### On a Duality Between Weakly Separated Subspaces of Topological Spaces

by
Zbigniew Karno

Copyright (c) 1992 Association of Mizar Users

MML identifier: TSEP_2
[ MML identifier index ]

```environ

vocabulary PRE_TOPC, BOOLE, SUBSET_1, TOPS_1, TSEP_1, CONNSP_1, TARSKI,
SETFAM_1, TSEP_2;
notation TARSKI, XBOOLE_0, SUBSET_1, PRE_TOPC, STRUCT_0, TOPS_1, BORSUK_1,
TSEP_1;
constructors CONNSP_1, TOPS_1, BORSUK_1, TSEP_1, MEMBERED;
clusters PRE_TOPC, STRUCT_0, MEMBERED, ZFMISC_1;
requirements BOOLE, SUBSET;
definitions BORSUK_1, TSEP_1, XBOOLE_0;
theorems PRE_TOPC, TOPS_1, CONNSP_1, BORSUK_1, TSEP_1, TDLAT_1, TDLAT_3,
SUBSET_1, XBOOLE_0, XBOOLE_1;

begin
:: 1. Certain Set-Decompositions of a Topological Space.
reserve X for non empty TopSpace;

theorem Th1:
for A, B being Subset of X holds ( A`) \  B` = B \ A
proof let A, B be Subset of X;
A` \  B` = ([#]X \ A) \  B` by PRE_TOPC:17;
then  A` \  B` = [#]X \ (A \/  B`) by XBOOLE_1:41;
then  A` \  B` = (A \/  B`)` by PRE_TOPC:17;
then  A` \  B` = (A \/  B`)`;
then  A` \  B` = ( A`) /\  B`` by SUBSET_1:29;
then  A` \  B` = B /\  A`;
hence thesis by SUBSET_1:32;
end;

::Complemented Subsets.
definition let X be TopSpace, A1, A2 be Subset of X;
pred A1,A2 constitute_a_decomposition means
:Def1: A1 misses A2 & A1 \/ A2 = the carrier of X;
symmetry;
antonym A1,A2 do_not_constitute_a_decomposition;
end;

reserve A, A1, A2, B1, B2 for Subset of X;

theorem Th2:
A1,A2 constitute_a_decomposition iff A1 misses A2 & A1 \/ A2 = [#]X
proof
thus A1,A2 constitute_a_decomposition implies
A1 misses A2 & A1 \/ A2 = [#]X
proof
assume A1 misses A2 & A1 \/ A2 = the carrier of X;
hence thesis by PRE_TOPC:12;
end;
assume A1 misses A2 & A1 \/ A2 = [#]X;
then A1 misses A2 & A1 \/ A2 = the carrier of X by PRE_TOPC:12;
hence thesis by Def1;
end;

canceled;

theorem Th4:
A1,A2 constitute_a_decomposition implies A1 =  A2` & A2 =  A1`
proof
assume A1,A2 constitute_a_decomposition;
then A1: A1 misses A2 & A1 \/ A2 = [#]X by Th2;
then  A2` c= A1 & A1 c=  A2` by TDLAT_1:1,2;
hence A1 =  A2` by XBOOLE_0:def 10;
A1` c= A2 & A2 c=  A1` by A1,TDLAT_1:1,2;
hence thesis by XBOOLE_0:def 10;
end;

theorem Th5:
A1 =  A2` or A2 =  A1` implies A1,A2 constitute_a_decomposition
proof
assume A1 =  A2` or A2 =  A1`;
then A1 misses A2 & A1 \/ A2 = [#]X by TDLAT_1:1,2;
hence thesis by Th2;
end;

theorem Th6:
A, A` constitute_a_decomposition
proof
thus A, A` constitute_a_decomposition
proof
A misses  A` & A \/  A` = [#]X by PRE_TOPC:18,26;
hence thesis by Th2;
end;
end;

theorem
{}X,[#]X constitute_a_decomposition
proof
thus {}X,[#]X constitute_a_decomposition
proof
[#]X = ({}X)` by PRE_TOPC:27;
hence thesis by Th6;
end;
end;

theorem Th8:
A,A do_not_constitute_a_decomposition
proof
assume A1: A,A constitute_a_decomposition;
per cases;
suppose A2: A is non empty;
A =  A` by A1,Th4;
then A misses A by PRE_TOPC:21;
then A /\ A = {} by XBOOLE_0:def 7;
suppose A is empty;
then {} = A \/ A
.= the carrier of X by A1,Def1;
end;

definition let X be non empty TopSpace, A1, A2 be Subset of X;
