Copyright (c) 1998 Association of Mizar Users
environ vocabulary FUNCT_1, RELAT_1, PRE_TOPC, EQREL_1, BORSUK_1, TARSKI, BOOLE, SUBSET_1, SETFAM_1, PROB_1, URYSOHN1, ORDINAL2, T_1TOPSP; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, RELAT_1, FUNCT_1, FUNCT_2, SETFAM_1, STRUCT_0, PRE_TOPC, EQREL_1, TOPS_2, URYSOHN1, BORSUK_1; constructors URYSOHN1, TOPS_2; clusters PRE_TOPC, BORSUK_1, STRUCT_0, FSM_1, FUNCT_1, RELSET_1, SUBSET_1, XBOOLE_0, EQREL_1, PARTFUN1; requirements BOOLE, SUBSET; definitions TARSKI, EQREL_1, PRE_TOPC, XBOOLE_0; theorems ZFMISC_1, PRE_TOPC, TARSKI, SETFAM_1, EQREL_1, URYSOHN1, BORSUK_1, FUNCT_2, FUNCT_1, TOPS_2, RELSET_1, RELAT_1, XBOOLE_0, XBOOLE_1; schemes FUNCT_1; begin reserve X for non empty set, x for Element of X, y,v,w for set; canceled; theorem Th2: for T being non empty TopSpace, A being non empty a_partition of the carrier of T, y being Subset of space A holds (Proj(A))"y = union y proof let T be non empty TopSpace; let A be non empty a_partition of the carrier of T; let y be Subset of space A; reconsider y as Subset of A by BORSUK_1:def 10; (Proj(A))"y = (proj A)"y by BORSUK_1:def 11 .= union y by BORSUK_1:31; hence thesis; end; theorem Th3: for X being non empty set, S being a_partition of X, A being Subset of S holds (union S) \ (union A) = union (S \ A) proof let X be non empty set; let S be a_partition of X; let A be Subset of S; thus (union S) \ (union A) c= union (S \ A) proof let y be set; assume y in (union S) \ (union A); then A1: y in (union S) & not y in (union A) by XBOOLE_0:def 4; then consider z being set such that A2: y in z & z in S by TARSKI:def 4; not z in A by A1,A2,TARSKI:def 4; then y in z & z in S \ A by A2,XBOOLE_0:def 4; hence thesis by TARSKI:def 4; end; thus union (S \ A) c= (union S) \ (union A) proof let y be set; assume y in union(S \ A); then consider z being set such that A3: y in z & z in S \ A by TARSKI:def 4; A4: y in z & z in S & not z in A by A3,XBOOLE_0:def 4; then A5: y in union S by TARSKI:def 4; A6: now let w be set; assume A7: w in A; then w in S; then z misses w by A4,A7,EQREL_1:def 6; hence not y in w by A3,XBOOLE_0:3; end; now assume y in union A; then consider v being set such that A8: y in v & v in A by TARSKI:def 4; thus contradiction by A6,A8; end; hence thesis by A5,XBOOLE_0:def 4; end; end; theorem Th4: for X being non empty set,A being Subset of X, S being a_partition of X holds A in S implies union(S \ {A}) = X \ A proof let X be non empty set; let A be Subset of X; let S be a_partition of X; assume A1: A in S; {A} c= S proof let y be set; assume y in {A}; hence thesis by A1,TARSKI:def 1; end; then reconsider AA = {A} as Subset of S; thus union (S \ {A}) = (union S) \ (union AA) by Th3 .= X \ (union {A}) by EQREL_1:def 6 .= X \ A by ZFMISC_1:31; end; theorem Th5: for T being non empty TopSpace, S being non empty a_partition of the carrier of T, A being Subset of space S, B being Subset of T holds B = union A implies (A is closed iff B is closed) proof let T be non empty TopSpace; let S be non empty a_partition of the carrier of T; let A be Subset of space S; let B be Subset of T; assume A1: B = union A; reconsider C = A as Subset of S by BORSUK_1:def 10; A2: [#](T) \ union A = (the carrier of T) \ union A by PRE_TOPC:def 3 .= (union S) \ (union C) by EQREL_1:def 6 .= union (S \ A) by Th3 .= union ((the carrier of space S) \ A) by BORSUK_1:def 10 .= union ([#](space S) \ A) by PRE_TOPC:def 3; thus A is closed implies B is closed proof assume A is closed; then A3: [#](space S) \ A is open by PRE_TOPC:def 6; reconsider om = [#](space S) \ A as Subset of S by BORSUK_1:def 10; om in the topology of space S by A3,PRE_TOPC:def 5; then [#](T) \ B in the topology of T by A1,A2,BORSUK_1:69; then [#](T) \ B is open by PRE_TOPC:def 5; hence thesis by PRE_TOPC:def 6; end; thus B is closed implies A is closed proof assume B is closed; then [#](T) \ B is open by PRE_TOPC:def 6; then A4: [#](T) \ union A in the topology of T by A1,PRE_TOPC: def 5; reconsider om = [#](space S) \ A as Subset of S by BORSUK_1:def 10; om in the topology of space S by A2,A4,BORSUK_1:69; then [#](space S) \ A is open by PRE_TOPC:def 5; hence thesis by PRE_TOPC:def 6; end; end; :: Classes of partitions of a set definition let X be non empty set,x be Element of X,S1 be a_partition of X; func EqClass(x,S1) -> Subset of