Components and Unions of Components

by
Yatsuka Nakamura, and
Andrzej Trybulec

Received February 5, 1996

Copyright (c) 1996 Association of Mizar Users

MML identifier: CONNSP_3
[ MML identifier index ]

```environ

vocabulary PRE_TOPC, RELAT_1, CONNSP_1, SETFAM_1, RELAT_2, TARSKI, BOOLE,
SUBSET_1, CONNSP_3;
notation TARSKI, XBOOLE_0, ZFMISC_1, SETFAM_1, STRUCT_0, PRE_TOPC, CONNSP_1;
constructors CONNSP_1, MEMBERED;
clusters PRE_TOPC, STRUCT_0, SUBSET_1, XBOOLE_0;
requirements SUBSET, BOOLE;
definitions TARSKI, XBOOLE_0;
theorems TARSKI, ZFMISC_1, SETFAM_1, PRE_TOPC, SUBSET_1, CONNSP_1, JORDAN1,
XBOOLE_0, XBOOLE_1;
schemes PRE_TOPC;

begin :: The component of a subset in a topological space

reserve x,X,X2,Y,Y2 for set;
reserve GX for non empty TopSpace;
reserve A2,B2 for Subset of GX;
reserve B for Subset of GX;

definition let GX be TopStruct, V be Subset of GX;
func skl V -> Subset of GX means
:Def1:ex F being Subset-Family of GX st
(for A being Subset of GX holds A in F iff
A is connected & V c= A) & union F = it;
existence
proof
defpred P[set] means
ex A1 being Subset of GX st A1 = \$1 & A1 is connected & V c= \$1;
consider F being Subset-Family of GX such that
A1:  for A being Subset of GX
holds A in F iff P[A] from SubFamExS;
take union F, F;
thus for A being Subset of GX holds A in F iff A is connected & V c= A
proof
let A be Subset of GX;
thus A in F implies A is connected & V c= A
proof
assume A in F;
then ex A1 being Subset of GX st A1 = A & A1 is connected & V c= A by
A1;
hence A is connected & V c= A;
end;
thus thesis by A1;
end;
thus thesis;
end;
uniqueness
proof let S,S' be Subset of GX; assume
A2:  (ex F being Subset-Family of GX st
(for A being Subset of GX
holds A in F iff A is connected &
V c= A) & union F = S) & ex F' being Subset-Family of GX st
(for A being Subset of GX
holds A in F' iff A is connected &
V c= A) & union F' = S';
then consider F being Subset-Family of GX such that
A3:  (for A being Subset of GX holds A in F iff A is connected
& V c= A) & union F = S;
consider F' being Subset-Family of GX such that
A4: (for A being Subset of GX holds A in F' iff A is connected
& V c= A) & union F' = S' by A2;
now let y be set;
A5:  now assume
y in S;
then consider B being set such that
A6: y in B & B in F by A3,TARSKI:def 4;
reconsider B as Subset of GX by A6;
B is connected & V c= B & y in B by A3,A6;
then B in F'& y in B by A4;
hence y in S' by A4,TARSKI:def 4;
end;
now assume
y in S';
then consider B being set such that
A7: y in B & B in F' by A4,TARSKI:def 4;
reconsider B as Subset of GX by A7;
B is connected & V c= B & y in B by A4,A7;
then B in F & y in B by A3;
hence y in S by A3,TARSKI:def 4;
end;
hence y in S iff y in S' by A5;
end;
hence S = S' by TARSKI:2;
end;
end;

theorem Th1:
for GX being TopSpace, V being Subset of GX st
(ex A being Subset of GX st A is connected & V c= A)
holds V c= skl V
proof
let GX be TopSpace, V be Subset of GX;
given A being Subset of GX such that
A1: A is connected & V c= A;
consider F being Subset-Family of GX such that
A2: (for A being Subset of GX holds A in F iff A is connected
& V c= A) and
A3: skl V = union F by Def1;
A4: F <> {} by A1,A2;
for A being set holds A in F implies V c= A by A2;
then A5: V c= meet F by A4,SETFAM_1:6;
meet F c= union F by SETFAM_1:3;
hence V c= skl V by A3,A5,XBOOLE_1:1;
end;

theorem
for GX being TopSpace, V being Subset of GX st
(not ex A being Subset of GX st A is connected & V c= A)
holds skl V = {}
proof
let GX be TopSpace, V be Subset of GX such that
A1: not ex A being Subset of GX st A is connected & V c= A;
consider F being Subset-Family of GX such that
A2: (for A being Subset of GX holds A in F iff A is connected
& V c= A) and
A3: skl V = union F by Def1;
now assume F <> {};
then consider A being Subset of GX such that
A4:   A in F by SUBSET_1:10;
reconsider A as Subset of GX;
A is connected & V c= A by A2,A4;
hence contradiction by A1;
