being non empty strict open SubSpace of X such that
A21: C2 = the carrier of Y2 by A18, Th20;
consider Y1 being non empty strict open SubSpace of X such that
A22: C1 = the carrier of Y1 by A17, Th20;
take Y1 ; ::_thesis: ex Y2 being non empty open SubSpace of X st
( Y1 is SubSpace of X1 & Y2 is SubSpace of X2 & ( X = Y1 union Y2 or ex Y being non empty closed SubSpace of X st
( X = (Y1 union Y2) union Y & Y is SubSpace of X1 meet X2 ) ) )
take Y2 ; ::_thesis: ( Y1 is SubSpace of X1 & Y2 is SubSpace of X2 & ( X = Y1 union Y2 or ex Y being non empty closed SubSpace of X st
( X = (Y1 union Y2) union Y & Y is SubSpace of X1 meet X2 ) ) )
now__::_thesis:_(_X_<>_Y1_union_Y2_implies_ex_Y_being_non_empty_closed_SubSpace_of_X_st_
(_X_=_(Y1_union_Y2)_union_Y_&_Y_is_SubSpace_of_X1_meet_X2_)_)
assume X <> Y1 union Y2 ; ::_thesis: ex Y being non empty closed SubSpace of X st
( X = (Y1 union Y2) union Y & Y is SubSpace of X1 meet X2 )
then consider C being non empty Subset of X such that
A23: A1 \/ A2 = (C1 \/ C2) \/ C and
A24: C c= A1 /\ A2 and
A25: C is closed by A1, A2, A20, A22, A21, Def2;
thus ex Y being non empty closed SubSpace of X st
( X = (Y1 union Y2) union Y & Y is SubSpace of X1 meet X2 ) ::_thesis: verum
proof
A26: C c= the carrier of (X1 meet X2) by A3, A24, Def4;
consider Y being non empty strict closed SubSpace of X such that
A27: C = the carrier of Y by A25, Th15;
take Y ; ::_thesis: ( X = (Y1 union Y2) union Y & Y is SubSpace of X1 meet X2 )
the carrier of X = the carrier of (Y1 union Y2) \/ C by A2, A22, A21, A23, Def2
.= the carrier of ((Y1 union Y2) union Y) by A27, Def2 ;
hence ( X = (Y1 union Y2) union Y & Y is SubSpace of X1 meet X2 ) by A1, A27, A26, Th4, Th5; ::_thesis: verum
end;
end;
hence ( Y1 is SubSpace of X1 & Y2 is SubSpace of X2 & ( X = Y1 union Y2 or ex Y being non empty closed SubSpace of X st
( X = (Y1 union Y2) union Y & Y is SubSpace of X1 meet X2 ) ) ) by A19, A22, A21, Th4; ::_thesis: verum
end;
end;
hence ( X1 is SubSpace of X2 or X2 is SubSpace of X1 or ex Y1, Y2 being non empty open SubSpace of X st
( Y1 is SubSpace of X1 & Y2 is SubSpace of X2 & ( X = Y1 union Y2 or ex Y being non empty closed SubSpace of X st
( X = (Y1 union Y2) union Y & Y is SubSpace of X1 meet X2 ) ) ) ) ; ::_thesis: verum
end;
hence ( X1,X2 are_weakly_separated iff ( X1 is SubSpace of X2 or X2 is SubSpace of X1 or ex Y1, Y2 being non empty open SubSpace of X st
( Y1 is SubSpace of X1 & Y2 is SubSpace of X2 & ( X = Y1 union Y2 or ex Y being non empty closed SubSpace of X st
( X = (Y1 union Y2) union Y & Y is SubSpace of X1 meet X2 ) ) ) ) ) by A4; ::_thesis: verum
end;
theorem :: TSEP_1:91
for X being non empty TopSpace
for X1, X2 being non empty SubSpace of X st X = X1 union X2 & X1 misses X2 holds
( X1,X2 are_weakly_separated iff ( X1 is open SubSpace of X & X2 is open SubSpace of X ) )
proof
let X be non empty TopSpace; ::_thesis: for X1, X2 being non empty SubSpace of X st X = X1 union X2 & X1 misses X2 holds