X means :Def1: x in it & it in S1; existence proof consider EQR being Equivalence_Relation of X such that A1: S1 = Class EQR by EQREL_1:43; take Class(EQR,x); thus x in Class(EQR,x) by EQREL_1:28; thus Class(EQR,x) in S1 by A1,EQREL_1:def 5; end; uniqueness proof let A,B be Subset of X; assume A2: x in A & A in S1 & x in B & B in S1; then A meets B by XBOOLE_0:3; hence A = B by A2,EQREL_1:def 6; end; end; theorem Th6: for S1,S2 being a_partition of X st (for x being Element of X holds EqClass(x,S1) = EqClass(x,S2)) holds S1=S2 proof let S1,S2 be a_partition of X; assume A1: for x being Element of X holds EqClass(x,S1) = EqClass(x,S2); now let P be Subset of X; thus P in S1 implies P in S2 proof assume A2: P in S1; then A3: P is non empty by EQREL_1:def 6; consider x being Element of P; x in P by A3; then reconsider x as Element of X; P = EqClass(x,S1) by A2,A3,Def1; then P = EqClass(x,S2) by A1; hence P in S2 by Def1; end; thus P in S2 implies P in S1 proof assume A4: P in S2; then A5: P <> {} by EQREL_1:def 6; consider x being Element of P; x in P by A5; then reconsider x as Element of X; P = EqClass(x,S2) by A4,A5,Def1; then P = EqClass(x,S1) by A1; hence P in S1 by Def1; end; end; hence thesis by SETFAM_1:44; end; theorem Th7: for X being non empty set holds {X} is a_partition of X proof let X be non empty set; {X} c= bool X by ZFMISC_1:80; then reconsider A1 = {X} as Subset-Family of X by SETFAM_1:def 7; A1: union A1 = X by ZFMISC_1:31; for A being Subset of X st A in A1 holds A <> {} & for B being Subset of X st B in A1 holds A = B or A misses B proof let A be Subset of X; assume A2: A in A1; hence A <> {} by TARSKI:def 1; let B be Subset of X; assume B in A1; then B = X by TARSKI:def 1; hence A = B or A misses B by A2,TARSKI:def 1; end; hence {X} is a_partition of X by A1,EQREL_1:def 6; end; definition let X be set; mode Family-Class of X means :Def2: it c= bool bool X; existence; end; Lm1: for X being set holds {} is Family-Class of X proof let X be set; {} c= bool bool X by XBOOLE_1:2; hence thesis by Def2; end; definition let X be set; let F be Family-Class of X; attr F is partition-membered means :Def3: for S being set st S in F holds S is a_partition of X; end; definition let X be set; cluster partition-membered Family-Class of X; existence proof reconsider E = {} as Family-Class of X by Lm1; take E; let S be set; assume S in E; hence thesis; end; end; definition let X be set; mode Part-Family of X is partition-membered Family-Class of X; end; reserve F for Part-Family of X; definition let X be non empty set; cluster non empty a_partition of X; existence proof take {X}; thus thesis by Th7; end; end; theorem Th8: for X being set, p being a_partition of X holds {p} is Part-Family of X proof let X be set; let p be a_partition of X; A1: {p} c= bool bool X proof let x be set; assume x in {p}; then x is a_partition of X by TARSKI:def 1; hence thesis; end; for x be set st x in {p} holds x is a_partition of X by TARSKI:def 1; hence thesis by A1,Def2,Def3; end; definition let X be set; cluster non empty Part-Family of X; existence proof consider p being a_partition of X; reconsider P = {p} as Part-Family of X by Th8; take P; thus thesis; end; end; theorem Th9: for S1 being a_partition of X, x,y being Element of X holds EqClass(x,S1) meets EqClass(y,S1) implies EqClass(x,S1) = EqClass(y,S1) proof let S1 be a_partition of X; let x,y be Element of X; assume A1: EqClass(x,S1) meets EqClass(y,S1); consider EQR being Equivalence_Relation of X such that A2: S1 = Class EQR by EQREL_1:43; A3: Class(EQR,x) = EqClass(x,S1) proof A4: x in Class(EQR,x) by EQREL_1:28; Class(EQR,x) in S1 by A2,EQREL_1:def 5; hence thesis by A4,Def1; end; Class(EQR,y) = EqClass(y,S1) proof A5: y in Class(EQR,y) by EQREL_1:28; Class(EQR,y) in S1 by A2,EQREL_1:def 5; hence thesis by A5,Def1; end; hence thesis by A1,A3,EQREL_1:32; end; Lm2: for A being set holds A in {EqClass(x,S) where S is a_partition of X : S in F} implies ex T being a_partition of X st T in F & A = EqClass(x,T) proof let A be set; assume A in {EqClass(x,S) where S is a_partition of X : S in F}; then consider S being a_partition of X such that A1: A = EqClass(x,S) & S in F; take S; thus thesis by A1; end; theorem Th10: for A being set,X being non empty set,S being a_partition of X holds A in S implies ex x being Element of X st A = EqClass(x,S) proof let A be set,X be non empty set,S be a_partition of X; assume A1: A in S; then A is non empty by EQREL_1:def 6; then consider x being set such that A2: x in A by XBOOLE_0:def 1; take x; thus thesis by A1,A2,Def1; end; definition let X be non empty set,F be non empty Part-Family of X; func Intersection F -> non empty a_partition of X means :Def4: for x being Element of X holds EqClass(x,it) = meet{EqClass(x,S) where S is a_partition of X : S in F}; uniqueness proof given S1,S2 being a_partition of X such that A1: for x being Element of X holds EqClass(x,S1) = meet{EqClass(x,S) where S is a_partition of X : S in F} and A2: for x being Element of X holds EqClass(x,S2) = meet{EqClass(x,S) where S is a_partition of X : S in F} and A3: not S1 = S2; now let x be Element of X; EqClass(x,S1) = meet{EqClass(x,S) where S is a_partition of X : S in F} by A1; hence EqClass(x,S1) = EqClass(x,S2) by A2; end; hence contradiction by A3,Th6; end; existence proof thus ex G being non empty a_partition of X st for x being Element of X holds EqClass(x,G) = meet{EqClass(x,S) where S is a_partition of X : S in F} proof set G = {meet{EqClass(x,S) where S is a_partition of X : S in F} where x is Element of X : not contradiction }; G c= bool X proof let y be set; assume y in G; then consider x being Element of X such that A4: y = meet{EqClass(x,S) where S is a_partition of X : S in F}; y c= X proof let e be set; assume A5: e in y; consider T being set such that A6: T in F by XBOOLE_0:def 1; reconsider T as a_partition of X by A6,Def3; EqClass(x,T) in {EqClass(x,S) where S is a_partition of X : S in F} by A6; then consider f being set such that A7: f in {EqClass(x,S) where S is a_partition of X : S in F}; consider S being a_partition of X such that A8: f = EqClass(x,S) & S in F by A7; e in EqClass(x,S) by A4,A5,A7,A8,SETFAM_1:def 1; hence e in X; end; hence y in bool X; end; then reconsider G as Subset-Family of X by SETFAM_1:def 7; G is a_partition of X proof per cases; case X <> {}; thus union G = X proof thus union G c= X; thus X c= union G proof let a be set; assume a in X; then reconsider a1=a as Element of X; consider T being set such that A9: T in F by XBOOLE_0:def 1; reconsider T as a_partition of X by A9,Def3; A10: EqClass(a1,T) in {EqClass(a1,S) where S is a_partition of X : S in F} by A9; for T being set st T in {EqClass(a1,S) where S is a_partition of X : S in F} holds a1 in T proof let T be set; assume T in {EqClass(a1,S) where S is a_partition of X : S in F}; then consider A being a_partition of X such that A11: T = EqClass(a1,A) and A in F; thus thesis by A11,Def1; end; then A12: a in meet{EqClass(a1,S) where S is a_partition of X : S in F} by A10,SETFAM_1:def 1; meet{EqClass(a1,S) where S is a_partition of X : S in F} in G; hence a in union G by A12,TARSKI:def 4; end; end; let A be Subset of X; assume A13: A in G; then consider x being Element of X such that A14: A = meet{EqClass(x,S) where S is a_partition of X : S in F}; consider T being set such that A15: T in F by XBOOLE_0:def 1; reconsider T as a_partition of X by A15,Def3; A16: EqClass(x,T) in {EqClass(x,S) where S is a_partition of X : S in F} by A15; for Y being set st Y in {EqClass(x,S) where S is a_partition of X : S in F} holds x in Y proof let Y be set; assume Y in {EqClass(x,S) where S is a_partition of X : S in F}; then ex T being a_partition of X st Y = EqClass(x,T) & T in F; hence x in Y by Def1; end; hence A<>{} by A14,A16,SETFAM_1:def 1; consider x being Element of X such that A17: A = meet{EqClass(x,S) where S is a_partition of X : S in F} by A13; let B be Subset of X; assume B in G; then consider y being Element of X such that A18: B = meet{EqClass(y,S) where S is a_partition of X : S in F}; thus A = B or A misses B proof A19: {EqClass(x,S) where S is a_partition of X : S in F} is non empty proof consider T being set such that A20: T in F by XBOOLE_0:def 1; reconsider T as a_partition of X by A20,Def3; EqClass(x,T) in {EqClass(x,S) where S is a_partition of X : S in F} by A20; hence thesis; end; A21: {EqClass(y,S) where S is a_partition of X : S in F} is non empty proof consider T being set such that A22: T in F by XBOOLE_0:def 1; reconsider T as a_partition of X by A22,Def3; EqClass(y,T) in {EqClass(y,S) where S is a_partition of X : S in F} by A22; hence thesis; end; now assume A meets