:: Components and Unions of Components
:: by Yatsuka Nakamura and Andrzej Trybulec
::
:: Received February 5, 1996
:: Copyright (c) 1996-2026 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies XBOOLE_0, PRE_TOPC, SUBSET_1, CONNSP_1, SETFAM_1, RELAT_2,
TARSKI, STRUCT_0, ZFMISC_1, RELAT_1, RUSUB_4, RCOMP_1, CONNSP_3;
notations TARSKI, XBOOLE_0, SUBSET_1, SETFAM_1, STRUCT_0, PRE_TOPC, CONNSP_1;
constructors SETFAM_1, CONNSP_1;
registrations SUBSET_1, STRUCT_0, PRE_TOPC, CONNSP_1;
requirements SUBSET, BOOLE;
definitions TARSKI, XBOOLE_0;
equalities XBOOLE_0, STRUCT_0;
expansions TARSKI, XBOOLE_0;
theorems TARSKI, ZFMISC_1, SETFAM_1, PRE_TOPC, SUBSET_1, CONNSP_1, XBOOLE_0,
XBOOLE_1, ORDERS_1;
schemes SUBSET_1;
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 Component_of 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 SUBSET_1:sch 3;
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 thesis;
end;
thus thesis by A1;
end;
thus thesis;
end;
uniqueness
proof
let S,S9 be Subset of GX;
assume that
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 and
A3: ex F9 being Subset-Family of GX st (for A being Subset of GX holds
A in F9 iff A is connected & V c= A) & union F9 = S9;
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: union F = S by A2;
consider F9 being Subset-Family of GX such that
A6: for A being Subset of GX holds A in F9 iff A is connected & V c= A and
A7: union F9 = S9 by A3;
now
let y be object;
A8: now
assume y in S9;
then consider B being set such that
A9: y in B and
A10: B in F9 by A7,TARSKI:def 4;
reconsider B as Subset of GX by A10;
B is connected & V c= B by A6,A10;
then B in F by A4;
hence y in S by A5,A9,TARSKI:def 4;
end;
now
assume y in S;
then consider B being set such that
A11: y in B and
A12: B in F by A5,TARSKI:def 4;
reconsider B as Subset of GX by A12;
B is connected & V c= B by A4,A12;
then B in F9 by A6;
hence y in S9 by A7,A11,TARSKI:def 4;
end;
hence y in S iff y in S9 by A8;
end;
hence thesis 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= Component_of 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: Component_of V = union F by Def1;
A4: for A being set holds A in F implies V c= A by A2;
F <> {} by A1,A2;
then
A5: V c= meet F by A4,SETFAM_1:5;
meet F c= union F by SETFAM_1:2;
hence thesis by A3,A5;
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 Component_of 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: Component_of V = union F by Def1;
now
assume F <> {};
then consider A being Subset of GX such that
A4: A in F by SUBSET_1:4;
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:
Component_of {}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 SUBSET_1:sch 3;
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 object;
hereby
assume x in the carrier of GX;
then reconsider p = x as Point of GX;
reconsider Y = Component_of p as set;
take Y;
thus x in Y by CONNSP_1:38;
Component_of p is connected & {}GX c= Y;
hence Y in F by A2;
end;
given Y being set such that
A3: x in Y & Y in F;
thus x in the carrier of GX by A3;
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 Component_of V <>{}
proof
let V be Subset of GX such that
A1: V is connected;
per cases;
suppose
V = {};
then V = {}GX;
hence thesis by Th3;
end;
suppose
V <>{};
hence thesis by A1,Th1,XBOOLE_1:3;
end;
end;
theorem Th5:
for GX being TopSpace, V being Subset of GX st V is connected & V
<> {} holds Component_of V is connected
proof
let GX be TopSpace;
let V be Subset of GX;
assume that
A1: V is connected and
A2: V<>{};
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 and
A4: Component_of V = union F by Def1;
A5: for A being set st A in F holds V c= A by A3;
F <> {} by A1,A3;
then V c= meet F by A5,SETFAM_1:5;
