### The Mizar article:

### Bases and Refinements of Topologies

**by****Grzegorz Bancerek**

- Received March 9, 1998
Copyright (c) 1998 Association of Mizar Users

- MML identifier: YELLOW_9
- [ MML identifier index ]

environ vocabulary ORDERS_1, SETFAM_1, WAYBEL_0, SUBSET_1, BOOLE, PRE_TOPC, FUNCT_1, RELAT_1, TARSKI, CANTOR_1, WAYBEL_9, REALSET1, RELAT_2, NATTRA_1, FINSET_1, BHSP_3, LATTICES, FUNCOP_1, YELLOW_0, ORDINAL2, FUNCT_3, SEQM_3, CAT_1, WELLORD1, OPPCAT_1, QUANTAL1, FUNCT_2, FRAENKEL, CONNSP_2, TOPS_1, LATTICE3, BORSUK_1, PRELAMB, WAYBEL11, PROB_1, YELLOW_6, YELLOW_9, RLVECT_3; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, SETFAM_1, RELAT_1, FUNCT_1, FINSET_1, MCART_1, STRUCT_0, RELAT_2, RELSET_1, REALSET1, FRAENKEL, FUNCT_2, WELLORD1, GRCAT_1, ORDERS_1, LATTICE3, ORDERS_3, YELLOW_0, WAYBEL_0, TDLAT_3, GROUP_1, PRE_TOPC, TOPS_1, TOPS_2, CONNSP_2, YELLOW_6, YELLOW_7, WAYBEL_9, WAYBEL11, CANTOR_1, BORSUK_1; constructors RELAT_2, WAYBEL11, CANTOR_1, TOPS_1, TDLAT_3, GROUP_1, TOLER_1, WAYBEL_3, WAYBEL_5, ORDERS_3, WAYBEL_1, GRCAT_1, TOPS_2, BORSUK_1, LATTICE3, PARTFUN1; clusters TDLAT_3, FRAENKEL, YELLOW_6, YELLOW_2, FINSET_1, RELSET_1, BORSUK_1, STRUCT_0, TEX_1, YELLOW_0, LATTICE3, WAYBEL_0, WAYBEL_7, WAYBEL_9, WAYBEL11, SUBSET_1, RLVECT_2, FUNCT_2, SETFAM_1, XBOOLE_0, ZFMISC_1, PARTFUN1; requirements BOOLE, SUBSET; definitions TARSKI, STRUCT_0, PRE_TOPC, CANTOR_1, ORDERS_1, GROUP_1, YELLOW_0, WAYBEL_0, WAYBEL_1, TOPS_2, WAYBEL11, XBOOLE_0; theorems STRUCT_0, SETFAM_1, FRAENKEL, ENUMSET1, REALSET1, PRE_TOPC, CANTOR_1, YELLOW_0, WAYBEL_0, FUNCT_1, RELAT_1, FUNCT_2, TEX_1, ORDERS_1, ZFMISC_1, TARSKI, FINSET_1, FUNCOP_1, RELSET_1, TDLAT_3, GRCAT_1, BORSUK_1, YELLOW_6, YELLOW_8, TOPS_1, CONNSP_2, WAYBEL_9, WAYBEL11, TOPS_2, LATTICE3, ORDERS_2, WELLORD1, XBOOLE_0, XBOOLE_1; schemes FRAENKEL, COMPTS_1; begin :: Subsets as nets scheme FraenkelInvolution {A() -> non empty set, X,Y() -> Subset of A(), F(set) -> Element of A()}: X() = {F(a) where a is Element of A(): a in Y()} provided A1: Y() = {F(a) where a is Element of A(): a in X()} and A2: for a being Element of A() holds F(F(a)) = a proof hereby let x be set; assume A3: x in X(); then reconsider a = x as Element of A(); F(a) in Y() & F(F(a)) = a by A1,A2,A3; hence x in {F(b) where b is Element of A(): b in Y()}; end; let x be set; assume x in {F(b) where b is Element of A(): b in Y()}; then consider b being Element of A() such that A4: x = F(b) & b in Y(); consider a being Element of A() such that A5: b = F(a) & a in X() by A1,A4; thus x in X() by A2,A4,A5; end; scheme FraenkelComplement1 {A() -> non empty RelStr, X() -> Subset-Family of A(), Y() -> set, F(set) -> Subset of A()}: COMPLEMENT X() = {F(a)` where a is Element of A(): a in Y()} provided A1: X() = {F(a) where a is Element of A(): a in Y()} proof hereby let x be set; assume A2: x in COMPLEMENT X(); then reconsider y = x as Subset of A(); y` in X() by A2,YELLOW_8:13; then consider b being Element of A() such that A3: y` = F(b) & b in Y() by A1; x = y`` & F(b) = F(b)``; hence x in {F(a)` where a is Element of A(): a in Y()} by A3; end; let x be set; assume x in {F(a)` where a is Element of A(): a in Y()}; then consider a being Element of A() such that A4: x = F(a)` & a in Y(); F(a) in X() by A1,A4; hence x in COMPLEMENT X() by A4,YELLOW_8:14; end; scheme FraenkelComplement2 {A() -> non empty RelStr, X() -> Subset-Family