redefine pred A1,A2 constitute_a_decomposition;
irreflexivity by Th8;
end;

theorem Th9:
A1,A constitute_a_decomposition & A,A2 constitute_a_decomposition
implies A1 = A2
proof
assume A1,A constitute_a_decomposition;
then A1: A1 =  A` by Th4;
assume A,A2 constitute_a_decomposition;
hence thesis by A1,Th4;
end;

theorem Th10:
A1,A2 constitute_a_decomposition implies
(Cl A1),(Int A2) constitute_a_decomposition &
(Int A1),(Cl A2) constitute_a_decomposition
proof
assume A1: A1,A2 constitute_a_decomposition;
thus (Cl A1),(Int A2) constitute_a_decomposition
proof
Cl A1 = (Int A1`)` by TDLAT_3:2;
then Cl A1 = (Int A2)` by A1,Th4;
hence thesis by Th5;
end;
thus (Int A1),(Cl A2) constitute_a_decomposition
proof
Int A1 = (Cl A1`)` by TOPS_1:def 1;
then Int A1 = (Cl A2)` by A1,Th4;
hence thesis by Th5;
end;
end;

theorem
(Cl A),(Int  A`) constitute_a_decomposition &
(Cl  A`),(Int A) constitute_a_decomposition &
(Int A),(Cl  A`) constitute_a_decomposition &
(Int  A`),(Cl A) constitute_a_decomposition
proof
thus A1: (Cl A),(Int  A`) constitute_a_decomposition
proof
A, A` constitute_a_decomposition by Th6;
hence thesis by Th10;
end;
thus (Cl  A`),(Int A) constitute_a_decomposition
proof
A`,A constitute_a_decomposition by Th6;
hence thesis by Th10;
end;
hence (Int A),(Cl  A`) constitute_a_decomposition;
thus (Int  A`),(Cl A) constitute_a_decomposition by A1;
end;

theorem Th12:
for A1, A2 being Subset of X st
A1,A2 constitute_a_decomposition holds (A1 is open iff A2 is closed)
proof
let A1, A2 be Subset of X;
assume A1,A2 constitute_a_decomposition;
then A1: A1 =  A2` & A2 =  A1` by Th4;
hence A1 is open implies A2 is closed by TOPS_1:30;
assume A2 is closed;
hence thesis by A1,TOPS_1:29;
end;

theorem
for A1, A2 being Subset of X st
A1,A2 constitute_a_decomposition holds (A1 is closed iff A2 is open) by Th12;

theorem Th14:
A1,A2 constitute_a_decomposition & B1,B2 constitute_a_decomposition implies
(A1 /\ B1),(A2 \/ B2) constitute_a_decomposition
proof
assume A1: A1,A2 constitute_a_decomposition;
assume B1,B2 constitute_a_decomposition;
then A2: B1 misses B2 & B1 \/ B2 = [#]X by Th2;
then A3: B1 /\ B2 = {}X & B1 \/ B2 = [#]X by XBOOLE_0:def 7;
A1 misses A2 by A1,Th2;
then A4: A1 /\ A2 = {} by XBOOLE_0:def 7;
(A1 /\ B1) /\ (A2 \/ B2) =((B1 /\ A1) /\ A2) \/ ((A1 /\ B1) /\
B2) by XBOOLE_1:23
.= (B1 /\ (A1 /\ A2)) \/ ((A1 /\ B1) /\ B2) by XBOOLE_1:16
.= (B1 /\ (A1 /\ A2)) \/ (A1 /\ (B1 /\ B2)) by XBOOLE_1:16
.= {}X by A3,A4;
then A5:    (A1 /\ B1) misses (A2 \/ B2) by XBOOLE_0:def 7;
(A1 /\ B1) \/ (A2 \/ B2) =
(A1 \/ (A2 \/ B2)) /\ (B1 \/ (A2 \/ B2)) by XBOOLE_1:24
.= ((A1 \/ A2) \/ B2) /\ (B1 \/ (B2 \/ A2)) by XBOOLE_1:4
.= ((A1 \/ A2) \/ B2) /\ ((B1 \/ B2) \/ A2) by XBOOLE_1:4
.= ([#]X \/ B2) /\ ([#]X \/ A2) by A1,A2,Th2
.= ([#]X \/ B2) /\ [#]X by TOPS_1:2
.= [#]X /\ [#]X by TOPS_1:2
.= [#]X;
hence thesis by A5,Th2;
end;

theorem
A1,A2 constitute_a_decomposition & B1,B2 constitute_a_decomposition implies
(A1 \/ B1),(A2 /\ B2) constitute_a_decomposition by Th14;

begin
:: 2. A Duality Between Pairs of Weakly Separated Subsets.