end;
hence thesis by A3,ZFMISC_1:2;
end;

theorem Th3:
skl {}GX = the carrier of GX
proof
defpred P[set] means
ex A1 being Subset of GX st A1 = \$1 & A1 is connected & {}GX c= \$1;
consider F being Subset-Family of GX such that
A1: for A being Subset of GX
holds A in F iff P[A] from SubFamExS;
A2: for A being Subset of GX holds A in F iff A is connected & {}GX c= A
proof
let A be Subset of GX;
thus A in F implies A is connected & {}GX c= A
proof
assume A in F;
then ex A1 being Subset of GX st A1 = A & A1 is connected & {}GX c= A
by A1;
hence thesis;
end;
thus thesis by A1;
end;
now let x be set;
hereby assume x in the carrier of GX;
then reconsider p = x as Point of GX;
reconsider Y = skl p as set;
take Y;
thus x in Y by CONNSP_1:40;
skl p is connected & {}GX c= Y by CONNSP_1:41,XBOOLE_1:2;
hence Y in F by A2;
end;
given Y being set such that
A3: x in Y and
A4: Y in F;
thus x in the carrier of GX by A3,A4;
end;
then union F = the carrier of GX by TARSKI:def 4;
hence thesis by A2,Def1;
end;

theorem
for V being Subset of GX st V is connected holds skl V <>{}
proof let V be Subset of GX such that
A1: V is connected;
per cases;
suppose V = {};
then V = {}GX;
then skl V = the carrier of GX by Th3;
hence thesis;
suppose
A2: V <>{};
now assume A3:skl V={};
V c= skl V by A1,Th1;
hence contradiction by A2,A3,XBOOLE_1:3;
end;
hence thesis;
end;

theorem Th5:
for GX being TopSpace, V being Subset of GX st
V is connected & V <> {} holds skl V is connected
proof let GX be TopSpace;let V be Subset of GX;
assume A1:V is connected & V<>{};
consider F being Subset-Family of GX such that
A2: for A being Subset of GX holds
A in F iff A is connected & V c= A and
A3: skl V = union F by Def1;
A4: for A being set st A in F holds V c= A by A2;
A5: for A being Subset of GX st A in F holds A is connected by A2;
F <> {} by A1,A2;
then V c= meet F by A4,SETFAM_1:6;
then meet F<>{}(GX) by A1,XBOOLE_1:3;
hence skl V is connected by A3,A5,CONNSP_1:27;
end;

theorem Th6:
for V,C being Subset of GX st V is connected & C is connected
holds skl V c= C implies C = skl V
proof let V,C be Subset of GX;
assume that
A1: V is connected and
A2: C is connected; assume
A3: skl V c= C;
consider F being Subset-Family of GX such that
A4: for A being Subset of GX holds
(A in F iff A is connected & V c= A) and
A5: skl V = union F by Def1;
V c= skl V by A1,Th1; then V c= C by A3,XBOOLE_1:1;
then C in F by A2,A4;
then C c= skl V by A5,ZFMISC_1:92;
hence C = skl V by A3,XBOOLE_0:def 10;
end;

theorem Th7:
for A being Subset of GX st A is_a_component_of GX
holds skl A=A
proof let A be Subset of GX;assume A1:A is_a_component_of GX;
then A2:A is connected by CONNSP_1:def 5;
then A3:A c= skl A by Th1;
A <>{}(GX) by A1,CONNSP_1:34;
then skl A is connected by A2,Th5;
hence thesis by A1,A3,CONNSP_1:def 5;
end;

theorem Th8:for A being Subset of GX holds
A is_a_component_of GX iff
ex V being Subset of GX st V is connected &
V <> {} & A = skl V
proof let A be Subset of GX;
A1:  now assume
A2:  A is_a_component_of GX;
then A3:   A <> {}(GX) & A is connected by CONNSP_1:34,def 5;
take V = A;
thus V is connected & V<>{} & A = skl V by A2,A3,Th7;
end;
now given V being Subset of GX such that
A4: V is connected and A5:V<>{} and
A6:   A = skl V;
A7:   A is connected by A4,A5,A6,Th5;
for B being Subset of GX st B is connected
holds A c= B implies A = B by A4,A6,Th6;
hence A is_a_component_of GX by A7,CONNSP_1:def 5;
end;
hence thesis by A1;
end;

theorem
for V being Subset of GX st V is connected & V<>{}
holds skl V is_a_component_of GX by Th8;

theorem Th10:
for A, V be Subset of GX st
A is_a_component_of GX & V is connected & V c= A & V<>{}
holds A = skl V
proof
let A, V be Subset of GX;
assume that
A1: A is_a_component_of GX and
A2:V is connected and
A3: V c= A and A4:V<>{}; assume
A5: A <> skl V;
reconsider A' = A as Subset of GX;
V c= skl V by A2,Th1;
then A6: A meets (skl V) by A3,A4,XBOOLE_1:67;