of A(), Y() -> set, F(set) -> Subset of A()}: COMPLEMENT X() = {F(a) where a is Element of A(): a in Y()} provided A1: X() = {F(a)` where a is Element of A(): a in Y()} proof hereby let x be set; assume A2: x in COMPLEMENT X(); then reconsider y = x as Subset of A(); y` in X() by A2,YELLOW_8:13; then consider b being Element of A() such that A3: y` = F(b)` & b in Y() by A1; x = y`` & F(b) = F(b)``; hence x in {F(a) where a is Element of A(): a in Y()} by A3; end; let x be set; assume x in {F(a) where a is Element of A(): a in Y()}; then consider a being Element of A() such that A4: x = F(a) & a in Y(); F(a)` in X() by A1,A4; hence x in COMPLEMENT X() by A4,YELLOW_8:13; end; theorem for R being non empty RelStr, x,y being Element of R holds y in (uparrow x)` iff not x <= y proof let R be non empty RelStr, x,y be Element of R; [#]R = the carrier of R by PRE_TOPC:12; then (uparrow x)` = (the carrier of R) \ uparrow x by PRE_TOPC:17; then y in (uparrow x)` iff not y in uparrow x by XBOOLE_0:def 4; hence y in (uparrow x)` iff not x <= y by WAYBEL_0:18; end; ABC {A() -> TopSpace, F(set) -> set, f() -> Function, P[set]}: f()"union {F(x) where x is Subset of A(): P[x]} = union {f()"(F(y)) where y is Subset of A(): P[y]} proof set X = {F(x) where x is Subset of A(): P[x]}; set Y = {f()"(F(y)) where y is Subset of A(): P[y]}; hereby let x be set; assume x in f()"union X; then A1: x in dom f() & f().x in union X by FUNCT_1:def 13; then consider y being set such that A2: f().x in y & y in X by TARSKI:def 4; consider a being Subset of A() such that A3: y = F(a) & P[a] by A2; x in f()"F(a) & f()"F(a) in Y by A1,A2,A3,FUNCT_1:def 13; hence x in union Y by TARSKI:def 4; end; let x be set; assume x in union Y; then consider y being set such that A4: x in y & y in Y by TARSKI:def 4; consider a being Subset of A() such that A5: y = f()"F(a) & P[a] by A4; f().x in F(a) & F(a) in X by A4,A5,FUNCT_1:def 13; then f().x in union X & x in dom f() by A4,A5,FUNCT_1:def 13,TARSKI:def 4; hence thesis by FUNCT_1:def 13; end; theorem Th2: for S being 1-sorted, T being non empty 1-sorted, f being map of S,T for X being Subset of T holds (f"X)` = f"X` proof let S be 1-sorted, T be non empty 1-sorted, f be map of S,T; let X be Subset of T; A1: the carrier of T = [#]T & the carrier of S = [#]S by PRE_TOPC:12; hence (f"X)` = (the carrier of S)\(f"X) by PRE_TOPC:17 .= f"(the carrier of T)\(f"X) by FUNCT_2:48 .= f"((the carrier of T)\X) by FUNCT_1:138 .= f"X` by A1,PRE_TOPC:17; end; theorem Th3: for T being 1-sorted, F being Subset-Family of T holds COMPLEMENT F = {a` where a is Subset of T: a in F} proof let T be 1-sorted, F be Subset-Family of T; set X = {a` where a is Subset of T: a in F}; hereby let x be set; assume A1: x in COMPLEMENT F; then reconsider P = x as Subset of T; P` in F & P`` = P by A1,YELLOW_8:13; hence x in X; end; let x be set; assume x in X; then ex P being Subset of T st x = P` & P in F; hence thesis by YELLOW_8:14; end; theorem Th4: for R being non empty RelStr for F being Subset of R holds uparrow F = union {uparrow x where x is Element of R: x in F} & downarrow F = union {downarrow x where x is Element of R: x in F} proof let R be non empty RelStr, F be Subset of R; A1: uparrow F = {x where x is Element of R: ex y being Element of R st x >= y & y in F} by WAYBEL_0:15; A2: downarrow F = {x where x is Element of R: ex y being Element of R st x <= y & y