( X1,X2 are_weakly_separated iff ( X1 is open SubSpace of X & X2 is open SubSpace of X ) )
let X1, X2 be non empty SubSpace of X; ::_thesis: ( X = X1 union X2 & X1 misses X2 implies ( X1,X2 are_weakly_separated iff ( X1 is open SubSpace of X & X2 is open SubSpace of X ) ) )
assume A1: X = X1 union X2 ; ::_thesis: ( not X1 misses X2 or ( X1,X2 are_weakly_separated iff ( X1 is open SubSpace of X & X2 is open SubSpace of X ) ) )
assume A2: X1 misses X2 ; ::_thesis: ( X1,X2 are_weakly_separated iff ( X1 is open SubSpace of X & X2 is open SubSpace of X ) )
thus ( X1,X2 are_weakly_separated implies ( X1 is open SubSpace of X & X2 is open SubSpace of X ) ) ::_thesis: ( X1 is open SubSpace of X & X2 is open SubSpace of X implies X1,X2 are_weakly_separated )
proof
reconsider A2 = the carrier of X2 as Subset of X by Th1;
reconsider A1 = the carrier of X1 as Subset of X by Th1;
assume X1,X2 are_weakly_separated ; ::_thesis: ( X1 is open SubSpace of X & X2 is open SubSpace of X )
then X1,X2 are_separated by A2, Th78;
then A3: A1,A2 are_separated by Def6;
A1 \/ A2 = [#] X by A1, Def2;
then ( A1 is open & A2 is open ) by A3, CONNSP_1:4;
hence ( X1 is open SubSpace of X & X2 is open SubSpace of X ) by Th16; ::_thesis: verum
end;
thus ( X1 is open SubSpace of X & X2 is open SubSpace of X implies X1,X2 are_weakly_separated ) by Th81; ::_thesis: verum
end;
theorem :: TSEP_1:92
for X being non empty TopSpace
for X1, X2 being non empty SubSpace of X holds
( X1,X2 are_separated iff ex Y1, Y2 being non empty SubSpace of X st
( Y1,Y2 are_weakly_separated & X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & ( Y1 misses Y2 or Y1 meet Y2 misses X1 union X2 ) ) )
proof
let X be non empty TopSpace; ::_thesis: for X1, X2 being non empty SubSpace of X holds
( X1,X2 are_separated iff ex Y1, Y2 being non empty SubSpace of X st
( Y1,Y2 are_weakly_separated & X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & ( Y1 misses Y2 or Y1 meet Y2 misses X1 union X2 ) ) )
let X1, X2 be non empty SubSpace of X; ::_thesis: ( X1,X2 are_separated iff ex Y1, Y2 being non empty SubSpace of X st
( Y1,Y2 are_weakly_separated & X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & ( Y1 misses Y2 or Y1 meet Y2 misses X1 union X2 ) ) )
thus ( X1,X2 are_separated implies ex Y1, Y2 being non empty SubSpace of X st
( Y1,Y2 are_weakly_separated & X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & ( Y1 misses Y2 or Y1 meet Y2 misses X1 union X2 ) ) ) ::_thesis: ( ex Y1, Y2 being non empty SubSpace of X st
( Y1,Y2 are_weakly_separated & X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & ( Y1 misses Y2 or Y1 meet Y2 misses X1 union X2 ) ) implies X1,X2 are_separated )
proof
assume X1,X2 are_separated ; ::_thesis: ex Y1, Y2 being non empty SubSpace of X st
( Y1,Y2 are_weakly_separated & X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & ( Y1 misses Y2 or Y1 meet Y2 misses X1 union X2 ) )
then consider Y1, Y2 being non empty open SubSpace of X such that