B; then consider c being set such that A23: c in A & c in B by XBOOLE_0:3; c in meet{EqClass(x,S) where S is a_partition of X : S in F} /\ meet{EqClass(y,S) where S is a_partition of X : S in F} by A17,A18,A23,XBOOLE_0:def 3; then A24: c in meet({EqClass(x,S) where S is a_partition of X : S in F } \/ {EqClass(y,S) where S is a_partition of X : S in F}) by A19,A21,SETFAM_1:10; A25: now let T be a_partition of X; assume T in F; then EqClass(x,T) in {EqClass(x,S) where S is a_partition of X : S in F}; then EqClass(x,T) in {EqClass(x,S) where S is a_partition of X : S in F} \/ {EqClass(y,S) where S is a_partition of X : S in F} by XBOOLE_0:def 2; hence c in EqClass(x,T) by A24,SETFAM_1:def 1; end; A26: now let T be a_partition of X; assume T in F; then EqClass(y,T) in {EqClass(y,S) where S is a_partition of X : S in F}; then EqClass(y,T) in {EqClass(x,S) where S is a_partition of X : S in F} \/ {EqClass(y,S) where S is a_partition of X : S in F} by XBOOLE_0:def 2; hence c in EqClass(y,T) by A24,SETFAM_1:def 1; end; A27: for T being a_partition of X st T in F holds ex A being set st A in EqClass(x,T) & A in EqClass(y,T) proof let T be a_partition of X; assume A28: T in F; take c; thus thesis by A25,A26,A28; end; A29: for T being a_partition of X st T in F holds (EqClass(x,T)) meets (EqClass(y,T)) proof let T be a_partition of X; assume T in F; then consider A being set such that A30: A in EqClass(x,T) & A in EqClass(y,T) by A27; thus thesis by A30,XBOOLE_0:3; end; A31: for T being a_partition of X st T in F holds EqClass(x,T) = EqClass(y,T) proof let T be a_partition of X; assume T in F; then EqClass(x,T) meets EqClass(y,T) by A29; hence thesis by Th9; end; {EqClass(x,S) where S is a_partition of X : S in F} = {EqClass(y,S) where S is a_partition of X : S in F} proof thus {EqClass(x,S) where S is a_partition of X : S in F} c= {EqClass(y,S) where S is a_partition of X : S in F} proof let A be set; assume A in {EqClass(x,S) where S is a_partition of X : S in F}; then consider T being a_partition of X such that A32: T in F & A = EqClass(x,T) by Lm2; A = EqClass(y,T) by A31,A32; hence thesis by A32; end; thus {EqClass(y,S) where S is a_partition of X : S in F} c= {EqClass(x,S) where S is a_partition of X : S in F} proof let A be set; assume A in {EqClass(y,S) where S is a_partition of X : S in F}; then consider T being a_partition of X such that A33: T in F & A = EqClass(y,T) by Lm2; A = EqClass(x,T) by A31,A33; hence thesis by A33; end; end; hence thesis by A17,A18; end; hence thesis; end; case X = {}; hence G = {}; end; then reconsider G as non empty a_partition of X; take G; for x being Element of X holds EqClass(x,G) = meet{EqClass(x,S) where S is a_partition of X : S in F} proof let x be Element of X; A34: meet{EqClass(x,S) where S is a_partition of X : S in F} in G; x in meet{EqClass(x,S) where S is a_partition of X : S in F} proof A35: {EqClass(x,S) where S is a_partition of X : S in F} is non empty proof consider T being set such that A36: T in F by XBOOLE_0:def 1; reconsider T as a_partition of X by A36,Def3; EqClass(x,T) in {EqClass(x,S) where S is a_partition of X : S in F} by A36; hence thesis; end; now let Y be set; assume Y in {EqClass(x,S) where S is a_partition of X : S in F}; then ex T being a_partition of X st Y = EqClass(x,T) & T in F; hence x in Y by Def1; end; hence thesis by A35,SETFAM_1:def 1; end; hence thesis by A34,Def1; end; hence thesis; end; end; end; :: Families of partitions of topological spaces reserve T for non empty TopSpace; theorem Th11: { A where A is a_partition of the carrier of T : A is closed } is Part-Family of the carrier of T proof set S = { A where A is a_partition of the carrier of T : A is closed }; A1: S c= bool bool the carrier of T proof let B be set; assume B in S; then consider A being a_partition of the carrier of T such that A2: B = A and A is closed; thus thesis by A2; end; now let B be set; assume B in { A where A is a_partition of the carrier of T : A is closed }; then consider A being a_partition of the carrier of T such that A3: B = A and A is closed; thus B is a_partition of the carrier of T by A3; end; hence thesis by A1,Def2,Def3; end; definition let T; func Closed_Partitions T -> non empty Part-Family of the carrier of T equals :Def5: { A where A is a_partition of the carrier of T : A is closed }; coherence proof