in F} by WAYBEL_0:14; hereby let a be set; assume a in uparrow F; then consider x being Element of R such that A3: a = x & ex y being Element of R st x >= y & y in F by A1; consider y being Element of R such that A4: x >= y & y in F by A3; uparrow y in {uparrow z where z is Element of R: z in F} & x in uparrow y by A4,WAYBEL_0:18; hence a in union {uparrow z where z is Element of R: z in F} by A3,TARSKI:def 4; end; hereby let a be set; assume a in union {uparrow x where x is Element of R: x in F}; then consider X being set such that A5: a in X & X in {uparrow x where x is Element of R: x in F} by TARSKI:def 4; consider x being Element of R such that A6: X = uparrow x & x in F by A5; reconsider y = a as Element of R by A5,A6; y >= x by A5,A6,WAYBEL_0:18; hence a in uparrow F by A1,A6; end; hereby let a be set; assume a in downarrow F; then consider x being Element of R such that A7: a = x & ex y being Element of R st x <= y & y in F by A2; consider y being Element of R such that A8: x <= y & y in F by A7; downarrow y in {downarrow z where z is Element of R: z in F} & x in downarrow y by A8,WAYBEL_0:17; hence a in union {downarrow z where z is Element of R: z in F} by A7,TARSKI:def 4; end; hereby let a be set; assume a in union {downarrow x where x is Element of R: x in F}; then consider X being set such that A9: a in X & X in {downarrow x where x is Element of R: x in F} by TARSKI:def 4; consider x being Element of R such that A10: X = downarrow x & x in F by A9; reconsider y = a as Element of R by A9,A10; y <= x by A9,A10,WAYBEL_0:17; hence a in downarrow F by A2,A10; end; end; theorem for R being non empty RelStr for A being Subset-Family of R, F being Subset of R st A = {(uparrow x)` where x is Element of R: x in F} holds Intersect A = (uparrow F)` proof let R be non empty RelStr; deffunc F(Element of R) = uparrow $1; let A be Subset-Family of R, F be Subset of R such that A1: A = {F(x)` where x is Element of R: x in F}; A2: COMPLEMENT A = {F(x) where x is Element of R: x in F} from FraenkelComplement2(A1); reconsider C = COMPLEMENT A as Subset-Family of R; COMPLEMENT C = A; hence Intersect A = (union C)` by YELLOW_8:15 .= (uparrow F)` by A2,Th4; end; Lm1: for tau being Subset-Family of {0}, r being Relation of {0} st tau = {{},{0}} & r = {[0,0]} holds TopRelStr (#{0},r,tau#) is trivial reflexive non empty discrete finite proof let tau be Subset-Family of {0}, r be Relation of {0} such that A1: tau = {{},{0}} and A2: r = {[0,0]}; set T = TopRelStr (#{0},r,tau#); thus T is trivial; thus T is reflexive proof let x be Element of T; x = 0 by TARSKI:def 1; then [x,x] in {[0,0]} by TARSKI:def 1; hence x <= x by A2,ORDERS_1:def 9; end; thus T is non empty; the topology of T = bool the carrier of T by A1,ZFMISC_1:30; hence T is discrete by TDLAT_3:def 1; thus the carrier of T is finite; end; definition cluster strict trivial reflexive non empty discrete finite TopRelStr; existence proof {{},{0}} = bool {0} & bool {0} c= bool {0} by ZFMISC_1:30; then reconsider tau = {{},{0}} as Subset of bool {0}; reconsider tau as Subset-Family of {0} by SETFAM_1:def 7; 0 in {0} by TARSKI:def 1; then reconsider r = {[0,0]} as Relation of {0} by RELSET_1:8; take TopRelStr (#{0},r,tau#); thus thesis by Lm1; end; end; definition cluster strict complete trivial TopLattice; existence proof consider T being strict trivial reflexive non empty discrete finite TopRelStr; take