reserve X for non empty TopSpace, A1, A2 for Subset of X;

theorem Th16:
for A1, A2, C1, C2 being Subset of X st
A1,C1 constitute_a_decomposition & A2,C2 constitute_a_decomposition holds
A1,A2 are_weakly_separated iff C1,C2 are_weakly_separated
proof let A1, A2, C1, C2 be Subset of X;
assume A1,C1 constitute_a_decomposition & A2,C2 constitute_a_decomposition;
then A1: C1 =  A1` & A1 =  C1` & C2 =  A2` & A2 =  C2` by Th4;
thus A1,A2 are_weakly_separated implies C1,C2 are_weakly_separated
proof
assume A1 \ A2,A2 \ A1 are_separated;
then C2 \ C1, C2` \  C1` are_separated by A1,Th1;
then C2 \ C1,C1 \ C2 are_separated by Th1;
hence thesis by TSEP_1:def 7;
end;
assume C1,C2 are_weakly_separated;
then C1 \ C2,C2 \ C1 are_separated by TSEP_1:def 7;
then  C2` \  C1`,C2 \ C1 are_separated by Th1;
then A2 \ A1,A1 \ A2 are_separated by A1,Th1;
hence thesis by TSEP_1:def 7;
end;

theorem
A1,A2 are_weakly_separated iff  A1`, A2` are_weakly_separated
proof
A1, A1` constitute_a_decomposition &
A2, A2` constitute_a_decomposition by Th6;
hence thesis by Th16;
end;

theorem
for A1, A2, C1, C2 being Subset of X st
A1,C1 constitute_a_decomposition & A2,C2 constitute_a_decomposition holds
A1,A2 are_separated implies C1,C2 are_weakly_separated
proof let A1, A2, C1, C2 be Subset of X;
assume A1: A1,C1 constitute_a_decomposition &
A2,C2 constitute_a_decomposition;
assume A1,A2 are_separated;
then A1,A2 are_weakly_separated by TSEP_1:51;
hence thesis by A1,Th16;
end;

theorem Th19:
for A1, A2, C1, C2 being Subset of X st
A1,C1 constitute_a_decomposition & A2,C2 constitute_a_decomposition holds
A1 misses A2 & C1,C2 are_weakly_separated implies A1,A2 are_separated
proof let A1, A2, C1, C2 be Subset of X;
assume A1: A1,C1 constitute_a_decomposition &
A2,C2 constitute_a_decomposition;
assume A1 /\ A2 = {};
then A2: A1 misses A2 by XBOOLE_0:def 7;
assume C1,C2 are_weakly_separated;
then A1,A2 are_weakly_separated by A1,Th16;
hence thesis by A2,TSEP_1:51;
end;

theorem
for A1, A2, C1, C2 being Subset of X st
A1,C1 constitute_a_decomposition & A2,C2 constitute_a_decomposition holds
C1 \/ C2 = the carrier of X & C1,C2 are_weakly_separated implies
A1,A2 are_separated
proof let A1, A2, C1, C2 be Subset of X;
assume A1: A1,C1 constitute_a_decomposition &
A2,C2 constitute_a_decomposition;
then A1 =  C1` & A2 =  C2` by Th4;
then A2: A1 = C1` & A2 = C2`;
assume C1 \/ C2 = the carrier of X;
then C1 \/ C2 = [#]X by PRE_TOPC:12;
then (C1 \/ C2)` = {}X by TOPS_1:8;
then (C1 \/ C2)` = {}X;
then A1 /\ A2 = {} by A2,SUBSET_1:29;
then A3: A1 misses A2 by XBOOLE_0:def 7;
assume C1,C2 are_weakly_separated;
hence thesis by A1,A3,Th19;
end;

theorem
A1,A2 constitute_a_decomposition implies
(A1,A2 are_weakly_separated iff A1,A2 are_separated)
proof
assume A1,A2 constitute_a_decomposition;
then A1 misses A2 by Def1;
hence A1,A2 are_weakly_separated iff A1,A2 are_separated by TSEP_1:51;
end;

theorem Th22:
A1,A2 are_weakly_separated iff
(A1 \/ A2) \ A1,(A1 \/ A2) \ A2 are_separated
proof
A1: (A1 \/ A2) \ A1 = A2 \ A1 & (A1 \/ A2) \ A2 = A1 \ A2 by XBOOLE_1:40;
thus A1,A2 are_weakly_separated implies
(A1 \/ A2) \ A1,(A1 \/ A2) \ A2 are_separated
proof
assume A1 \ A2,A2 \ A1 are_separated;
hence thesis by A1;
end;
assume (A1 \/ A2) \ A1,(A1 \/ A2) \ A2 are_separated;
hence thesis by A1,TSEP_1:def 7;
end;

::An Enlargement Theorem for Subsets.
theorem Th23:
for A1, A2, C1, C2 being Subset of X st
C1 c= A1 & C2 c= A2 & C1 \/ C2 = A1 \/ A2 holds
C1,C2 are_weakly_separated implies A1,A2 are_weakly_separated
proof let A1, A2, C1, C2 be Subset of X;
assume A1: C1 c= A1 & C2 c= A2;
assume A2: C1 \/ C2 = A1 \/ A2;
assume C1,C2 are_weakly_separated;
then A3: (A1 \/ A2) \ C1,(A1 \/ A2) \ C2 are_separated by A2,Th22;
(A1 \/ A2) \ A1 c= (A1 \/ A2) \ C1 &
(A1 \/ A2) \ A2 c= (A1 \/ A2) \ C2 by A1,XBOOLE_1:34;
then (A1 \/ A2) \ A1,(A1 \/ A2) \ A2 are_separated by A3,CONNSP_1:8;
hence thesis by Th22;
end;

theorem Th24:
A1,A2 are_weakly_separated iff
A1 \ (A1 /\ A2),A2 \ (A1 /\ A2) are_separated
proof
A1: A1 \ (A1 /\ A2) = A1 \ A2 & A2 \ (A1 /\ A2) = A2 \ A1 by XBOOLE_1:47;
thus A1,A2 are_weakly_separated implies
A1 \ (A1 /\ A2),A2 \ (A1 /\ A2) are_separated
proof
assume A1 \ A2,A2 \ A1 are_separated;
hence thesis by A1;
end;
assume A1 \ (A1 /\ A2),A2 \ (A1 /\ A2) are_separated;
hence thesis by A1,TSEP_1:def 7;
end;

::An Extenuation Theorem for Subsets.
