### The Mizar article:

### 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;

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