skl V is_a_component_of GX by A2,A4,Th8;
then A',skl V are_separated by A1,A5,CONNSP_1:36;
hence contradiction by A6,CONNSP_1:2;
end;

theorem
Th11: for V being Subset of GX st V is connected & V<>{} holds
skl (skl V)=skl V
proof let V be Subset of GX;assume A1:V is connected & V<>{};
then A2:skl V is_a_component_of GX by Th8;
then skl V <> {}(GX) by CONNSP_1:34;
then skl V is connected & skl V <> {} by A1,Th5;
hence thesis by A2,Th10;
end;

theorem
Th12: for A,B being Subset of GX st A is connected &
B is connected & A<>{} & A c= B
holds skl A = skl B
proof let A,B be Subset of GX;assume
A1: A is connected & B is connected & A<>{} & A c= B;
then A2:B<>{} by XBOOLE_1:3;
A3: B c= skl B by A1,Th1;
then A4: A c= skl B by A1,XBOOLE_1:1;
A5: skl B is connected by A1,A2,Th5;
A6: skl A is connected by A1,Th5;
A7:skl B c= skl A
proof
consider F being Subset-Family of GX such that
A8:   (for D being Subset of GX holds D in F iff
D is connected & A c= D) & union F = skl A by Def1;
skl B in F by A4,A5,A8;
hence thesis by A8,ZFMISC_1:92;
end;
skl A c= skl B
proof
consider F being Subset-Family of GX such that
A9:   (for D being Subset of GX holds D in F iff
D is connected & B c= D) & union F = skl B by Def1;
B c= skl A by A3,A7,XBOOLE_1:1;
then skl A in F by A6,A9;
hence thesis by A9,ZFMISC_1:92;
end;
hence thesis by A7,XBOOLE_0:def 10;
end;

theorem
Th13: for A,B being Subset of GX st A is connected &
B is connected & A<>{} & A c= B
holds B c= skl A
proof let A,B be Subset of GX;
assume A1: A is connected & B is connected & A<>{} & A c= B;
then skl A = skl B by Th12;
hence thesis by A1,Th1;
end;

theorem
Th14: for A being Subset of GX,B being Subset of GX st
A is connected & A \/ B is connected & A<>{}
holds A \/ B c= skl A
proof
let A be Subset of GX,B be Subset of GX;assume that
A1: A is connected & A \/ B is connected & A<>{};
A c= A \/ B by XBOOLE_1:7;
then skl (A \/ B) = skl A by A1,Th12;
hence A \/ B c= skl A by A1,Th1;
end;

theorem
Th15: for A being Subset of GX, p being Point of GX
st A is connected & p in A holds
skl p=skl A
proof let A be Subset of GX, p be Point of GX;
assume A1:A is connected & p in A;
then A2: A c= skl A by Th1;
skl A is_a_component_of GX by A1,Th8;
hence thesis by A1,A2,CONNSP_1:44;
end;

theorem
for A,B being Subset of GX st A is connected &
B is connected & A meets B
holds A \/ B c= skl A & A \/ B c= skl B & A c= skl B & B c= skl A
proof let A,B be Subset of GX;
assume A1: A is connected & B is connected & A /\ B <>{};
for C,D being Subset of GX st C is connected &
D is connected & C /\ D <>{}
holds C \/ D c= skl C
proof let C,D be Subset of GX;
assume A2: C is connected & D is connected & C /\ D <>{};
then A3: C <>{};
C meets D by A2,XBOOLE_0:def 7;
then not C,D are_separated by CONNSP_1:2;
then C \/ D is connected by A2,CONNSP_1:18;
hence thesis by A2,A3,Th14;
end;
then A4: A \/ B c= skl A & A \/ B c= skl B by A1;
A c= A \/ B & B c= A \/ B by XBOOLE_1:7;
hence thesis by A4,XBOOLE_1:1;
end;

theorem
for A being Subset of GX st A is connected & A<>{}
holds Cl A c= skl A
proof let A be Subset of GX;
assume that A1: A is connected and
A2: A<>{};
A3: A c= Cl A by PRE_TOPC:48;
Cl A is connected by A1,CONNSP_1:20;
hence thesis by A1,A2,A3,Th13;
end;

theorem
for A,B being Subset of GX
st A is_a_component_of GX & B is connected & B<>{} & A misses B
holds A misses skl B
proof let A,B be Subset of GX;assume
A1: A is_a_component_of GX & B is connected & B<>{} &
A /\ B ={};
now assume A /\ skl B <>{};
then consider x being Point of GX such that A2:x in A /\ skl B by SUBSET_1:10
;
A3: x in A & x in skl B by A2,XBOOLE_0:def 3;
A4:skl A=A by A1,Th7;
A is connected by A1,CONNSP_1:def 5;
then A5:skl x=skl A by A3,Th15;
A6:skl B=skl skl B by A1,Th11;
skl B is connected by A1,Th5;
then A7:(skl B) /\ B={} by A1,A3,A4,A5,A6,Th15;
B c= skl B by A1,Th1;
hence contradiction by A1,A7,XBOOLE_1:28;
end;
:: then A /\ skl B ={};