then
A6: meet F<>{}(GX) by A2;
for A being Subset of GX st A in F holds A is connected by A3;
hence thesis by A4,A6,CONNSP_1:26;
end;
theorem Th6:
for V,C being Subset of GX st V is connected & C is connected
holds Component_of V c= C implies C = Component_of V
proof
let V,C be Subset of GX;
assume that
A1: V is connected and
A2: C is connected;
assume
A3: Component_of 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: Component_of V = union F by Def1;
V c= Component_of V by A1,Th1;
then V c= C by A3;
then C in F by A2,A4;
then C c= Component_of V by A5,ZFMISC_1:74;
hence thesis by A3;
end;
theorem Th7:
for A being Subset of GX st A is a_component holds Component_of A=A
proof
let A be Subset of GX;
assume
A1: A is a_component;
then
A2: A is connected;
then
A3: A c= Component_of A by Th1;
A <>{}(GX) by A1;
then Component_of 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 iff ex V
being Subset of GX st V is connected & V <> {} & A = Component_of V
proof
let A be Subset of GX;
A1: now
assume
A2: A is a_component;
take V = A;
thus V is connected & V<>{} & A = Component_of V by A2,Th7;
end;
now
given V being Subset of GX such that
A3: V is connected & V<>{} & A = Component_of V;
A is connected & for B being Subset of GX st B is connected holds A c=
B implies A = B by A3,Th5,Th6;
hence A is a_component by CONNSP_1:def 5;
end;
hence thesis by A1;
end;
theorem
for V being Subset of GX st V is connected & V<>{} holds Component_of
V is a_component by Th8;
theorem
for A, V be Subset of GX st A is a_component & V is connected &
V c= A & V<>{} holds A = Component_of V
proof
let A, V be Subset of GX;
assume that
A1: A is a_component and
A2: V is connected and
A3: V c= A and
A4: V<>{};
V c= Component_of V by A2,Th1;
then
A5: A meets (Component_of V) by A3,A4,XBOOLE_1:67;
assume
A6: A <> Component_of V;
Component_of V is a_component by A2,A4,Th8;
hence contradiction by A1,A6,A5,CONNSP_1:1,34;
end;
theorem Th11:
for V being Subset of GX st V is connected & V<>{} holds
Component_of (Component_of V)=Component_of V
proof
let V be Subset of GX;
assume V is connected & V<>{};
then Component_of V is a_component by Th8;
hence thesis by Th7;
end;
theorem Th12:
for A,B being Subset of GX st A is connected & B is connected &
A<>{} & A c= B holds Component_of A = Component_of B
proof
let A,B be Subset of GX;
assume that
A1: A is connected and
A2: B is connected and
A3: A<>{} and
A4: A c= B;
B<>{} by A3,A4;
then
A5: Component_of B is connected by A2,Th5;
A6: B c= Component_of B by A2,Th1;
then
A7: A c= Component_of B by A4;
A8: Component_of B c= Component_of A
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 & A c= D and
A10: union F = Component_of A by Def1;
Component_of B in F by A7,A5,A9;
hence thesis by A10,ZFMISC_1:74;
end;
A11: Component_of A is connected by A1,A3,Th5;
Component_of A c= Component_of B
proof
consider F being Subset-Family of GX such that
A12: for D being Subset of GX holds D in F iff D is connected & B c= D and
A13: union F = Component_of B by Def1;
B c= Component_of A by A6,A8;
then Component_of A in F by A11,A12;
hence thesis by A13,ZFMISC_1:74;
end;
hence thesis by A8;
end;
theorem Th13:
for A,B being Subset of GX st A is connected & B is connected &
A<>{} & A c= B holds B c= Component_of A
proof
let A,B be Subset of GX;
assume that
A1: A is connected and
A2: B is connected and
A3: A<>{} & A c= B;
Component_of A = Component_of B by A1,A2,A3,Th12;
hence thesis by A2,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= Component_of A
proof
let A be Subset of GX,B be Subset of GX;
assume that
A1: A is connected and
A2: A \/ B is connected and
A3: A<>{};
Component_of (A \/ B) = Component_of A by A1,A2,A3,Th12,XBOOLE_1:7;