set F = { A where A is a_partition of the carrier of T : A is closed }; reconsider ct = {the carrier of T} as a_partition of the carrier of T by Th7; for A being Subset of T st A in ct holds A is closed proof let A be Subset of T; assume A in ct; then A = the carrier of T by TARSKI:def 1; then A = [#](T) by PRE_TOPC:def 3; hence thesis; end; then ct is closed by TOPS_2:def 2; then ct in F; hence thesis by Th11; end; end; :: T_1 reflex of a topological space definition let T be non empty TopSpace; func T_1-reflex T -> TopSpace equals :Def6: space (Intersection Closed_Partitions T); correctness; end; definition let T; cluster T_1-reflex T -> strict non empty; coherence proof T_1-reflex T = space (Intersection Closed_Partitions T) by Def6; hence thesis; end; end; theorem Th12: for T being non empty TopSpace holds T_1-reflex T is being_T1 proof let T be non empty TopSpace; now let p be Point of T_1-reflex T; T_1-reflex T = space (Intersection Closed_Partitions T) by Def6; then A1: the carrier of T_1-reflex T = Intersection Closed_Partitions T by BORSUK_1:def 10; then A2: p in Intersection Closed_Partitions T; consider x being Element of T such that A3: p = EqClass(x,Intersection Closed_Partitions T) by A1,Th10; A4: p = meet { EqClass(x,S) where S is a_partition of the carrier of T : S in Closed_Partitions T } by A3,Def4; reconsider q=p as Subset of T by A3; A5: { EqClass(x,S) where S is a_partition of the carrier of T : S in Closed_Partitions T } is non empty proof consider Y being set such that A6: Y in Closed_Partitions T by XBOOLE_0:def 1; reconsider Y as a_partition of the carrier of T by A6,Def3; EqClass(x,Y) in {EqClass(x,S) where S is a_partition of the carrier of T : S in Closed_Partitions T} by A6; hence thesis; end; { EqClass(x,S) where S is a_partition of the carrier of T : S in Closed_Partitions T } c= bool the carrier of T proof let Z be set; assume Z in { EqClass(x,S) where S is a_partition of the carrier of T : S in Closed_Partitions T }; then consider Y being a_partition of the carrier of T such that A7: Z = EqClass(x,Y) & Y in Closed_Partitions T; thus thesis by A7; end; then reconsider m = { EqClass(x,S) where S is a_partition of the carrier of T : S in Closed_Partitions T } as non empty Subset-Family of T by A5,SETFAM_1:def 7; reconsider m as non empty Subset-Family of T; for A being Subset of T st A in m holds A is closed proof let A be Subset of T; assume A in m; then consider S being a_partition of the carrier of T such that A8: A = EqClass(x,S) & S in Closed_Partitions T; S in { B where B is a_partition of the carrier of T : B is closed } by A8,Def5; then consider B being a_partition of the carrier of T such that A9: S = B & B is closed; A in S by A8,Def1; hence thesis by A9,TOPS_2:def 2; end; then q is closed by A4,PRE_TOPC:44; then [#](T) \ q is open by PRE_TOPC:def 6; then [#](T) \ p in the topology of T by PRE_TOPC:def 5; then (the carrier of T) \ p in the topology of T by PRE_TOPC:def 3; then A10: union((Intersection Closed_Partitions T) \ {p}) in the topology of T by A2,Th4; reconsider I = (Intersection Closed_Partitions T) \ {p} as Subset of (Intersection Closed_Partitions T) by XBOOLE_1:36; A11: I in {A where A is Subset of (Intersection Closed_Partitions T) : union A in the topology of T} by A10; the topology of space(Intersection Closed_Partitions T) = {A where A is Subset of (Intersection Closed_Partitions T) : union A in the topology of T} by BORSUK_1:def 10; then A12: I in the topology of T_1-reflex T by A11,Def6; reconsider I as Subset of space(Intersection Closed_Partitions T) by BORSUK_1:def 10; reconsider I as Subset of T_1-reflex T by Def6; I = (the carrier of space(Intersection Closed_Partitions T)) \ {p} by BORSUK_1:def 10 .= (the carrier of T_1-reflex T) \ {p} by Def6 .