T; thus thesis; end; end; definition let S be non empty RelStr, T be upper-bounded non empty reflexive antisymmetric RelStr; cluster infs-preserving map of S,T; existence proof take f = S --> Top T; let A be Subset of S; assume ex_inf_of A,S; A1: f = (the carrier of S) --> Top T by BORSUK_1:def 3; then rng f = {Top T} & f.:A c= rng f by FUNCOP_1:14,RELAT_1:144; then A2: f.:A = {} or f.:A = {Top T} by ZFMISC_1:39; hence ex_inf_of f.:A,T by YELLOW_0:38,43; thus inf (f.:A) = Top T by A2,YELLOW_0:39,def 12 .= f.inf A by A1,FUNCOP_1:13; end; end; definition let S be non empty RelStr, T be lower-bounded non empty reflexive antisymmetric RelStr; cluster sups-preserving map of S,T; existence proof take f = S --> Bottom T; let A be Subset of S; assume ex_sup_of A,S; A1: f = (the carrier of S) --> Bottom T by BORSUK_1:def 3; then rng f = {Bottom T} & f.:A c= rng f by FUNCOP_1:14,RELAT_1:144; then A2: f.:A = {} or f.:A = {Bottom T} by ZFMISC_1:39; hence ex_sup_of f.:A,T by YELLOW_0:38,42; thus sup (f.:A) = Bottom T by A2,YELLOW_0:39,def 11 .= f.sup A by A1,FUNCOP_1:13; end; end; definition let R,S be 1-sorted; assume A1: the carrier of S c= the carrier of R; dom id the carrier of S = the carrier of S & rng id the carrier of S = the carrier of S by RELAT_1:71; then A2: id the carrier of S is Function of the carrier of S, the carrier of R by A1,FUNCT_2:4; func incl(S,R) -> map of S,R equals: Def1: id the carrier of S; coherence by A2; end; definition let R be non empty RelStr; let S be non empty SubRelStr of R; cluster incl(S,R) -> monotone; coherence proof let x,y be Element of S; reconsider a = x, b = y as Element of R by YELLOW_0:59; the carrier of S c= the carrier of R by YELLOW_0:def 13; then incl(S,R) = id the carrier of S by Def1; then incl(S,R).x = a & incl(S,R).y = b by FUNCT_1:35; hence thesis by YELLOW_0:60; end; end; definition let R be non empty RelStr, X be non empty Subset of R; func X+id -> strict non empty NetStr over R equals: Def2: (incl(subrelstr X, R))*((subrelstr X)+id); coherence; func X opp+id -> strict non empty NetStr over R equals: Def3: (incl(subrelstr X, R))*((subrelstr X)opp+id); coherence; end; theorem for R being non empty RelStr, X being non empty Subset of R holds the carrier of X+id = X & X+id is full SubRelStr of R & for x being Element of X+id holds X+id.x = x proof let R be non empty RelStr, X be non empty Subset of R; X+id = (incl(subrelstr X, R))*((subrelstr X)+id) by Def2; then A1: the RelStr of X+id = the RelStr of (subrelstr X)+id & the mapping of X+id = incl(subrelstr X, R)*the mapping of (subrelstr X)+id by WAYBEL_9:def 8; A2: the carrier of subrelstr X = X by YELLOW_0:def 15; hence A3: the carrier of X+id = X by A1,WAYBEL_9:def 5; A4: the RelStr of (subrelstr X)+id = subrelstr X by WAYBEL_9:def 5; the InternalRel of subrelstr X c= the InternalRel of R by YELLOW_0:def 13; then reconsider S = X+id as SubRelStr of R by A1,A2,A4,YELLOW_0:def 13; S is full proof thus the InternalRel of S = (the InternalRel of R)|_2 the carrier of S by A1,A4,YELLOW_0:def 14; end; hence X+id is full SubRelStr of R; let x be Element of X+id; id subrelstr X = id X by A2,GRCAT_1:def 11; then the mapping of (subrelstr X)+id = id X & dom id X = X & incl(subrelstr X, R) = id X by A2,Def1,RELAT_1:71,WAYBEL_9:def 5; then the mapping of X+id = id X by A1,FUNCT_1:42; hence X+id.x = (id X).x by WAYBEL_0:def 8 .