hence thesis by A2,Th1;
end;
theorem Th15:
for A being Subset of GX, p being Point of GX st A is connected
& p in A holds Component_of p=Component_of A
proof
let A be Subset of GX, p be Point of GX;
assume that
A1: A is connected and
A2: p in A;
A c= Component_of A & Component_of A is a_component by A1,A2,Th1,Th8;
hence thesis by A2,CONNSP_1:41;
end;
theorem
for A,B being Subset of GX st A is connected & B is connected & A
meets B holds A \/ B c= Component_of A & A \/ B c= Component_of B & A c=
Component_of B & B c= Component_of A
proof
let A,B be Subset of GX;
A1: A c= A \/ B & B c= A \/ B by XBOOLE_1:7;
A2: for C,D being Subset of GX st C is connected & D is connected & C /\ D
<>{} holds C \/ D c= Component_of C
proof
let C,D be Subset of GX;
assume that
A3: C is connected and
A4: D is connected and
A5: C /\ D <>{};
C meets D by A5;
then
A6: C \/ D is connected by A3,A4,CONNSP_1:1,17;
C <>{} by A5;
hence thesis by A3,A6,Th14;
end;
assume A is connected & B is connected & A /\ B <>{};
then A \/ B c= Component_of A & A \/ B c= Component_of B by A2;
hence thesis by A1;
end;
theorem
for A being Subset of GX st A is connected & A<>{} holds Cl A c=
Component_of A
proof
let A be Subset of GX;
assume that
A1: A is connected and
A2: A<>{};
Cl A is connected by A1,CONNSP_1:19;
hence thesis by A1,A2,Th13,PRE_TOPC:18;
end;
theorem
for A,B being Subset of GX st A is a_component & B is connected
& B<>{} & A misses B holds A misses Component_of B
proof
let A,B be Subset of GX;
assume that
A1: A is a_component and
A2: B is connected & B<>{} and
A3: A /\ B ={};
A4: A is connected by A1;
assume A /\ Component_of B <>{};
then consider x being Point of GX such that
A5: x in A /\ Component_of B by SUBSET_1:4;
x in A by A5,XBOOLE_0:def 4;
then
A6: Component_of x=Component_of A by A4,Th15;
A7: x in Component_of B by A5,XBOOLE_0:def 4;
Component_of A=A & Component_of B=Component_of Component_of B by A1,A2,Th7
,Th11;
then (Component_of B) /\ B={} by A2,A3,A7,A6,Th5,Th15;
hence contradiction by A2,Th1,XBOOLE_1:28;
end;
begin :: On unions of components
Lm1: now
let GX be TopStruct;
reconsider S={} as Subset-Family of GX by XBOOLE_1:2;
for B being Subset of GX st B in S holds B is a_component;
hence ex F being Subset-Family of GX st (for B being Subset of GX st B in F
holds B is a_component) & {}(GX)=union F by ZFMISC_1:2;
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) & 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) & {}(GX) = union F by Lm1;
end;
theorem Th20:
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;
{B : B is a_component} c= bool (the carrier of GX)
proof
let x be object;
assume x in {B : B is a_component};
then ex B being Subset of GX st x=B & B is a_component;
hence thesis;
end;
then reconsider S={B: B is a_component} as Subset-Family of GX;
A1: for B being Subset of GX st B in S holds B is a_component
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;
hence thesis;
end;
the carrier of GX c= union S
proof
let x be object;
assume x in the carrier of GX;
then reconsider p=x as Point of GX;
set B3=Component_of p;
B3 is a_component by CONNSP_1:40;
then p in Component_of p & B3 in S by CONNSP_1:38;
hence thesis by TARSKI:def 4;
end;
then
A2: the carrier of GX=union S;
assume A=(the carrier of GX);
hence thesis by A2,A1,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 Component_of p c= A
proof
let A be Subset of GX,p be Point of GX;
assume that
A1: p in A and
A2: A is a_union_of_components of GX;
consider F being Subset-Family of GX such that
A3: for B being Subset of GX st B in F holds B is a_component and
A4: A=union F by A2,Def2;
consider X such that
A5: p in X and
A6: X in F by A1,A4,TARSKI:def 4;
reconsider B2=X as Subset of GX by A6;