theorem Th25:
for A1, A2, C1, C2 being Subset of X st
C1 c= A1 & C2 c= A2 & C1 /\ C2 = A1 /\ A2 holds
A1,A2 are_weakly_separated implies C1,C2 are_weakly_separated
proof let A1, A2, C1, C2 be Subset of X;
assume A1: C1 c= A1 & C2 c= A2;
assume A2: C1 /\ C2 = A1 /\ A2;
assume A1,A2 are_weakly_separated;
then A3: A1 \ (C1 /\ C2),A2 \ (C1 /\ C2) are_separated by A2,Th24;
C1 \ (C1 /\ C2) c= A1 \ (C1 /\ C2) &
C2 \ (C1 /\ C2) c= A2 \ (C1 /\ C2) by A1,XBOOLE_1:33;
then C1 \ (C1 /\ C2),C2 \ (C1 /\ C2) are_separated by A3,CONNSP_1:8;
hence thesis by Th24;
end;

::Separated and Weakly Separated Subsets in Subspaces.
reserve X0 for non empty SubSpace of X, B1, B2 for Subset of X0;

theorem Th26:
B1 = A1 & B2 = A2 implies
(A1,A2 are_separated iff B1,B2 are_separated)
proof
assume A1: B1 = A1 & B2 = A2;
then A2: Cl B1 = (Cl A1) /\ [#]X0 & Cl B2 = (Cl A2) /\ [#]X0 by PRE_TOPC:47;
thus A1,A2 are_separated implies B1,B2 are_separated
proof assume A1,A2 are_separated;
then Cl A1 misses A2 & A1 misses Cl A2 by CONNSP_1:def 1;
then Cl A1 /\ A2 = {} & A1 /\ Cl A2 = {} by XBOOLE_0:def 7;
then Cl B1 /\ B2 = {} /\ [#]X0 & B1 /\ Cl B2 = {} /\ [#]X0
by A1,A2,XBOOLE_1:16;
then Cl B1 misses B2 & B1 misses Cl B2 by XBOOLE_0:def 7;
hence B1,B2 are_separated by CONNSP_1:def 1;
end;
thus B1,B2 are_separated implies A1,A2 are_separated
proof assume B1,B2 are_separated;
then ((Cl A1) /\ [#]X0) misses A2 & A1 misses ((Cl A2) /\ [#]X0)
by A1,A2,CONNSP_1:def 1;
then ((Cl A1) /\ [#]X0) /\ A2 = {} & A1 /\ ((Cl A2) /\ [#]X0) = {}
by XBOOLE_0:def 7;
then A3: (Cl A1 /\ A2) /\ [#]X0 = {} & (A1 /\ Cl A2) /\
[#]X0 = {} by XBOOLE_1:16;
Cl A1 /\ A2 c= A2 & A1 /\ Cl A2 c= A1 & A2 c= [#]X0 & A1 c= [#]X0
by A1,PRE_TOPC:14,XBOOLE_1:17;
then Cl A1 /\ A2 c= [#]X0 & A1 /\ Cl A2 c= [#]X0 by XBOOLE_1:1;
then Cl A1 /\ A2 = {} & A1 /\ Cl A2 = {} by A3,XBOOLE_1:28;
then Cl A1 misses A2 & A1 misses Cl A2 by XBOOLE_0:def 7;
hence thesis by CONNSP_1:def 1;
end;
end;

theorem Th27:
B1 = (the carrier of X0) /\ A1 & B2 = (the carrier of X0) /\ A2 implies
(A1,A2 are_separated implies B1,B2 are_separated)
proof
reconsider A1' = A1, A2' = A2 as Subset of X;
assume B1 = (the carrier of X0) /\ A1;
then A1: B1 c= A1 by XBOOLE_1:17;
then B1 is Subset of X by XBOOLE_1:1;
then reconsider C1 = B1 as Subset of X;
assume B2 = (the carrier of X0) /\ A2;
then A2: B2 c= A2 by XBOOLE_1:17;
then B2 is Subset of X by XBOOLE_1:1;
then reconsider C2 = B2 as Subset of X;
assume A1,A2 are_separated;
then A1',A2' are_separated;
then C1,C2 are_separated by A1,A2,CONNSP_1:8;
hence thesis by Th26;
end;

theorem Th28:
B1 = A1 & B2 = A2 implies
(A1,A2 are_weakly_separated iff B1,B2 are_weakly_separated)
proof
assume A1: B1 = A1 & B2 = A2;
thus A1,A2 are_weakly_separated implies B1,B2 are_weakly_separated
proof assume A1,A2 are_weakly_separated;
then A1 \ A2,A2 \ A1 are_separated by TSEP_1:def 7;
then B1 \ B2,B2 \ B1 are_separated by A1,Th26;
hence thesis by TSEP_1:def 7;
end;
thus B1,B2 are_weakly_separated implies A1,A2 are_weakly_separated
proof assume B1,B2 are_weakly_separated;
then B1 \ B2,B2 \ B1 are_separated by TSEP_1:def 7;
then A1 \ A2,A2 \ A1 are_separated by A1,Th26;
hence thesis by TSEP_1:def 7;
end;
end;

theorem Th29:
B1 = (the carrier of X0) /\ A1 & B2 = (the carrier of X0) /\ A2 implies
(A1,A2 are_weakly_separated implies B1,B2 are_weakly_separated)
proof
assume A1: B1 = (the carrier of X0) /\ A1 & B2 = (the carrier of X0) /\ A2;
assume A2: A1 \ A2,A2 \ A1 are_separated;
B1 \ B2 = (the carrier of X0) /\ (A1 \ A2) &
B2 \ B1 = (the carrier of X0) /\ (A2 \ A1) by A1,XBOOLE_1:50;
then B1 \ B2,B2 \ B1 are_separated by A2,Th27;
hence thesis by TSEP_1:def 7;
end;

begin
:: 3. Certain Subspace-Decompositions of a Topological Space.