= x by A3,FUNCT_1:35; end; theorem for R being non empty RelStr, X being non empty Subset of R holds the carrier of X opp+id = X & X opp+id is full SubRelStr of R opp & for x being Element of X opp+id holds X opp+id.x = x proof let R be non empty RelStr, X be non empty Subset of R; X opp+id = (incl(subrelstr X, R))*((subrelstr X)opp+id) by Def3; then A1: the RelStr of X opp+id = the RelStr of (subrelstr X)opp+id & the mapping of X opp+id = incl(subrelstr X, R)*the mapping of (subrelstr X) opp+id by WAYBEL_9:def 8; A2: the carrier of subrelstr X = X by YELLOW_0:def 15; A3: the carrier of (subrelstr X) opp+id = the carrier of subrelstr X & the RelStr of (subrelstr X) opp+id = (subrelstr X) opp & the InternalRel of (subrelstr X) opp+id = (the InternalRel of subrelstr X)~ by WAYBEL_9:11,def 6; thus the carrier of X opp+id = X by A1,A2,WAYBEL_9:def 6; A4: R opp = RelStr(#the carrier of R, (the InternalRel of R)~#) by LATTICE3:def 5; the InternalRel of subrelstr X = (the InternalRel of R)|_2 X by A2,YELLOW_0:def 14; then A5: the InternalRel of (subrelstr X)opp+id = (the InternalRel of R)~|_2 X by A3,ORDERS_2:95; then the InternalRel of (subrelstr X)opp+id c= the InternalRel of R opp by A4,WELLORD1:15; then reconsider S = X opp+id as SubRelStr of R opp by A1,A2,A3,A4,YELLOW_0: def 13; S is full proof thus the InternalRel of S = (the InternalRel of R opp)|_2 the carrier of S by A1,A3,A4,A5, YELLOW_0:def 15; end; hence X opp+id is full SubRelStr of R opp; let x be Element of X opp+id; id subrelstr X = id X by A2,GRCAT_1:def 11; then the mapping of (subrelstr X)opp+id = id X & dom id X = X & incl(subrelstr X, R) = id X by A2,Def1,RELAT_1:71,WAYBEL_9:def 6; then the mapping of X opp+id = id X by A1,FUNCT_1:42; hence X opp+id.x = (id X).x by WAYBEL_0:def 8 .= x by A1,A2,A3,FUNCT_1:35; end; definition let R be non empty reflexive RelStr; let X be non empty Subset of R; cluster X +id -> reflexive; coherence proof X+id = (incl(subrelstr X, R))*((subrelstr X)+id) by Def2; hence thesis; end; cluster X opp+id -> reflexive; coherence proof X opp+id = (incl(subrelstr X, R))*((subrelstr X) opp+id) by Def3; hence thesis; end; end; definition let R be non empty transitive RelStr; let X be non empty Subset of R; cluster X +id -> transitive; coherence proof X+id = (incl(subrelstr X, R))*((subrelstr X)+id) by Def2; hence thesis; end; cluster X opp+id -> transitive; coherence proof X opp+id = (incl(subrelstr X, R))*((subrelstr X) opp+id) by Def3; hence thesis; end; end; definition let R be non empty reflexive RelStr; let I be directed Subset of R; cluster subrelstr I -> directed; coherence proof let x,y be Element of subrelstr I; A1: [#]subrelstr I = the carrier of subrelstr I by PRE_TOPC:12; A2: the carrier of subrelstr I = I by YELLOW_0:def 15; assume A3: x in [#]subrelstr I & y in [#]subrelstr I; then reconsider a = x, b = y as Element of R by A1,A2; consider c being Element of R such that A4: c in I & a <= c & b <= c by A2,A3,WAYBEL_0:def 1; reconsider z = c as Element of subrelstr I by A2,A4; take z; thus thesis by A1,A3,A4,YELLOW_0:61; end; end; definition let R be non empty reflexive RelStr; let I be directed non empty Subset of R; cluster I+id -> directed; coherence proof I+id = (incl(subrelstr I, R))*((subrelstr I)+id) by Def2; hence thesis; end; end; definition let R be non empty reflexive RelStr; let F be filtered non empty Subset of R; cluster (subrelstr F) opp+id -> directed; coherence