B2=Component_of p by A3,A5,A6,CONNSP_1:41;
hence thesis by A4,A6,ZFMISC_1:74;
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 that
A1: A is a_union_of_components of GX and
A2: B is a_union_of_components of GX;
consider Fa being Subset-Family of GX such that
A3: for Ba being Subset of GX st Ba in Fa holds Ba is a_component and
A4: A=union Fa by A1,Def2;
consider Fb being Subset-Family of GX such that
A5: for Bb being Subset of GX st Bb in Fb holds Bb is a_component and
A6: B=union Fb by A2,Def2;
A7: for B2 being Subset of GX st B2 in Fa \/ Fb holds B2 is a_component
proof
let B2 be Subset of GX;
assume B2 in Fa \/ Fb;
then B2 in Fa or B2 in Fb by XBOOLE_0:def 3;
hence thesis by A3,A5;
end;
A8: A /\ B is a_union_of_components of GX
proof
reconsider Fd= Fa /\ Fb as Subset-Family of GX;
A9: for B4 being Subset of GX st B4 in Fd holds B4 is a_component
proof
let B4 be Subset of GX;
assume B4 in Fd;
then B4 in Fa by XBOOLE_0:def 4;
hence thesis by A3;
end;
A10: A /\ B c= union Fd
proof
let x be object;
assume
A11: x in A /\ B;
then x in A by XBOOLE_0:def 4;
then consider F1 being set such that
A12: x in F1 and
A13: F1 in Fa by A4,TARSKI:def 4;
reconsider C1=F1 as Subset of GX by A13;
x in B by A11,XBOOLE_0:def 4;
then consider F2 being set such that
A14: x in F2 and
A15: F2 in Fb by A6,TARSKI:def 4;
reconsider C2=F2 as Subset of GX by A15;
A16: C2 is a_component by A5,A15;
C1 is a_component by A3,A13;
then
A17: C1 = C2 or C1 misses C2 by A16,CONNSP_1:35;
F1 /\ F2 <>{} by A12,A14,XBOOLE_0:def 4;
then C1 in Fa /\ Fb by A13,A15,A17,XBOOLE_0:def 4;
hence thesis by A12,TARSKI:def 4;
end;
union Fd c= A /\ B
proof
let x be object;
assume x in union Fd;
then consider X4 being set such that
A18: x in X4 and
A19: X4 in Fd by TARSKI:def 4;
X4 in Fb by A19,XBOOLE_0:def 4;
then
A20: x in union Fb by A18,TARSKI:def 4;
X4 in Fa by A19,XBOOLE_0:def 4;
then x in union Fa by A18,TARSKI:def 4;
hence thesis by A4,A6,A20,XBOOLE_0:def 4;
end;
then A /\ B =union Fd by A10;
hence thesis by A9,Def2;
end;
reconsider Fc = Fa \/ Fb as Subset-Family of GX;
A \/ B =union Fc by A4,A6,ZFMISC_1:78;
hence thesis by A7,A8,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;
{B: ex B2 st B2 in Fu & B c= B2 & B is a_component} c= bool (the
carrier of GX)
proof
let x be object;
assume x in {B: ex B2 st B2 in Fu & B c= B2 & B is a_component};
then ex B st x=B & ex B2 st B2 in Fu & B c= B2 & B is a_component;
hence thesis;
end;
then reconsider
F2={B: ex B2 st B2 in Fu & B c= B2 & B is a_component} as
Subset-Family of GX;
A1: for B being Subset of GX st B in F2 holds B is a_component
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;
hence thesis;
end;
assume
A2: for A being Subset of GX st A in Fu holds A is a_union_of_components of GX;
A3: union Fu c= union F2
proof
let x be object;
assume x in union Fu;
then consider X2 such that
A4: x in X2 and
A5: X2 in Fu by TARSKI:def 4;
reconsider B3=X2 as Subset of GX by A5;
B3 is a_union_of_components of GX by A2,A5;
then consider F being Subset-Family of GX such that
A6: for B being Subset of GX st B in F holds B is a_component and
A7: B3=union F by Def2;
consider Y2 such that
A8: x in Y2 and
A9: Y2 in F by A4,A7,TARSKI:def 4;
reconsider A3=Y2 as Subset of GX by A9;
A3 is a_component & Y2 c= B3 by A6,A7,A9,ZFMISC_1:74;
then A3 in F2 by A5;
hence thesis by A8,TARSKI:def 4;
end;
union F2 c= union Fu
proof
let x be object;
assume x in union F2;
then consider X such that
A10: x in X and
A11: 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 by A11;
hence thesis by A10,TARSKI:def 4;
end;
then union Fu = union F2 by A3;
hence thesis by A1,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 & for A2 st A2 in Fu holds B c= A2} c=