definition let X be non empty TopSpace, X1, X2 be SubSpace of X;
pred X1,X2 constitute_a_decomposition means
:Def2: for A1, A2 being Subset of X st
A1 = the carrier of X1 & A2 = the carrier of X2 holds
A1,A2 constitute_a_decomposition;
symmetry;
antonym X1,X2 do_not_constitute_a_decomposition;
end;
reserve X0, X1, X2, Y1, Y2 for non empty SubSpace of X;

theorem Th30:
X1,X2 constitute_a_decomposition iff
X1 misses X2 & the TopStruct of X = X1 union X2
proof
reconsider A1 = the carrier of X1,
A2 = the carrier of X2 as Subset of X by TSEP_1:1;
thus X1,X2 constitute_a_decomposition implies
X1 misses X2 & the TopStruct of X = X1 union X2
proof
assume for A1, A2 being Subset of X st
A1 = the carrier of X1 & A2 = the carrier of X2 holds
A1,A2 constitute_a_decomposition;
then A1: A1,A2 constitute_a_decomposition;
then A1 misses A2 by Def1;
hence X1 misses X2 by TSEP_1:def 3;
A1 \/ A2 = the carrier of X by A1,Def1;
then the carrier of X1 union X2 = the carrier of X &
X is SubSpace of X by TSEP_1:2,def 2;
hence thesis by TSEP_1:5;
end;
assume A2: X1 misses X2;
assume A3: the TopStruct of X = X1 union X2;
now let A1, A2 be Subset of X;
assume A4: A1 = the carrier of X1 & A2 = the carrier of X2;
then A5:A1 misses A2 by A2,TSEP_1:def 3;
A1 \/ A2 = the carrier of X by A3,A4,TSEP_1:def 2;
hence A1,A2 constitute_a_decomposition by A5,Def1;
end;
hence X1,X2 constitute_a_decomposition by Def2;
end;

canceled;

theorem Th32:
X0,X0 do_not_constitute_a_decomposition
proof
reconsider A0 = the carrier of X0 as Subset of X by TSEP_1:1;
now take A1 = A0, A2 = A0;
thus A1 = the carrier of X0 & A2 = the carrier of X0 &
A1,A2 do_not_constitute_a_decomposition by Th8;
end;
hence thesis by Def2;
end;

definition let X be non empty TopSpace, A1, A2 be non empty SubSpace of X;
redefine pred A1,A2 constitute_a_decomposition;
irreflexivity by Th32;
end;

theorem
X1,X0 constitute_a_decomposition & X0,X2 constitute_a_decomposition
implies the TopStruct of X1 = the TopStruct of X2
proof
assume A1: for A1, A0 being Subset of X st
A1 = the carrier of X1 & A0 = the carrier of X0 holds
A1,A0 constitute_a_decomposition;
assume A2: for A0, A2 being Subset of X st
A0 = the carrier of X0 & A2 = the carrier of X2 holds
A0,A2 constitute_a_decomposition;
reconsider A0 = the carrier of X0,
A1 = the carrier of X1,
A2 = the carrier of X2 as Subset of X by TSEP_1:1;
A1,A0 constitute_a_decomposition &
A0,A2 constitute_a_decomposition by A1,A2;
then A1 = A2 by Th9;
hence thesis by TSEP_1:5;
end;

theorem Th34:
for X1, X2, Y1, Y2 being non empty SubSpace of X st
X1,Y1 constitute_a_decomposition & X2,Y2 constitute_a_decomposition holds
Y1 union Y2 = the TopStruct of X iff X1 misses X2
proof let X1, X2, Y1, Y2 be non empty SubSpace of X;
reconsider A1 = the carrier of X1,
A2 = the carrier of X2 as Subset of X by TSEP_1:1;
reconsider B1 = the carrier of Y1,
B2 = the carrier of Y2 as Subset of X by TSEP_1:1;
assume X1,Y1 constitute_a_decomposition &
X2,Y2 constitute_a_decomposition;
then A1,B1 constitute_a_decomposition &
A2,B2 constitute_a_decomposition by Def2;
then B1 =  A1` & A1 =  B1` & B2 =  A2` & A2 =  B2` by Th4;
then A1: B1 = A1` & A1 = B1` & B2 = A2` & A2 = B2`;
thus Y1 union Y2 = the TopStruct of X implies X1 misses X2
proof
assume Y1 union Y2 = the TopStruct of X;
then B1 \/ B2 = the carrier of X by TSEP_1:def 2;
then B1 \/ B2 = [#]X by PRE_TOPC:12;
then (B1 \/ B2)` = {}X by TOPS_1:8;
then (B1 \/ B2)` = {}X;
then A1 /\ A2 = {} by A1,SUBSET_1:29;
then A1 misses A2 by XBOOLE_0:def 7;
hence thesis by TSEP_1:def 3;
end;
assume X1 misses X2;
then A1 misses A2 by TSEP_1:def 3;
then A1 /\ A2 = {}X by XBOOLE_0:def 7;
then (B1 \/ B2)` = {}X by A1,SUBSET_1:29;
then B1 \/ B2 = ({}X)`;
then B1 \/ B2 = [#]X by PRE_TOPC:27;