proof set I = F, A = (subrelstr I)opp+id; let x,y be Element of (subrelstr I)opp+id; A1: [#]A = the carrier of A by PRE_TOPC:12; A2: the carrier of subrelstr I = I by YELLOW_0:def 15; A3: the carrier of A = the carrier of subrelstr F by WAYBEL_9:def 6; assume x in [#]A & y in [#]A; then reconsider a = x, b = y as Element of R by A1,A2,A3; A4: the RelStr of A = the RelStr of (subrelstr F) opp by WAYBEL_9:11; consider c being Element of R such that A5: c in I & a >= c & b >= c by A2,A3,WAYBEL_0:def 2; reconsider a1 = x, b1 = y, c1 = c as Element of subrelstr F by A2,A3,A5; reconsider z = c as Element of A by A2,A3,A5; take z; A6: a1~ = a1 & b1~ = b1 & c1~ = c1 by LATTICE3:def 6; a1 >= c1 & b1 >= c1 by A5,YELLOW_0:61; then a1~ <= c1~ & b1~ <= c1~ by LATTICE3:9; hence thesis by A1,A4,A6,YELLOW_0:1; end; end; definition let R be non empty reflexive RelStr; let F be filtered non empty Subset of R; cluster F opp+id -> directed; coherence proof F opp+id = (incl(subrelstr F, R))*((subrelstr F) opp+id) by Def3; hence thesis; end; end; begin :: Operations on families of open sets theorem Th8: for T being TopSpace st T is empty holds the topology of T = {{}} proof let T1 be TopSpace; assume T1 is empty; then the carrier of T1 = {} by STRUCT_0:def 1; then the topology of T1 c= {{}} & {} in the topology of T1 by PRE_TOPC:def 1,ZFMISC_1:1; hence thesis by ZFMISC_1:39; end; theorem Th9: for T being trivial non empty TopSpace holds the topology of T = bool the carrier of T & for x being Point of T holds the carrier of T = {x} & the topology of T = {{},{x}} proof let T be trivial non empty TopSpace; thus the topology of T c= bool the carrier of T; consider x being Point of T such that A1: the carrier of T = {x} by TEX_1:def 1; {} in the topology of T & the carrier of T in the topology of T by PRE_TOPC:5,def 1; then A2: {{},the carrier of T} c= the topology of T & bool {x} = {{},{x}} by ZFMISC_1:30,38; hence bool the carrier of T c= the topology of T by A1; let a be Point of T; a = x by REALSET1:def 20; hence the carrier of T = {a} & the topology of T c= {{},{a}} & {{},{a}} c= the topology of T by A1,A2; end; theorem for T being trivial non empty TopSpace holds {the carrier of T} is Basis of T & {} is prebasis of T & {{}} is prebasis of T proof let T be trivial non empty TopSpace; set BB = {the carrier of T}; the carrier of T c= the carrier of T; then A1: the topology of T = bool the carrier of T & the carrier of T in bool the carrier of T by Th9; then BB c= the topology of T by ZFMISC_1:37; then BB is Subset-Family of T by A1,SETFAM_1:def 7; then reconsider BB as Subset-Family of T; consider x being Element of T; A2: the topology of T = {{}, {x}} & the carrier of T = {x} by Th9; A3: {} c= bool the carrier of T & {} c= BB & {} c= the carrier of T by XBOOLE_1:2; then {} is Subset-Family of T by SETFAM_1:def 7; then reconsider C = {} as Subset-Family of T; the topology of T c= UniCl BB proof let a be set; assume a in the topology of T; then A4: a = {x} & union {{x}} = {x} & BB c= BB & BB c= bool the carrier of T or a = {} by A2,TARSKI:def 2,ZFMISC_1:31; {{x}} is Subset of bool the carrier of T by ZFMISC_1:37; then {{x}} is Subset-Family of T by SETFAM_1:def 7; then {{x}} is Subset-Family of T & C is Subset-Family of T; hence thesis by A2,A3,A4,CANTOR_1:def 1,ZFMISC_1:2; end; hence A5: {the carrier of T} is Basis of T by A1,CANTOR_1:def 2; A6: {} c= the topology