hence thesis by XBOOLE_0:def 7;
end;

begin :: On unions of components

Lm1:
now let GX be TopStruct;
{} c= bool (the carrier of GX) by XBOOLE_1:2;
then {} is Subset-Family of GX by SETFAM_1:def 7;
then reconsider S={} as Subset-Family of GX;
(for B being Subset of GX st B in S holds B is_a_component_of GX) &
{}(GX)=union S by ZFMISC_1:2;
hence ex F being Subset-Family of GX st (for B being
Subset of GX st B in F holds B is_a_component_of GX) &
{}(GX)=union F;
end;

definition let GX be TopStruct;
mode a_union_of_components of GX -> Subset of GX means
:Def2:ex F being Subset-Family of GX st (for B being
Subset of GX st B in F holds B is_a_component_of GX) &
it = union F;
existence
proof
take {}GX;
thus thesis by Lm1;
end;
end;

theorem
Th19: {}(GX) is a_union_of_components of GX
proof
thus ex F being Subset-Family of GX st (for B being
Subset of GX st B in F holds B is_a_component_of GX) &
{}(GX) = union F by Lm1;
end;

theorem
for A being Subset of GX st A=(the carrier of GX) holds
A is a_union_of_components of GX
proof let A be Subset of GX;assume A1: A=(the carrier of GX);
{B : B is_a_component_of GX} c=
bool (the carrier of GX)
proof let x;assume x in {B : B is_a_component_of GX};
then ex B being Subset of GX st x=B & B is_a_component_of GX;
hence thesis;
end;
then {B: B is_a_component_of GX} is Subset-Family of (the carrier of GX)
by SETFAM_1:def 7;
then reconsider S={B: B is_a_component_of GX} as Subset-Family of GX
;
A2:the carrier of GX=union S
proof
the carrier of GX c= union S
proof let x;assume x in the carrier of GX;
then reconsider p=x as Point of GX;
A3:p in skl p by CONNSP_1:40;
set B3=skl p;
B3 is_a_component_of GX by CONNSP_1:43;
then B3 in S;
hence x in union S by A3,TARSKI:def 4;
end;
hence thesis by XBOOLE_0:def 10;
end;
for B being Subset of GX st B in S holds B is_a_component_of GX
proof let B be Subset of GX;assume B in S;
then ex B2 being Subset of GX st B=B2 & B2 is_a_component_of GX;
hence thesis;
end;
hence thesis by A1,A2,Def2;
end;

theorem
Th21: for A being Subset of GX,p being Point of GX st p in A
& A is a_union_of_components of GX holds
skl p c= A
proof let A be Subset of GX,p be Point of GX;
assume A1:p in A & A is a_union_of_components of GX;
then consider F being Subset-Family of GX such that
A2:(for B being Subset of GX st B in F holds B is_a_component_of GX) &
A=union F by Def2;
consider X such that A3:p in X & X in F by A1,A2,TARSKI:def 4;
reconsider B2=X as Subset of GX by A3;
B2 is_a_component_of GX by A2,A3;
then B2=skl p by A3,CONNSP_1:44;
hence thesis by A2,A3,ZFMISC_1:92;
end;

theorem
for A,B being Subset of GX st
A is a_union_of_components of GX &
B is a_union_of_components of GX holds
A \/ B is a_union_of_components of GX &
A /\ B is a_union_of_components of GX
proof let A,B be Subset of GX;assume
A1: A is a_union_of_components of GX &
B is a_union_of_components of GX;
then consider Fa being Subset-Family of GX such that
A2:   (for Ba being Subset of GX st Ba in Fa holds Ba is_a_component_of GX) &
A=union Fa by Def2;
consider Fb being Subset-Family of GX such that
A3:   (for Bb being Subset of GX st Bb in Fb holds Bb is_a_component_of GX) &
B=union Fb by A1,Def2;
reconsider Fc = Fa \/ Fb as Subset-Family of GX;
A4:A \/ B =union Fc by A2,A3,ZFMISC_1:96;
A5: for B2 being Subset of GX st B2 in Fa \/ Fb
holds B2 is_a_component_of GX
proof let B2 be Subset of GX;assume
B2 in Fa \/ Fb;
then B2 in Fa or B2 in Fb by XBOOLE_0:def 2;
hence B2 is_a_component_of GX by A2,A3;
end;
A /\ B is a_union_of_components of GX
proof
reconsider Fd= Fa /\ Fb as Subset-Family of GX;
A6: for B4 being Subset of GX st B4 in Fd holds B4 is_a_component_of GX
proof let B4 be Subset of GX;assume B4 in Fd;
then B4 in Fa & B4 in Fb by XBOOLE_0:def 3;
hence B4 is_a_component_of GX by A2;
end;
A /\ B =union Fd
proof
A7: A /\ B c= union Fd
proof let x be set;assume x in A /\ B;
then A8: x in A & x in B by XBOOLE_0:def 3;