= ([#] T_1-reflex T) \ {p} by PRE_TOPC:def 3; then ([#] T_1-reflex T) \ {p} is open by A12,PRE_TOPC:def 5; hence {p} is closed by PRE_TOPC:def 6; end; hence thesis by URYSOHN1:27; end; definition let T; cluster T_1-reflex T -> being_T1; coherence by Th12; end; definition let T be non empty TopSpace; func T_1-reflect T -> continuous map of T,T_1-reflex T equals :Def7: Proj Intersection Closed_Partitions T; correctness proof space Intersection Closed_Partitions T = T_1-reflex T by Def6; hence thesis; end; end; theorem Th13: for T,T1 being non empty TopSpace,f being continuous map of T,T1 holds T1 is being_T1 implies ({f"{z} where z is Element of T1 : z in rng f} is a_partition of the carrier of T) & (for A being Subset of T st A in {f"{z} where z is Element of T1 : z in rng f} holds A is closed) proof let T,T1 be non empty TopSpace; let f be continuous map of T,T1; A1: dom f = the carrier of T & rng f c= the carrier of T1 by FUNCT_2:def 1,RELSET_1:12; assume A2: T1 is being_T1; thus {f"{z} where z is Element of T1 : z in rng f} is a_partition of the carrier of T proof {f"{z} where z is Element of T1 : z in rng f} c= bool the carrier of T proof let y; assume y in {f"{z} where z is Element of T1 : z in rng f}; then consider z being Element of T1 such that A3: y = f"{z} & z in rng f; thus thesis by A3; end; then reconsider fz = {f"{z} where z is Element of T1 : z in rng f} as Subset-Family of T by SETFAM_1:def 7; reconsider fz as Subset-Family of T; A4: union fz = the carrier of T proof thus union fz c= the carrier of T; thus the carrier of T c= union fz proof let y; assume A5: y in the carrier of T; consider z being set such that A6: z = f.y; A7: z in rng f by A1,A5,A6,FUNCT_1:def 5; then reconsider z as Element of T1 by A1; A8: f"{z} in fz by A7; f.y in {f.y} by TARSKI:def 1; then y in f"{z} by A1,A5,A6,FUNCT_1:def 13; hence thesis by A8,TARSKI:def 4; end; end; for A being Subset of T st A in fz holds A <> {} & for B being Subset of T st B in fz holds A = B or A misses B proof let A be Subset of T; assume A in fz; then consider z being Element of T1 such that A9: A = f"{z} & z in rng f; consider y being set such that A10: y in dom f & z = f.y by A9,FUNCT_1:def 5; f.y in {f.y} by TARSKI:def 1; hence A <> {} by A9,A10,FUNCT_1:def 13; let B be Subset of T; assume B in fz; then consider w being Element of T1 such that A11: B = f"{w} & w in rng f; thus A = B or A misses B proof now assume not A misses B; then consider v being set such that A12: v in A & v in B by XBOOLE_0:3; f.v in {z} & f.v in {w} by A9,A11,A12,FUNCT_1:def 13; then f.v = z & f.v = w by TARSKI:def 1; hence A = B by A9,A11; end; hence thesis; end; end; hence thesis by A4,EQREL_1:def 6; end; thus for A being Subset of T st A in {f"{z} where z is Element of T1 : z in rng f} holds A is closed proof let A be Subset of T; assume A in {f"{z} where z is Element of T1 : z in rng f}; then consider z being Element of T1 such that A13: A = f"{z} & z in rng f; {z} is closed by A2,URYSOHN1:27; hence thesis by A13,PRE_TOPC:def 12; end; end; theorem Th14: for T,T1 being non empty TopSpace,f being continuous map of T,T1 holds T1 is being_T1 implies for w for x being Element of T holds w = EqClass(x,(Intersection Closed_Partitions T)) implies w c= f"{f.x} proof let T,T1 be non empty TopSpace; let f be continuous map of T,T1; A1: dom f = the carrier of T & rng f c= the carrier of T1 by FUNCT_2:def 1,RELSET_1:12; assume A2: T1 is being_T1; let w be set; let x be Element of T; assume A3: w = EqClass(x,(Intersection Closed_Partitions T)); let y be set; assume A4: y in w; reconsider fz = {f"{z} where z is Element of T1 : z in rng f} as a_partition of the carrier of T by A2,Th13; for A being Subset of T st A in fz holds A is closed by A2,Th13; then fz is closed by TOPS_2:def 2; then fz in {B where B is a_partition of the carrier of T : B is closed}; then fz in Closed_Partitions T by Def5; then A5: EqClass(x,fz) in {EqClass(x,S) where S is a_partition of the carrier of T : S in Closed_Partitions T}; A6: f"{f.x} = EqClass(x,fz) proof f.x in {f.x} by TARSKI:def 1; then A7: x in f"{f.x} by A1,FUNCT_1:def 13; A8: f.x in rng f by A1,FUNCT_1:def 5; reconsider fx = f.x as Element of T1; f"{fx} in fz by A8; hence thesis by A7,Def1; end; EqClass(x,(Intersection Closed_Partitions T)) = meet{EqClass(x,S) where S is a_partition of the carrier of T : S in Closed_Partitions T} by Def4; hence y in f"{f.x} by A3,A4,A5,A6,SETFAM_1:def 1; end; theorem Th15: for T,T1 being non empty TopSpace,f being continuous map of T,T1 holds T1 is being_T1 implies for w st w in the carrier of T_1-reflex T ex z being Element of T1 st z in rng f & w c= f"{z} proof let T,T1 be non empty TopSpace; let f be continuous map of T,T1; assume A1: T1 is being_T1; let w be set; assume w in the carrier of T_1-reflex T; then w in the carrier of space (Intersection Closed_Partitions T) by Def6 ; then w in Intersection (Closed_Partitions T) by