of T & C c= C & C is finite by XBOOLE_1:2; BB c= FinMeetCl C proof let x be set; assume x in BB; then x = the carrier of T by TARSKI:def 1; then Intersect C = x by CANTOR_1:def 3; hence thesis by CANTOR_1:def 4; end; hence {} is prebasis of T by A5,A6,CANTOR_1:def 5; {} c= the carrier of T by XBOOLE_1:2; then {{}} is Subset of bool the carrier of T by ZFMISC_1:37; then {{}} is Subset-Family of T by SETFAM_1:def 7; then reconsider D = {{}} as Subset-Family of T; A7: D c= the topology of T by A2,ZFMISC_1:12; BB c= FinMeetCl D proof let x be set; assume x in BB; then x = the carrier of T & Intersect C = the carrier of T & C c= D by CANTOR_1:def 3,TARSKI:def 1,XBOOLE_1:2; hence thesis by CANTOR_1:def 4; end; hence {{}} is prebasis of T by A5,A7,CANTOR_1:def 5; end; theorem Th11: for X,Y being set, A being Subset-Family of X st A = {Y} holds FinMeetCl A = {Y,X} & UniCl A = {Y,{}} proof let X,Z be set, A be Subset-Family of X such that A1: A = {Z}; thus FinMeetCl A c= {Z,X} proof let x be set; assume x in FinMeetCl A; then consider Y being Subset-Family of X such that A2: Y c= A & Y is finite & x = Intersect Y by CANTOR_1:def 4; Y = {} or Y = {Z} by A1,A2,ZFMISC_1:39; then x = X or x = meet {Z} by A2,CANTOR_1:def 3; then x = X or x = Z by SETFAM_1:11; hence thesis by TARSKI:def 2; end; reconsider E = {} as Subset of bool X by XBOOLE_1:2; reconsider E as Subset-Family of X by SETFAM_1:def 7; hereby let x be set; assume x in {Z,X}; then x = X or x = Z by TARSKI:def 2; then x = X or x = meet {Z} by SETFAM_1:11; then x = Intersect E & E c= A or x = Intersect A & A c= A by A1,CANTOR_1:def 3,XBOOLE_1:2; hence x in FinMeetCl A by A1,CANTOR_1:def 4; end; hereby let x be set; assume x in UniCl A; then consider Y being Subset-Family of X such that A3: Y c= A & x = union Y by CANTOR_1:def 1; Y = {} or Y = {Z} by A1,A3,ZFMISC_1:39; then x = {} or x = Z by A3,ZFMISC_1:2,31; hence x in {Z,{}} by TARSKI:def 2; end; let x be set; assume x in {Z,{}}; then x = {} or x = Z by TARSKI:def 2; then x = union E & E c= A or x = union A & A c= A by A1,XBOOLE_1:2,ZFMISC_1 :2,31 ; hence thesis by CANTOR_1:def 1; end; theorem for X being set, A,B being Subset-Family of X st A = B \/ {X} or B = A \ {X} holds Intersect A = Intersect B proof let X be set, A,B be Subset-Family of X; assume A1: A = B \/ {X} or B = A \ {X}; hereby let x be set; assume A2: x in Intersect A; now let y be set; assume y in B; then y in A by A1,XBOOLE_0:def 2,def 4; hence x in y by A2,CANTOR_1:10; end; hence x in Intersect B by A2,CANTOR_1:10; end; let x be set; assume A3: x in Intersect B; now let y be set; assume y in A; then y in B & not y in {X} or y in {X} by A1,XBOOLE_0:def 2,def 4; then y in B or y = X by TARSKI:def 1; hence x in y by A3,CANTOR_1:10; end; hence thesis by A3,CANTOR_1:10; end; theorem for X being set, A,B being Subset-Family of X st A = B \/ {X} or B = A \ {X} holds FinMeetCl A = FinMeetCl B proof let X be set, A,B be Subset-Family of X; assume A1: A = B \/ {X} or B = A \ {X}; X in FinMeetCl B by CANTOR_1:8; then {X} c= FinMeetCl B & B c= FinMeetCl B & (A \ {X}) \/ {X} = A \/ {X} by CANTOR_1:4,XBOOLE_1:39,ZFMISC_1:37; then A c= B \/ {X} & B \/ {X} c= FinMeetCl B by A1,XBOOLE_1:7,8 ; then A c= FinMeetCl B by XBOOLE_1:1; then FinMeetCl A c= FinMeetCl FinMeetCl B by CANTOR_1:16; hence FinMeetCl A c= FinMeetCl B by CANTOR_1:13; B c= A by A1,XBOOLE_1:7,36; hence thesis by CANTOR_1:16; end; theorem Th14: for X being set, A being Subset-Family of X st X in A for x being set holds x in FinMeetCl A iff ex Y being finite non empty Subset-Family of X st Y c= A & x = Intersect Y proof let X be set, A be Subset-Family of X; assume A1: X in A; then A2: {X} c= A by ZFMISC_1:37; reconsider Z = {X} as finite non empty Subset of bool X by A1,ZFMISC_1:37; reconsider Z as finite non empty Subset-Family of X by SETFAM_1:def 7; A3: Intersect Z = meet Z by CANTOR_1:def 3 .