then consider F1 being set such that
A9: x in F1 & F1 in Fa by A2,TARSKI:def 4;
consider F2 being set such that
A10: x in F2 & F2 in Fb by A3,A8,TARSKI:def 4;
A11: F1 /\ F2 <>{} by A9,A10,XBOOLE_0:def 3;
reconsider C1=F1 as Subset of GX by A9;
reconsider C2=F2 as Subset of GX by A10;
A12: C1 is_a_component_of GX by A2,A9;
C2 is_a_component_of GX by A3,A10;
then C1 = C2 or C1 misses C2 by A12,CONNSP_1:37;
then C1 in Fa /\ Fb by A9,A10,A11,XBOOLE_0:def 3,def 7;
hence thesis by A9,TARSKI:def 4;
end;
union Fd c= A /\ B
proof let x be set;assume x in union Fd;
then consider X4 being set such that
A13: x in X4 & X4 in Fd by TARSKI:def 4;
A14:X4 in Fa & X4 in Fb by A13,XBOOLE_0:def 3;
then A15: x in union Fa by A13,TARSKI:def 4;
x in union Fb by A13,A14,TARSKI:def 4;
hence thesis by A2,A3,A15,XBOOLE_0:def 3;
end;
hence thesis by A7,XBOOLE_0:def 10;
end;
hence thesis by A6,Def2;
end;
hence thesis by A4,A5,Def2;
end;

theorem
for Fu being Subset-Family of GX st
(for A being Subset of GX st A in Fu
holds A is a_union_of_components of GX)
holds union Fu is a_union_of_components of GX
proof let Fu be Subset-Family of GX;assume
A1: (for A being Subset of GX st
A in Fu holds A is a_union_of_components of GX);
{B: ex B2 st B2 in Fu & B c= B2 & B is_a_component_of GX}
c= bool (the carrier of GX)
proof let x;assume x in
{B: ex B2 st B2 in Fu & B c= B2 & B is_a_component_of GX};
then ex B st x=B & ex B2 st B2 in Fu & B c= B2 & B is_a_component_of GX;
hence x in bool the carrier of GX;
end;
then {B: ex B2 st B2 in Fu & B c= B2 & B is_a_component_of GX}
is Subset-Family of (the carrier of GX) by SETFAM_1:def 7;
then reconsider F2={B: ex B2 st B2 in Fu & B c= B2 & B is_a_component_of GX}
as Subset-Family of GX;
A2:union Fu = union F2
proof
A3:union Fu c= union F2
proof let x;assume x in union Fu;
then consider X2 such that A4: x in X2 & X2 in Fu by TARSKI:def 4;
reconsider B3=X2 as Subset of GX by A4;
B3 is a_union_of_components of GX by A1,A4;
then consider F being Subset-Family of GX such that
A5: (for B being Subset of GX st B in F holds B is_a_component_of GX) &
B3=union F by Def2;
consider Y2 such that A6:x in Y2 & Y2 in F by A4,A5,TARSKI:def 4;
reconsider A3=Y2 as Subset of GX by A6;
A7: A3 is_a_component_of GX by A5,A6;
Y2 c= B3 by A5,A6,ZFMISC_1:92;
then A3 in F2 by A4,A7;
hence x in union F2 by A6,TARSKI:def 4;
end;
union F2 c= union Fu
proof let x;assume x in union F2;
then consider X such that A8:x in X & X in F2 by TARSKI:def 4;
ex B st X=B & ex B2 st B2 in Fu & B c= B2 & B is_a_component_of GX
by A8;
hence x in union Fu by A8,TARSKI:def 4;
end;
hence thesis by A3,XBOOLE_0:def 10;
end;
for B being Subset of GX st B in F2
holds B is_a_component_of GX
proof let B be Subset of GX;assume B in F2;
then ex A2 being Subset of GX st B=A2 &
ex B2 st B2 in Fu & A2 c= B2 & A2 is_a_component_of GX;
hence B is_a_component_of GX;
end;
hence union Fu is a_union_of_components of GX by A2,Def2;
end;

theorem
for Fu being Subset-Family of GX st
(for A being Subset of GX st A in Fu
holds A is a_union_of_components of GX)
holds meet Fu is a_union_of_components of GX
proof
let Fu be Subset-Family of GX;assume A1:
(for A being Subset of GX st A in Fu
holds A is a_union_of_components of GX);
now per cases;
case A2:Fu<>{};
{B:B is_a_component_of GX & for A2 st A2 in Fu holds B c= A2}
c= bool(the carrier of GX)
proof let x;assume
x in {B:B is_a_component_of GX & for A2 st A2 in Fu holds B c= A2};
then ex B st x=B &
B is_a_component_of GX & for A2 st A2 in Fu holds B c= A2;
hence x in bool(the carrier of GX);
end;
then {B:B is_a_component_of GX & for A2 st A2 in Fu holds B c= A2}
is Subset-Family of GX by SETFAM_1:def 7;
then reconsider F1={B:B is_a_component_of GX & for A2 st A2 in Fu holds B c=
A2}
as Subset-Family of GX;
A3: meet Fu=union F1
proof
A4:meet Fu c= union F1
proof
meet Fu c= union F1
proof let x;assume A5: x in meet Fu;