BORSUK_1:def 10; then consider x being Element of T such that A2: w = EqClass(x,Intersection (Closed_Partitions T)) by Th10; reconsider x as Element of T; A3: dom f = the carrier of T & rng f c= the carrier of T1 by FUNCT_2:def 1,RELSET_1:12; reconsider fx = f.x as Element of T1; take fx; thus thesis by A1,A2,A3,Th14,FUNCT_1:def 5; end; :: The theorem on factorization theorem Th16: for T,T1 being non empty TopSpace,f being continuous map of T,T1 holds T1 is being_T1 implies ex h being continuous map of T_1-reflex T, T1 st f = h*T_1-reflect T proof let T,T1 be non empty TopSpace; let f be continuous map of T,T1; assume A1: T1 is being_T1; then reconsider fx = {f"{x} where x is Element of T1 : x in rng f} as a_partition of the carrier of T by Th13; set g = T_1-reflect T; A2: dom f = the carrier of T & rng f c= the carrier of T1 by FUNCT_2:def 1,RELSET_1:12; A3: dom g = the carrier of T & rng g c= the carrier of T_1-reflex T by FUNCT_2:def 1,RELSET_1:12; defpred X[set,set] means for z being Element of T1 holds (z in rng f & $1 c= f"{z}) implies $2 = f"{z}; A4: for y,w,v st y in the carrier of T_1-reflex T & X[y,w] & X[y,v] holds w = v proof let y,w,v; assume A5: y in the carrier of T_1-reflex T & (for z being Element of T1 holds (z in rng f & y c= f"{z}) implies w = f"{z}) & (for z being Element of T1 holds (z in rng f & y c= f"{z}) implies v = f"{z}); then y in the carrier of space Intersection(Closed_Partitions T) by Def6; then y in Intersection(Closed_Partitions T) by BORSUK_1:def 10; then consider x being Element of T such that A6: y = EqClass(x,Intersection(Closed_Partitions T)) by Th10; reconsider x as Element of T; reconsider fix = f.x as Element of T1; fix in rng f & y c= f"{fix} by A1,A2,A6,Th14,FUNCT_1:def 5; then w = f"{fix} & v = f"{fix} by A5; hence w = v; end; A7: for y st y in the carrier of T_1-reflex T ex w st X[y,w] proof let y; assume y in the carrier of T_1-reflex T; then y in the carrier of space Intersection(Closed_Partitions T) by Def6; then y in Intersection(Closed_Partitions T) by BORSUK_1:def 10; then consider x being Element of T such that A8: y = EqClass(x,Intersection(Closed_Partitions T)) by Th10; reconsider x as Element of T; set w = f"{f.x}; take w; let z be Element of T1; assume A9: z in rng f & y c= f"{z}; then A10: f"{z} in fx; A11: f.x in rng f by A2,FUNCT_1:def 5; reconsider fix = f.x as Element of T1; f"{fix} in fx by A11; then A12: w misses f"{z} or w = f"{z} by A10,EQREL_1:def 6; A13: y c= w & y c= f"{z} by A1,A8,A9,Th14; y is non empty by A8,Def1; then consider z1 being set such that A14: z1 in y by XBOOLE_0:def 1; thus w = f"{z} by A12,A13,A14,XBOOLE_0:3; end; consider h1 being Function such that A15: dom h1 = the carrier of T_1-reflex T & for y st y in the carrier of T_1-reflex T holds X[y,h1.y] from FuncEx(A4,A7); defpred X[set,set] means for z being Element of T1 holds (z in rng f & $1 = f"{z}) implies $2 = z; A16: for y,w,v st y in fx & X[y,w] & X[y,v] holds w = v proof let y,w,v; assume A17: y in fx & (for z being Element of T1 holds (z in rng f & y = f"{z}) implies w = z) & (for z being Element of T1 holds (z in rng f & y = f"{z}) implies v = z); then consider z1 being Element of T1 such that A18: y = f"{z1} & z1 in rng f; w = z1 & v = z1 by A17,A18; hence thesis; end; A19: for y st y in fx ex w st X[y,w] proof let y be set; assume y in fx; then consider w being Element of T1 such that A20: y = f"{w} & w in rng f; take w; let z be Element of T1; assume A21: z in rng f & y = f"{z}; now assume A22: z <> w; consider v being set such that A23: v in dom f & z = f.v by A21,FUNCT_1:def 5; z in {z} by TARSKI:def 1; then v in f"{w} by A20,A21,A23,FUNCT_1:def 13; then f.v in {w} by FUNCT_1:def 13; hence contradiction by A22,A23,TARSKI:def 1; end; hence w = z; end; consider h2 being Function such that A24: dom h2 = fx & for y st y in fx holds X[y,h2.y] from FuncEx(A16,A19); set h = h2*h1; A25: dom h = the carrier of T_1-reflex T proof thus dom h c= the carrier of T_1-reflex T by A15,RELAT_1:44; let z be set; assume A26: z in the carrier of T_1-reflex T; then consider w being Element of T1 such that A27: w in rng f & z c= f"{w} by A1,Th15; h1.z = f"{w} by A15,A26,A27; then h1.z in dom h2 by A24,A27; hence z in dom h by A15,A26,FUNCT_1:21; end; A28: rng h c= rng h2 proof let z be set; thus thesis by FUNCT_1:25; end; rng h2 c= the