then B1 \/ B2 = the carrier of X by PRE_TOPC:12;
then the carrier of Y1 union Y2 = the carrier of X &
X is SubSpace of X by TSEP_1:2,def 2;
hence thesis by TSEP_1:5;
end;

theorem Th35:
X1,X2 constitute_a_decomposition implies (X1 is open iff X2 is closed)
proof
the carrier of X1 is Subset of X &
the carrier of X2 is Subset of X by TSEP_1:1;
then reconsider A1 = the carrier of X1,
A2 = the carrier of X2 as Subset of X;
assume A1: for A1, A2 being Subset of X st
A1 = the carrier of X1 & A2 = the carrier of X2 holds
A1,A2 constitute_a_decomposition;
thus X1 is open implies X2 is closed
proof
assume
A2:  for A1 being Subset of X st A1 = the carrier of X1 holds
A1 is open;
now let A2 be Subset of X;
assume A2 = the carrier of X2;
then A1,A2 constitute_a_decomposition & A1 is open by A1,A2;
hence A2 is closed by Th12;
end;
hence thesis by BORSUK_1:def 14;
end;
assume
A3:  for A2 being Subset of X st A2 = the carrier of X2 holds
A2 is closed;
now let A1 be Subset of X;
assume A1 = the carrier of X1;
then A1,A2 constitute_a_decomposition & A2 is closed by A1,A3;
hence A1 is open by Th12;
end;
hence thesis by TSEP_1:def 1;
end;

theorem
X1,X2 constitute_a_decomposition implies (X1 is closed iff X2 is open)
by Th35;

theorem Th37:
X1 meets Y1 &
X1,X2 constitute_a_decomposition & Y1,Y2 constitute_a_decomposition implies
(X1 meet Y1),(X2 union Y2) constitute_a_decomposition
proof
reconsider A1 = the carrier of X1,
A2 = the carrier of X2 as Subset of X by TSEP_1:1;
reconsider B1 = the carrier of Y1,
B2 = the carrier of Y2 as Subset of X by TSEP_1:1;
assume A1: X1 meets Y1;
assume for A1, A2 being Subset of X st
A1 = the carrier of X1 & A2 = the carrier of X2 holds
A1,A2 constitute_a_decomposition;
then A2: A1,A2 constitute_a_decomposition;
assume for B1, B2 being Subset of X st
B1 = the carrier of Y1 & B2 = the carrier of Y2 holds
B1,B2 constitute_a_decomposition;
then A3: B1,B2 constitute_a_decomposition;
now let C, D be Subset of X;
assume C = the carrier of X1 meet Y1 & D = the carrier of X2 union Y2;
then C = A1 /\ B1 & D = A2 \/ B2 by A1,TSEP_1:def 2,def 5;
hence C,D constitute_a_decomposition by A2,A3,Th14;
end;
hence thesis by Def2;
end;

theorem
X2 meets Y2 &
X1,X2 constitute_a_decomposition & Y1,Y2 constitute_a_decomposition implies
(X1 union Y1),(X2 meet Y2) constitute_a_decomposition by Th37;

begin
:: 4. A Duality Between Pairs of Weakly Separated Subspaces.
reserve X for non empty TopSpace;

theorem Th39:
for X1, X2, Y1, Y2 being SubSpace of X st
X1,Y1 constitute_a_decomposition & X2,Y2 constitute_a_decomposition holds
X1,X2 are_weakly_separated implies Y1,Y2 are_weakly_separated
proof let X1, X2, Y1, Y2 be SubSpace of X;
assume A1: for A1, B1 being Subset of X st
A1 = the carrier of X1 & B1 = the carrier of Y1 holds
A1,B1 constitute_a_decomposition;
assume A2: for A2, B2 being Subset of X st
A2 = the carrier of X2 & B2 = the carrier of Y2 holds
A2,B2 constitute_a_decomposition;
assume A3: for A1, A2 being Subset of X st
A1 = the carrier of X1 & A2 = the carrier of X2 holds
A1,A2 are_weakly_separated;
now let B1, B2 be Subset of X;
reconsider A1 = the carrier of X1,
A2 = the carrier of X2 as Subset of X
by TSEP_1:1;
assume B1 = the carrier of Y1 & B2 = the carrier of Y2;
then A1,B1 constitute_a_decomposition &
A2,B2 constitute_a_decomposition &
A1,A2 are_weakly_separated by A1,A2,A3;
hence B1,B2 are_weakly_separated by Th16;
end;
hence thesis by TSEP_1:def 9;
end;

theorem
for X1, X2, Y1, Y2 being non empty SubSpace of X st
X1,Y1 constitute_a_decomposition & X2,Y2 constitute_a_decomposition holds
X1,X2 are_separated implies Y1,Y2 are_weakly_separated
proof let X1, X2, Y1, Y2 be non empty SubSpace of X;
assume A1: X1,Y1 constitute_a_decomposition &