consider Y2 such that A6:Y2 in Fu by A2,XBOOLE_0:def 1;
A7:x in Y2 by A5,A6,SETFAM_1:def 1;
reconsider B2=Y2 as Subset of GX by A6;
B2 is a_union_of_components of GX by A1,A6;
then consider F being Subset-Family of GX such that
A8: (for B being Subset of GX st B in F
holds B is_a_component_of GX) &
B2=union F by Def2;
consider Y3 being set such that A9:x in Y3 & Y3 in F
by A7,A8,TARSKI:def 4;
reconsider B3=Y3 as Subset of GX by A9;
A10:B3 is_a_component_of GX by A8,A9;
for A2 st A2 in Fu holds B3 c= A2
proof let A2;assume A11:A2 in Fu;
then A12:A2 is a_union_of_components of GX by A1;
A13:B3 is_a_component_of GX by A8,A9;
A14:x in A2 by A5,A11,SETFAM_1:def 1;
reconsider p=x as Point of GX by A5;
skl p c= A2 by A12,A14,Th21;
hence thesis by A9,A13,CONNSP_1:44;
end;
then Y3 in F1 by A10;
hence x in union F1 by A9,TARSKI:def 4;
end;
hence thesis;
end;
union F1 c= meet Fu
proof let x;assume x in union F1;
then consider X such that A15: x in X & X in F1 by TARSKI:def 4;
consider B such that A16:X=B &
(B is_a_component_of GX & for A2 st A2 in Fu holds B c= A2) by A15;
for Y st Y in Fu holds x in Y
proof let Y;assume Y in Fu;
then B c= Y by A16;
hence x in Y by A15,A16;
end;
hence x in meet Fu by A2,SETFAM_1:def 1;
end;
hence thesis by A4,XBOOLE_0:def 10;
end;
for B being Subset of GX st B in F1 holds B is_a_component_of GX
proof let B be Subset of GX;assume B in F1;
then ex B1 be Subset of GX st B=B1 &
B1 is_a_component_of GX & for A2 st A2 in Fu holds B1 c= A2;
hence B is_a_component_of GX;
end;
hence meet Fu is a_union_of_components of GX by A3,Def2;
case Fu={};
then meet Fu={}(GX) by SETFAM_1:def 1;
hence meet Fu is a_union_of_components of GX by Th19;
end;
hence meet Fu is a_union_of_components of GX;
end;

theorem
for A,B being Subset of GX st A is a_union_of_components of
GX
& B is a_union_of_components of GX holds
A \ B is a_union_of_components of GX
proof let A,B be Subset of GX;assume
A1: A is a_union_of_components of GX
& B is a_union_of_components of GX;
then consider Fa being Subset-Family of GX such that
A2: (for B2 being
Subset of GX st B2 in Fa holds B2 is_a_component_of GX) &
A=union Fa by Def2;
consider Fb being Subset-Family of GX such that
A3: (for B3 being Subset of GX st B3 in Fb holds B3 is_a_component_of GX) &
B=union Fb by A1,Def2;
reconsider Fd=Fa\Fb as Subset-Family of GX;
A4:A \ B=union Fd
proof
A5:A \ B c= union Fd
proof let x;assume A6: x in A \ B;
then x in A & not x in B by XBOOLE_0:def 4;
then consider X such that
A7:x in X & X in Fa by A2,TARSKI:def 4;
reconsider A2=X as Subset of GX by A7;
now assume A2 in Fb; then A2 c= B by A3,ZFMISC_1:92;
hence contradiction by A6,A7,XBOOLE_0:def 4;
end;
then A2 in Fd by A7,XBOOLE_0:def 4;
hence thesis by A7,TARSKI:def 4;
end;
union Fd c= A \ B
proof let x;assume x in union Fd;
then consider X such that A8:x in X & X in Fd by TARSKI:def 4;
A9:X in Fa & not X in Fb by A8,XBOOLE_0:def 4;
reconsider A2=X as Subset of GX by A8;
A10:A2 is_a_component_of GX by A2,A9;
A11: A2 c= A by A2,A9,ZFMISC_1:92;
now assume x in B;
then consider Y3 being set such that
A12: x in Y3 & Y3 in Fb by A3,TARSKI:def 4;
reconsider B3=Y3 as Subset of GX by A12;
A13:B3 is_a_component_of GX by A3,A12;
A2 /\ B3 <>{} by A8,A12,XBOOLE_0:def 3;
then A2 meets B3 by XBOOLE_0:def 7;
hence contradiction by A9,A10,A12,A13,CONNSP_1:37;
end;
hence thesis by A8,A11,XBOOLE_0:def 4;
end;
hence thesis by A5,XBOOLE_0:def 10;
end;
for B4 being Subset of GX st B4 in Fd holds B4 is_a_component_of GX
proof let B4 be Subset of GX;assume B4 in Fd;
then B4 in Fa by XBOOLE_0:def 4;
hence thesis by A2;
end;
hence thesis by A4,Def2;
end;

begin :: Operations Down and Up

definition let GX be TopStruct, B be Subset of GX,
p be Point of GX;
assume A1: p in B;
func Down(p,B) -> Point of GX|B equals
:Def3:p;
coherence
proof B=the carrier of (GX|B)
proof
B=[#](GX|B) by PRE_TOPC:def 10;
hence thesis by PRE_TOPC:12;
end;
hence p is Point of GX|B by A1;