carrier of T1 proof let y; assume y in rng h2; then consider w being set such that A29: w in dom h2 & y = h2.w by FUNCT_1:def 5; consider x being Element of T1 such that A30: w = f"{x} & x in rng f by A24,A29; h2.w = x by A24,A29,A30; hence y in the carrier of T1 by A29; end; then A31: rng h c= the carrier of T1 by A28,XBOOLE_1:1; A32: dom (h*g) = the carrier of T proof thus dom (h*g) c= the carrier of T by A3,RELAT_1:44; let y be set; assume A33: y in the carrier of T; then g.y in rng g by A3,FUNCT_1:def 5; hence y in dom (h*g) by A3,A25,A33,FUNCT_1:21; end; A34: f = h*g proof for x being set st x in dom f holds f.x = (h*g).x proof let x be set; assume A35: x in dom f; then g.x in rng g by A2,A3,FUNCT_1:def 5; then g.x in the carrier of T_1-reflex T by A3; then g.x in the carrier of space (Intersection Closed_Partitions T) by Def6; then g.x in Intersection (Closed_Partitions T) by BORSUK_1:def 10; then consider y being Element of T such that A36: g.x = EqClass(y,Intersection (Closed_Partitions T)) by Th10; reconsider x as Element of T by A2,A35; g = Proj (Intersection Closed_Partitions T) by Def7; then g = proj (Intersection Closed_Partitions T) by BORSUK_1:def 11 ; then A37: x in g.x by BORSUK_1:def 1; x in EqClass(x,Intersection (Closed_Partitions T)) by Def1; then A38: EqClass(x,Intersection (Closed_Partitions T)) meets EqClass(y,Intersection (Closed_Partitions T)) by A36,A37,XBOOLE_0:3; reconsider fix = f.x as Element of T1; g.x = EqClass(x,Intersection (Closed_Partitions T)) by A36,A38,Th9 ; then A39: fix in rng f & g.x c= f"{fix} by A1,A35,Th14,FUNCT_1 :def 5; then A40: f"{fix} in fx; (h*g).x = (h2*h1).(g.x) by A32,FUNCT_1:22 .= h2.(h1.(g.x)) by A15,FUNCT_1:23 .= h2.(f"{fix}) by A15,A39 .= f.x by A24,A39,A40; hence thesis; end; hence thesis by A2,A32,FUNCT_1:9; end; reconsider h as Function of the carrier of T_1-reflex T,the carrier of T1 by A25,A31,FUNCT_2:def 1,RELSET_1:11; reconsider h as map of T_1-reflex T,T1; h is continuous proof let y be Subset of T1; assume A41: y is closed; reconsider hy = h"y as Subset of space Intersection(Closed_Partitions T) by Def6; union hy c= the carrier of T proof let z1 be set; assume z1 in union hy; then consider z2 being set such that A42: z1 in z2 & z2 in hy by TARSKI:def 4; z2 in the carrier of space Intersection(Closed_Partitions T) by A42 ; then z2 in Intersection(Closed_Partitions T) by BORSUK_1:def 10; hence thesis by A42; end; then reconsider uhy = union hy as Subset of T; A43: T_1-reflex T = space (Intersection Closed_Partitions T) by Def6; (h*g)"y is closed by A34,A41,PRE_TOPC:def 12; then g"(h"y) is closed by RELAT_1:181; then (Proj (Intersection Closed_Partitions T))"hy is closed by Def7; then uhy is closed by Th2; hence h"y is closed by A43,Th5; end; then reconsider h as continuous map of T_1-reflex T,T1; take h; thus thesis by A34; end; definition let T,S be non empty TopSpace; let f be continuous map of T,S; func T_1-reflex f -> continuous map of T_1-reflex T, T_1-reflex S means (T_1-reflect S) * f = it * (T_1-reflect T); existence by Th16; uniqueness proof let g1,g2 be continuous map of T_1-reflex T, T_1-reflex S; assume that A1: (T_1-reflect S) * f = g1 * (T_1-reflect T) and A2: (T_1-reflect S) * f = g2 * (T_1-reflect T); A3: dom g1 = the carrier of (T_1-reflex T) & dom g2 = the carrier of (T_1-reflex T) by FUNCT_2:def 1; now let x be set; assume x in dom g1; then A4: x in the carrier of T_1-reflex T by FUNCT_2:def 1; A5: the carrier of T_1-reflex T = the carrier of space (Intersection Closed_Partitions T) by Def6 .= (Intersection Closed_Partitions T) by BORSUK_1:def 10; then consider y being Element of T such that A6: x = EqClass(y,(Intersection (Closed_Partitions T))) by A4,Th10; A7: dom (T_1-reflect T) = the carrier of T & rng (T_1-reflect T) c= the carrier of T_1-reflex T by FUNCT_2:def 1,RELSET_1:12; reconsider y as Element of T; T_1-reflect T = Proj (Intersection Closed_Partitions T) by Def7 .= proj (Intersection Closed_Partitions T) by BORSUK_1:def 11; then A8: y in (T_1-reflect T).y & y in x by A6,Def1,BORSUK_1:def 1; set ty=(T_1-reflect T).y; ty in (Intersection Closed_Partitions T) by A5; then A9: ty misses x or ty = x by A4,A5,EQREL_1:def 6; hence g2.x = (g2 * (T_1-reflect T)).y by A7,A8,FUNCT_1:23,XBOOLE_0:3 .= g1.x by A1,A2,A7,A8,A9,FUNCT_1:23,XBOOLE_0:3; end; hence g1=g2 by A3,FUNCT_1:9; end; end;