X2,Y2 constitute_a_decomposition;
assume X1,X2 are_separated;
then X1,X2 are_weakly_separated by TSEP_1:85;
hence thesis by A1,Th39;
end;

theorem Th41:
for X1, X2, Y1, Y2 being non empty SubSpace of X st
X1,Y1 constitute_a_decomposition & X2,Y2 constitute_a_decomposition holds
X1 misses X2 & Y1,Y2 are_weakly_separated implies X1,X2 are_separated
proof let X1, X2, Y1, Y2 be non empty SubSpace of X;
assume A1: X1,Y1 constitute_a_decomposition &
X2,Y2 constitute_a_decomposition;
assume A2: X1 misses X2;
assume Y1,Y2 are_weakly_separated;
then X1,X2 are_weakly_separated by A1,Th39;
hence thesis by A2,TSEP_1:85;
end;

theorem
for X1, X2, Y1, Y2 being non empty SubSpace of X st
X1,Y1 constitute_a_decomposition & X2,Y2 constitute_a_decomposition holds
Y1 union Y2 = the TopStruct of X & Y1,Y2 are_weakly_separated implies
X1,X2 are_separated
proof let X1, X2, Y1, Y2 be non empty SubSpace of X;
assume A1: X1,Y1 constitute_a_decomposition &
X2,Y2 constitute_a_decomposition;
assume Y1 union Y2 = the TopStruct of X;
then A2: X1 misses X2 by A1,Th34;
assume Y1,Y2 are_weakly_separated;
hence thesis by A1,A2,Th41;
end;

theorem
for X1, X2 being non empty SubSpace of X st X1,X2 constitute_a_decomposition
holds (X1,X2 are_weakly_separated iff X1,X2 are_separated)
proof let X1, X2 be non empty SubSpace of X;
assume X1,X2 constitute_a_decomposition;
then X1 misses X2 by Th30;
hence X1,X2 are_weakly_separated iff X1,X2 are_separated by TSEP_1:85;
end;

::An Enlargement Theorem for Subspaces.
theorem
for X1, X2, Y1, Y2 being non empty SubSpace of X st
Y1 is SubSpace of X1 & Y2 is SubSpace of X2 & Y1 union Y2 = X1 union X2 holds
Y1,Y2 are_weakly_separated implies X1,X2 are_weakly_separated
proof let X1, X2, Y1, Y2 be non empty SubSpace of X;
reconsider A1 = the carrier of X1,
A2 = the carrier of X2 as Subset of X by TSEP_1:
1;
reconsider C1 = the carrier of Y1,
C2 = the carrier of Y2 as Subset of X by TSEP_1:
1;
assume Y1 is SubSpace of X1 & Y2 is SubSpace of X2;
then A1: C1 c= A1 & C2 c= A2 by TSEP_1:4;
assume A2: Y1 union Y2 = X1 union X2;
assume Y1,Y2 are_weakly_separated;
then A3: C1,C2 are_weakly_separated by TSEP_1:def 9;
now let A1, A2 be Subset of X;
assume A4: A1 = the carrier of X1 & A2 = the carrier of X2;
then A1 \/ A2 = the carrier of X1 union X2 by TSEP_1:def 2;
then A1 \/ A2 = C1 \/ C2 by A2,TSEP_1:def 2;
hence A1,A2 are_weakly_separated by A1,A3,A4,Th23;
end;
hence thesis by TSEP_1:def 9;
end;

::An Extenuation Theorem for Subspaces.
theorem
for X1, X2, Y1, Y2 being non empty SubSpace of X st
Y1 is SubSpace of X1 & Y2 is SubSpace of X2 &
Y1 meets Y2 & Y1 meet Y2 = X1 meet X2 holds
X1,X2 are_weakly_separated implies Y1,Y2 are_weakly_separated
proof let X1, X2, Y1, Y2 be non empty SubSpace of X;
reconsider A1 = the carrier of X1,
A2 = the carrier of X2 as Subset of X
by TSEP_1:1;
reconsider C1 = the carrier of Y1,
C2 = the carrier of Y2 as Subset of X
by TSEP_1:1;
assume Y1 is SubSpace of X1 & Y2 is SubSpace of X2;
then A1: C1 c= A1 & C2 c= A2 by TSEP_1:4;
then A2: C1 /\ C2 c= A1 /\ A2 by XBOOLE_1:27;
assume A3: Y1 meets Y2;
assume A4: Y1 meet Y2 = X1 meet X2;
assume X1,X2 are_weakly_separated;
then A5: A1,A2 are_weakly_separated by TSEP_1:def 9;
now let C1, C2 be Subset of X;
assume A6: C1 = the carrier of Y1 & C2 = the carrier of Y2;
then C1 meets C2 by A3,TSEP_1:def 3;
then C1 /\ C2 <> {} by XBOOLE_0:def 7;
then A1 /\ A2 <> {} by A2,A6,XBOOLE_1:3;
then A1 meets A2 by XBOOLE_0:def 7;
then X1 meets X2 by TSEP_1:def 3;
then A1 /\ A2 = the carrier of X1 meet X2 by TSEP_1:def 5;
then A1 /\ A2 = C1 /\ C2 by A3,A4,A6,TSEP_1:def 5;
hence C1,C2 are_weakly_separated by A1,A5,A6,Th25;
end;
hence thesis by TSEP_1:def 9;
end;

::Separated and Weakly Separated Subspaces in Subspaces.