end;
end;

definition let GX be TopStruct, B be Subset of GX,
p be Point of GX|B;
assume A1:B<>{};
func Up(p) -> Point of GX equals
p;
coherence
proof B=the carrier of (GX|B) by JORDAN1:1;
then p in B by A1;
hence p is Point of GX;
end;
end;

definition let GX be TopStruct, V,B be Subset of GX;
func Down(V,B) -> Subset of GX|B equals
:Def5:V /\ B;
coherence
proof A1:V /\ B c= B by XBOOLE_1:17;
B=the carrier of (GX|B)
proof
B=[#](GX|B) by PRE_TOPC:def 10;
hence thesis by PRE_TOPC:12;
end;
hence thesis by A1;
end;
end;

definition let GX be TopStruct, B be Subset of GX;
let V be Subset of GX|B;
func Up(V) -> Subset of GX equals
:Def6:V;
coherence
proof
B=the carrier of (GX|B) by JORDAN1:1;
then V c= the carrier of GX by XBOOLE_1:1;
hence thesis;
end;
end;

definition let GX be TopStruct, B be Subset of GX,
p be Point of GX;
assume A1: p in B;
func skl(p,B) -> Subset of GX means
:Def7:for q being Point of GX|B st q=p holds it=skl q;
existence
proof
A2:B=the carrier of (GX|B)
proof
B=[#](GX|B) by PRE_TOPC:def 10;
hence thesis by PRE_TOPC:12;
end;
then reconsider q3=p as Point of GX|B by A1;
reconsider C=skl q3 as Subset of GX by A2,XBOOLE_1:1;
take C;
thus thesis;
end;
uniqueness
proof let S,S' be Subset of GX; assume
A3:(for q being Point of GX|B st q=p holds S=skl q)&
(for q2 being Point of GX|B st q2=p holds S'=skl q2);
B=the carrier of (GX|B)
proof B=[#](GX|B) by PRE_TOPC:def 10;
hence thesis by PRE_TOPC:12;
end;
then reconsider q3=p as Point of GX|B by A1;
S=skl q3 & S'=skl q3 by A3;
hence S = S';
end;
end;

theorem
for B being Subset of GX, p be Point of GX st p in B
holds skl(p,B)<>{}
proof let B be Subset of GX, p be Point of GX;assume
A1: p in B;
then reconsider B' = B as non empty Subset of GX;
reconsider q=p as Point of GX|B' by A1,JORDAN1:1;
q in skl q by CONNSP_1:40;
hence skl(p,B)<>{} by A1,Def7;
end;

theorem
Th27: for B being Subset of GX, p be Point of GX st p in B
holds skl(p,B)=skl Down(p,B)
proof let B be Subset of GX, p be Point of GX;
assume A1: p in B;
then p=Down(p,B) by Def3;
hence thesis by A1,Def7;
end;

theorem
Th28: for V,B being Subset of GX st V c= B holds Down(V,B)=V
proof let V,B be Subset of GX; assume A1:V c= B;
Down(V,B)=V /\ B by Def5;
hence Down(V,B)=V by A1,XBOOLE_1:28;
end;

theorem
Th29: for GX being TopSpace
for V,B being Subset of GX st V is open holds
Down(V,B) is open
proof let GX be TopSpace;
let V,B be Subset of GX;assume V is open;
then A1: V in the topology of GX by PRE_TOPC:def 5;
Down(V,B)=V /\ B by Def5;
then Down(V,B)=V /\ [#](GX|B) by PRE_TOPC:def 10;
then Down(V,B) in the topology of GX|B by A1,PRE_TOPC:def 9;
hence Down(V,B) is open by PRE_TOPC:def 5;
end;

theorem
Th30:for V,B being Subset of GX st V c= B holds
Cl Down(V,B) =(Cl V) /\ B
proof let V,B be Subset of GX;
assume V c= B;
then Down(V,B)=V by Th28;
then Cl Down(V,B) =(Cl V) /\ ([#](GX|B)) by PRE_TOPC:47;
hence thesis by PRE_TOPC:def 10;
end;

theorem
for B being Subset of GX,V being Subset of GX|
B
holds Cl V =(Cl Up(V)) /\ B
proof let B be Subset of GX,
V be Subset of GX|B;
A1: B=the carrier of (GX|B)
proof
B=[#](GX|B) by PRE_TOPC:def 10;
hence thesis by PRE_TOPC:12;
end;
A2:Up(V)=V by Def6;
then Cl Down(Up(V),B)=(Cl Up(V))/\ B by A1,Th30;
hence Cl V =(Cl Up(V)) /\ B by A1,A2,Th28;
end;

theorem
for V,B being Subset of GX st V c= B holds
Cl Down(V,B) c= Cl V
proof let V,B be Subset of GX;assume V c= B;
then Cl Down(V,B) = (Cl V) /\ B by Th30;
hence Cl Down(V,B) c= Cl V by XBOOLE_1:17;
end;

theorem
for B being Subset of GX,
V being Subset of GX|B st
V c= B holds Down(Up(V),B)=V
proof let B be Subset of GX,
V be Subset of GX|B;
assume A1:V c= B;
V=Up(V) by Def6;
hence Down(Up(V),B)=V by A1,Th28;
end;

theorem
Th34: for GX being TopSpace
for V,B being Subset of GX,
W being Subset of GX|B st V=W & W is connected holds V is connected
proof let GX be TopSpace;
let V,B be Subset of GX, W be Subset of GX|B;