bool(the carrier of GX)
proof
let x be object;
assume
x in {B:B is a_component & for A2 st A2 in Fu holds B c= A2};
then
ex B st x=B & B is a_component & for A2 st A2 in Fu holds B
c= A2;
hence thesis;
end;
then reconsider
F1={B:B is a_component & for A2 st A2 in Fu holds B c=
A2} as Subset-Family of GX;
A3: meet Fu c= union F1
proof
let x be object;
consider Y2 being object such that
A4: Y2 in Fu by A2,XBOOLE_0:def 1;
reconsider Y2 as set by TARSKI:1;
reconsider B2=Y2 as Subset of GX by A4;
B2 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 and
A6: B2=union F by Def2;
assume
A7: x in meet Fu;
then x in Y2 by A4,SETFAM_1:def 1;
then consider Y3 being set such that
A8: x in Y3 and
A9: Y3 in F by A6,TARSKI:def 4;
reconsider B3=Y3 as Subset of GX by A9;
A10: for A2 st A2 in Fu holds B3 c= A2
proof
reconsider p=x as Point of GX by A7;
let A2;
assume
A11: A2 in Fu;
then x in A2 by A7,SETFAM_1:def 1;
then Component_of p c= A2 by A1,A11,Th21;
hence thesis by A5,A8,A9,CONNSP_1:41;
end;
B3 is a_component by A5,A9;
then Y3 in F1 by A10;
hence thesis by A8,TARSKI:def 4;
end;
A12: for B being Subset of GX st B in F1 holds B is a_component
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 & for A2
st A2 in Fu holds B1 c= A2;
hence thesis;
end;
union F1 c= meet Fu
proof
let x be object;
assume x in union F1;
then consider X such that
A13: x in X and
A14: X in F1 by TARSKI:def 4;
consider B such that
A15: X=B and
B is a_component and
A16: for A2 st A2 in Fu holds B c= A2 by A14;
for Y st Y in Fu holds x in Y
proof
let Y;
assume Y in Fu;
then B c= Y by A16;
hence thesis by A13,A15;
end;
hence thesis by A2,SETFAM_1:def 1;
end;
then meet Fu=union F1 by A3;
hence thesis by A12,Def2;
end;
case
Fu={};
then meet Fu={}(GX) by SETFAM_1:def 1;
hence thesis by Th19;
end;
end;
hence thesis;
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 that
A1: A is a_union_of_components of GX and
A2: B is a_union_of_components of GX;
consider Fa being Subset-Family of GX such that
A3: for B2 being Subset of GX st B2 in Fa holds B2 is a_component and
A4: A=union Fa by A1,Def2;
consider Fb being Subset-Family of GX such that
A5: for B3 being Subset of GX st B3 in Fb holds B3 is a_component and
A6: B=union Fb by A2,Def2;
reconsider Fd=Fa\Fb as Subset-Family of GX;
A7: union Fd c= A \ B
proof
let x be object;
assume x in union Fd;
then consider X such that
A8: x in X and
A9: X in Fd by TARSKI:def 4;
reconsider A2=X as Subset of GX by A9;
A10: not X in Fb by A9,XBOOLE_0:def 5;
A11: X in Fa by A9,XBOOLE_0:def 5;
then
A12: A2 is a_component by A3;
A13: now
assume x in B;
then consider Y3 being set such that
A14: x in Y3 and
A15: Y3 in Fb by A6,TARSKI:def 4;
reconsider B3=Y3 as Subset of GX by A15;
A2 /\ B3 <>{} by A8,A14,XBOOLE_0:def 4;
then
A16: A2 meets B3;
B3 is a_component by A5,A15;
hence contradiction by A10,A12,A15,A16,CONNSP_1:35;
end;
A2 c= A by A4,A11,ZFMISC_1:74;
hence thesis by A8,A13,XBOOLE_0:def 5;
end;
A17: for B4 being Subset of GX st B4 in Fd holds B4 is a_component
proof
let B4 be Subset of GX;
assume B4 in Fd;
then B4 in Fa by XBOOLE_0:def 5;
hence thesis by A3;
end;
A \ B c= union Fd
proof
let x be object;
assume
A18: x in A \ B;
then x in A by XBOOLE_0:def 5;
then consider X such that
A19: x in X and
A20: X in Fa by A4,TARSKI:def 4;
reconsider A2=X as Subset of GX by A20;
now
assume A2 in Fb;
then A2 c= B by A6,ZFMISC_1:74;
hence contradiction by A18,A19,XBOOLE_0:def 5;
end;
then A2 in Fd by A20,XBOOLE_0:def 5;
hence thesis by A19,TARSKI:def 4;
end;
then A \ B=union Fd by A7;
hence thesis by A17,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=[#](GX|B) by PRE_TOPC:def 5;
hence thesis 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 PRE_TOPC:8;
then p in B by A1;