reserve X0 for non empty SubSpace of X;

theorem Th46:
for X1, X2 being SubSpace of X, Y1, Y2 being SubSpace of X0 st
the carrier of X1 = the carrier of Y1 &
the carrier of X2 = the carrier of Y2 holds
X1,X2 are_separated iff Y1,Y2 are_separated
proof let X1, X2 be SubSpace of X, X01, X02 be SubSpace of X0;
assume A1: the carrier of X1 = the carrier of X01 &
the carrier of X2 = the carrier of X02;
reconsider A1 = the carrier of X1,
A2 = the carrier of X2 as Subset of X
by TSEP_1:1;
reconsider B1 = the carrier of X01,
B2 = the carrier of X02 as Subset of X0
by TSEP_1:1;
thus X1,X2 are_separated implies X01,X02 are_separated
proof assume X1,X2 are_separated;
then A1,A2 are_separated by TSEP_1:def 8;
then for B1,B2 be Subset of X0 holds
B1 = the carrier of X01 & B2 = the carrier of X02 implies B1,B2 are_separated
by A1,Th26;
hence thesis by TSEP_1:def 8;
end;
thus X01,X02 are_separated implies X1,X2 are_separated
proof assume X01,X02 are_separated;
then B1,B2 are_separated by TSEP_1:def 8;
then for A1,A2 be Subset of X holds
A1 = the carrier of X1 & A2 = the carrier of X2 implies A1,A2 are_separated
by A1,Th26;
hence thesis by TSEP_1:def 8;
end;
end;

theorem
for X1, X2 being non empty SubSpace of X st X1 meets X0 & X2 meets X0
for Y1, Y2 being SubSpace of X0 st Y1 = X1 meet X0 & Y2 = X2 meet X0 holds
X1,X2 are_separated implies Y1,Y2 are_separated
proof let X1, X2 be non empty SubSpace of X;
assume A1: X1 meets X0 & X2 meets X0;
let Y1, Y2 be SubSpace of X0;
assume A2: Y1 = X1 meet X0 & Y2 = X2 meet X0;
assume X1,X2 are_separated;
then (X1 meet X0),(X2 meet X0) are_separated by A1,TSEP_1:76;
hence thesis by A2,Th46;
end;

theorem
for X1, X2 being SubSpace of X, Y1, Y2 being SubSpace of X0 st
the carrier of X1 = the carrier of Y1 &
the carrier of X2 = the carrier of Y2 holds
X1,X2 are_weakly_separated iff Y1,Y2 are_weakly_separated
proof let X1, X2 be SubSpace of X, X01, X02 be SubSpace of X0;
assume A1: the carrier of X1 = the carrier of X01 &
the carrier of X2 = the carrier of X02;
reconsider A1 = the carrier of X1 as Subset of X
by TSEP_1:1;
reconsider A2 = the carrier of X2 as Subset of X
by TSEP_1:1;
reconsider B1 = the carrier of X01 as Subset of X0
by TSEP_1:1;
reconsider B2 = the carrier of X02 as Subset of X0
by TSEP_1:1;
thus X1,X2 are_weakly_separated implies X01,X02 are_weakly_separated
proof assume X1,X2 are_weakly_separated;
then A1,A2 are_weakly_separated by TSEP_1:def 9;

then for B1,B2 be Subset of X0 holds
B1 = the carrier of X01 & B2 = the carrier of X02 implies
B1,B2 are_weakly_separated by A1,Th28;
hence thesis by TSEP_1:def 9;
end;
thus X01,X02 are_weakly_separated implies X1,X2 are_weakly_separated
proof assume X01,X02 are_weakly_separated;
then B1,B2 are_weakly_separated by TSEP_1:def 9;

then for A1,A2 be Subset of X holds
A1 = the carrier of X1 & A2 = the carrier of X2 implies
A1,A2 are_weakly_separated by A1,Th28;
hence thesis by TSEP_1:def 9;
end;
end;

theorem
for X1, X2 being non empty SubSpace of X st X1 meets X0 & X2 meets X0
for Y1, Y2 being SubSpace of X0 st Y1 = X1 meet X0 & Y2 = X2 meet X0 holds
X1,X2 are_weakly_separated implies Y1,Y2 are_weakly_separated
proof let X1, X2 be non empty SubSpace of X;
reconsider A1 = the carrier of X1,
A2 = the carrier of X2 as Subset of X
by TSEP_1:1;
assume A1: X1 meets X0 & X2 meets X0;
let Y1, Y2 be SubSpace of X0;
assume A2: Y1 = X1 meet X0 & Y2 = X2 meet X0;
assume X1,X2 are_weakly_separated;
then A3: A1,A2 are_weakly_separated by TSEP_1:def 9;
now let C1, C2 be Subset of X0;
assume C1 = the carrier of Y1 & C2 = the carrier of Y2;
then C1 = (the carrier of X0) /\ A1 &
C2 = (the carrier of X0) /\ A2 by A1,A2,TSEP_1:def 5;
hence C1,C2 are_weakly_separated by A3,Th29;
end;
hence thesis by TSEP_1:def 9;
end;
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