assume A1:V=W & W is connected;
then (GX|B)|W is connected by CONNSP_1:def 3;
then A2:for A2,B2 being Subset of (GX|B)|W
st [#]((GX|B)|W) = A2 \/ B2 & A2 <> {}((GX|B)|W)
& B2 <> {}((GX|B)|W) & A2 is open &
B2 is open holds A2 meets B2 by CONNSP_1:12;
for A3,B3 being Subset of GX|V
st [#](GX|V) = A3 \/ B3 & A3 <> {}(GX|V)
& B3 <> {}(GX|V) & A3 is open & B3 is open holds A3 meets B3
proof let A3,B3 be Subset of GX|V;assume
A3:[#](GX|V) = A3 \/ B3 & A3 <> {}(GX|V)
& B3 <> {}(GX|V) & A3 is open &
B3 is open;
then A4: A3 in the topology of GX|V &
B3 in the topology of GX|V by PRE_TOPC:def 5;
then consider Qa being Subset of GX such that
A5:  Qa in the topology of GX &
A3 = Qa /\ [#](GX|V) by PRE_TOPC:def 9;
reconsider Qa as Subset of GX;
A6: Qa is open by A5,PRE_TOPC:def 5;
consider Qb being Subset of GX such that
A7:  Qb in the topology of GX &
B3 = Qb /\ [#](GX|V) by A4,PRE_TOPC:def 9;
reconsider Qb as Subset of GX;
[#](GX|V) c= (Qa \/ Qb)/\ [#](GX|V) by A3,A5,A7,XBOOLE_1:23;
then for x st x in [#](GX|V) holds x in (Qa \/ Qb) by XBOOLE_0:def 3
;
then A8:[#](GX|V) c= Qa \/ Qb by TARSKI:def 3;
A9:V=[#](GX|V) by PRE_TOPC:def 10;
A10: Qb is open by A7,PRE_TOPC:def 5;
reconsider A4=Down(Qa,B), B4=Down(Qb,B) as Subset of (GX|B);
reconsider A6=Down(A4,W), B6=Down(B4,W) as Subset of (GX|B)|W;
A11: A4 is open by A6,Th29;
A12: B4 is open by A10,Th29;
A13: A4=Qa /\ B by Def5;
A14: B4=Qb /\ B by Def5;
then A15: A4 \/ B4 =(Qa \/ Qb)/\ B by A13,XBOOLE_1:23;
A16: A6 is open by A11,Th29;
A17: B6 is open by A12,Th29;
W c= the carrier of GX|B;
then A18: W c= B by JORDAN1:1;
A3 c= the carrier of GX|V;
then A19: A3 c= V by JORDAN1:1;
then A20: A3 c=B by A1,A18,XBOOLE_1:1;
ex x being Point of GX|V st x in A3 by A3,SUBSET_1:10;
then consider x0 being set such that
A21: x0 in A3;
A22: x0 in Qa by A5,A21,XBOOLE_0:def 3;
A4=Qa /\ B by Def5;
then A23: x0 in A4 by A20,A21,A22,XBOOLE_0:def 3;
A24: W c= A4 \/ B4 by A1,A8,A9,A15,A18,XBOOLE_1:19;
A25: A6=A4 /\ W by Def5; then A26: x0 in A6 by A1,A19,A21,A23,XBOOLE_0:
def 3;
B3 c= the carrier of GX|V;
then A27: B3 c= V by JORDAN1:1;
then A28: B3 c=B by A1,A18,XBOOLE_1:1;
ex x being Point of GX|V st x in B3 by A3,SUBSET_1:10;
then consider y0 being set such that
A29: y0 in B3;
A30: y0 in Qb by A7,A29,XBOOLE_0:def 3;
B4=Qb /\ B by Def5;
then A31: y0 in B4 by A28,A29,A30,XBOOLE_0:def 3;
A32: B6=B4 /\ W by Def5;
then A33: B6<>{} by A1,A27,A29,A31,XBOOLE_0:def 3;
A6 \/ B6=(A4 \/ B4)/\ W by A25,A32,XBOOLE_1:23;
then A34:W c= A6 \/ B6 by A24,XBOOLE_1:19;
A6 \/ B6 c= the carrier of (GX|B)|W;
then A6 \/ B6 c= W by JORDAN1:1;
then A6 \/ B6=W by A34,XBOOLE_0:def 10
.= [#]((GX|B)|W) by PRE_TOPC:def 10;
then A6 meets B6 by A2,A16,A17,A26,A33;
then A6 /\ B6 <> {} by XBOOLE_0:def 7;
then consider x1 being set such that A35: x1 in A6 /\ B6 by XBOOLE_0:
def 1
;
x1 in A6 & x1 in B6 by A35,XBOOLE_0:def 3;
then x1 in A4 & x1 in W & x1 in B4 by A25,A32,XBOOLE_0:def 3;
then x1 in Qa & x1 in B & x1 in W & x1 in Qb & x1 in B
by A13,A14,XBOOLE_0:def 3;
then x1 in Qa /\ Qb & x1 in W by XBOOLE_0:def 3;
then A36:Qa /\ Qb /\ [#](GX|V)<>{} by A1,A9,XBOOLE_0:def 3;
Qa /\ Qb /\ [#](GX|V) c=(Qa /\ [#](GX|V))/\(Qb /\ [#](GX|V))
proof let x;assume x in Qa /\ Qb /\ [#](GX|V);
then x in Qa /\ Qb & x in [#](GX|V) by XBOOLE_0:def 3;
then x in Qa & x in Qb & x in [#](GX|V) by XBOOLE_0:def 3;
then x in Qa /\ [#](GX|V) & x in Qb /\ [#](GX|V) by XBOOLE_0:def 3;
hence x in (Qa /\ [#](GX|V))/\(Qb /\ [#](GX|V)) by XBOOLE_0:def 3;
end;
hence A3 /\ B3 <> {} by A5,A7,A36,XBOOLE_1:3;
end;
then GX|V is connected by CONNSP_1:12;
hence thesis by CONNSP_1:def 3;
end;

theorem
for B being Subset of GX, p be Point of GX st p in B
holds skl(p,B) is connected
proof let B be Subset of GX, p be Point of GX;
assume A1: p in B;
then reconsider B' = B as non empty Subset of GX;
A2:skl Down(p,B') is connected by CONNSP_1:41;
skl(p,B)=skl Down(p,B) by A1,Th27;
hence thesis by A2,Th34;
end;
```