hence thesis;
end;
end;
definition
let GX be TopStruct, V,B be Subset of GX;
func Down(V,B) -> Subset of GX|B equals
V /\ B;
coherence
proof
B=[#](GX|B) by PRE_TOPC:def 5;
hence thesis by XBOOLE_1:17;
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
V;
coherence
proof
B=the carrier of (GX|B) by PRE_TOPC:8;
hence thesis by XBOOLE_1:1;
end;
end;
definition
let GX be TopStruct, B be Subset of GX, p be Point of GX;
assume
A1: p in B;
func Component_of(p,B) -> Subset of GX means
:Def7:
for q being Point of GX| B st q=p holds it=Component_of q;
existence
proof
A2: B=[#](GX|B) by PRE_TOPC:def 5;
then reconsider q3=p as Point of GX|B by A1;
reconsider C=Component_of q3 as Subset of GX by A2,XBOOLE_1:1;
take C;
thus thesis;
end;
uniqueness
proof
B=[#](GX|B) by PRE_TOPC:def 5;
then reconsider q3=p as Point of GX|B by A1;
let S,S9 be Subset of GX;
assume that
A3: for q being Point of GX|B st q=p holds S=Component_of q and
A4: for q2 being Point of GX|B st q2=p holds S9=Component_of q2;
S=Component_of q3 by A3;
hence thesis by A4;
end;
end;
theorem
for B being Subset of GX, p be Point of GX st p in B holds p in
Component_of(p,B)
proof
let B be Subset of GX, p be Point of GX;
assume
A1: p in B;
then reconsider B9 = B as non empty Subset of GX;
reconsider q=p as Point of GX|B9 by A1,PRE_TOPC:8;
q in Component_of q by CONNSP_1:38;
hence thesis by A1,Def7;
end;
theorem Th27:
for B being Subset of GX, p be Point of GX st p in B holds
Component_of(p,B)=Component_of 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
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 2;
Down(V,B)=V /\ [#](GX|B) by PRE_TOPC:def 5;
then Down(V,B) in the topology of GX|B by A1,PRE_TOPC:def 4;
hence thesis by PRE_TOPC:def 2;
end;
theorem Th29:
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 XBOOLE_1:28;
then Cl Down(V,B) =(Cl V) /\ ([#](GX|B)) by PRE_TOPC:17;
hence thesis by PRE_TOPC:def 5;
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=[#](GX|B) by PRE_TOPC:def 5;
then Cl Down(Up(V),B)=(Cl Up(V))/\ B by Th29;
hence thesis by A1,XBOOLE_1:28;
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 Th29;
hence thesis 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 by XBOOLE_1:28;
theorem
for B being Subset of GX, p be Point of GX st p in B holds
Component_of(p,B) is connected
proof
let B be Subset of GX, p be Point of GX;
assume
A1: p in B;
then reconsider B9 = B as non empty Subset of GX;
Component_of Down(p,B9) is connected & Component_of(p,B)=Component_of
Down(p,B) by A1,Th27;
hence thesis by CONNSP_1:23;
end;
:: Moved from JORDAN1B, AK, 22.02.2006
registration
let T be non empty TopSpace;
cluster non empty for a_union_of_components of T;
existence
proof
reconsider A = [#]T as a_union_of_components of T by Th20;
A is non empty;
hence thesis;
end;
end;
theorem
for T being non empty TopSpace for A being non empty
a_union_of_components of T st A is connected holds A is a_component
proof
let T be non empty TopSpace;
let A be non empty a_union_of_components of T;
consider F being Subset-Family of T such that
A1: for B being Subset of T st B in F holds B is a_component and
A2: A = union F by Def2;
consider X being set such that
X <> {} and
A3: X in F by A2,ORDERS_1:6;
reconsider X as Subset of T by A3;
assume
A4: A is connected;
F={X}
proof
thus F c= {X}
proof
let x be object;
assume
A5: x in F;
then reconsider Y=x as Subset of T;
A6: X is a_component & X c= A by A1,A2,A3,ZFMISC_1:74;
Y is a_component & Y c= A by A1,A2,A5,ZFMISC_1:74;
then Y = A by A4,CONNSP_1:def 5
.= X by A4,A6,CONNSP_1:def 5;
hence thesis by TARSKI:def 1;
end;
let x be object;
assume x in {X};
hence thesis by A3,TARSKI:def 1;
end;
then A = X by A2,ZFMISC_1:25;
hence thesis by A1,A3;
end;