A1: ( X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & ( Y1 misses Y2 or Y1 meet Y2 misses X1 union X2 ) ) by Th77;
take Y1 ; ::_thesis: ex Y2 being non empty SubSpace of X st
( Y1,Y2 are_weakly_separated & X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & ( Y1 misses Y2 or Y1 meet Y2 misses X1 union X2 ) )
take Y2 ; ::_thesis: ( Y1,Y2 are_weakly_separated & X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & ( Y1 misses Y2 or Y1 meet Y2 misses X1 union X2 ) )
thus ( Y1,Y2 are_weakly_separated & X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & ( Y1 misses Y2 or Y1 meet Y2 misses X1 union X2 ) ) by A1, Th81; ::_thesis: verum
end;
given Y1, Y2 being non empty SubSpace of X such that A2: Y1,Y2 are_weakly_separated and
A3: ( X1 is SubSpace of Y1 & X2 is SubSpace of Y2 ) and
A4: ( Y1 misses Y2 or Y1 meet Y2 misses X1 union X2 ) ; ::_thesis: X1,X2 are_separated
reconsider C2 = the carrier of Y2 as Subset of X by Th1;
reconsider C1 = the carrier of Y1 as Subset of X by Th1;
now__::_thesis:_for_A1,_A2_being_Subset_of_X_st_A1_=_the_carrier_of_X1_&_A2_=_the_carrier_of_X2_holds_
A1,A2_are_separated
let A1, A2 be Subset of X; ::_thesis: ( A1 = the carrier of X1 & A2 = the carrier of X2 implies A1,A2 are_separated )
assume A5: ( A1 = the carrier of X1 & A2 = the carrier of X2 ) ; ::_thesis: A1,A2 are_separated
now__::_thesis:_A1,A2_are_separated
percases ( Y1 misses Y2 or not Y1 misses Y2 ) ;
suppose Y1 misses Y2 ; ::_thesis: A1,A2 are_separated
then Y1,Y2 are_separated by A2, Th78;
then A6: C1,C2 are_separated by Def6;
( A1 c= C1 & A2 c= C2 ) by A3, A5, Th4;
hence A1,A2 are_separated by A6, CONNSP_1:7; ::_thesis: verum
end;
supposeA7: not Y1 misses Y2 ; ::_thesis: A1,A2 are_separated
ex B1, B2 being Subset of X st
( B1,B2 are_weakly_separated & A1 c= B1 & A2 c= B2 & B1 /\ B2 misses A1 \/ A2 )
proof
take C1 ; ::_thesis: ex B2 being Subset of X st
( C1,B2 are_weakly_separated & A1 c= C1 & A2 c= B2 & C1 /\ B2 misses A1 \/ A2 )
take C2 ; ::_thesis: ( C1,C2 are_weakly_separated & A1 c= C1 & A2 c= C2 & C1 /\ C2 misses A1 \/ A2 )
( the carrier of (Y1 meet Y2) = C1 /\ C2 & the carrier of (X1 union X2) = A1 \/ A2 ) by A5, A7, Def2, Def4;
hence ( C1,C2 are_weakly_separated & A1 c= C1 & A2 c= C2 & C1 /\ C2 misses A1 \/ A2 ) by A2, A3, A4, A5, A7, Def3, Def7, Th4; ::_thesis: verum
end;
hence A1,A2 are_separated by Th62; ::_thesis: verum
end;
end;
end;
hence A1,A2 are_separated ; ::_thesis: verum
end;
hence X1,X2 are_separated by Def6; ::_thesis: verum
end;
theorem :: TSEP_1:93
for T being TopStruct holds T | ([#] T) = TopStruct(# the carrier of T, the topology of T #)
proof
let T be TopStruct ; ::_thesis: T | ([#] T) = TopStruct(# the carrier of T, the topology of T #)
( TopStruct(# the carrier of T, the topology of T #) is strict SubSpace of T & the carrier of T = [#] TopStruct(# the carrier of T, the topology of T #) ) by Th2, PRE_TOPC:10;
hence T | ([#] T) = TopStruct(# the carrier of T, the topology of T #) by PRE_TOPC:def_5; ::_thesis: verum
end;