= X by SETFAM_1:11; let x be set; hereby assume x in FinMeetCl A; then consider Y being Subset-Family of X such that A4: Y c= A & Y is finite & x = Intersect Y by CANTOR_1:def 4; per cases; suppose Y = {}; then x = X by A4,CANTOR_1:def 3; hence ex Y being finite non empty Subset-Family of X st Y c= A & x = Intersect Y by A2,A3; suppose Y <> {}; hence ex Y being finite non empty Subset-Family of X st Y c= A & x = Intersect Y by A4; end; thus thesis by CANTOR_1:def 4; end; theorem Th15: for X being set, A being Subset-Family of X holds UniCl UniCl A = UniCl A proof let X be set, A be Subset-Family of X; hereby let x be set; assume x in UniCl UniCl A; then consider Y being Subset-Family of X such that A1: Y c= UniCl A & x = union Y by CANTOR_1:def 1; set Z = {y where y is Subset of X: y in A & y c= x}; Z c= bool X proof let a be set; assume a in Z; then ex y being Subset of X st a = y & y in A & y c= x; hence thesis; end; then reconsider Z as Subset-Family of X by SETFAM_1:def 7; A2: x = union Z proof hereby let a be set; assume a in x; then consider s being set such that A3: a in s & s in Y by A1,TARSKI:def 4; consider t being Subset-Family of X such that A4: t c= A & s = union t by A1,A3,CANTOR_1:def 1; consider u being set such that A5: a in u & u in t by A3,A4,TARSKI:def 4; reconsider u as Subset of X by A5; u c= s & s c= x by A1,A3,A4,A5,ZFMISC_1:92; then u c= x by XBOOLE_1:1; then u in Z by A4,A5; hence a in union Z by A5,TARSKI:def 4; end; let a be set; assume a in union Z; then consider u being set such that A6: a in u & u in Z by TARSKI:def 4; ex y being Subset of X st u = y & y in A & y c= x by A6; hence thesis by A6; end; Z c= A proof let a be set; assume a in Z; then ex y being Subset of X st a = y & y in A & y c= x; hence thesis; end; hence x in UniCl A by A2,CANTOR_1:def 1; end; thus thesis by CANTOR_1:1; end; theorem Th16: for X being set, A being empty Subset-Family of X holds UniCl A = {{}} proof let X be set, A be empty Subset-Family of X; hereby let x be set; assume x in UniCl A; then consider B being Subset-Family of X such that A1: B c= A & x = union B by CANTOR_1:def 1; B = {} by A1,XBOOLE_1:3; hence x in {{}} by A1,TARSKI:def 1,ZFMISC_1:2; end; consider B being empty Subset-Family of X; let x be set; assume x in {{}}; then x = {} & B c= A & union B = {} by TARSKI:def 1,ZFMISC_1:2; hence thesis by CANTOR_1:def 1; end; theorem Th17: for X being set, A being empty Subset-Family of X holds FinMeetCl A = {X} proof let X be set, A be empty Subset-Family of X; hereby let x be set; assume x in FinMeetCl A; then consider B being Subset-Family of X such that A1: B c= A & B is finite & x = Intersect B by CANTOR_1:def 4; B = {} by A1,XBOOLE_1:3; then x = X by A1,CANTOR_1:def 3; hence x in {X} by TARSKI:def 1; end; consider B being empty Subset-Family of X; let x be set; assume x in {X}; then x = X & B c= A & Intersect B = X by CANTOR_1:def 3,TARSKI:def 1; hence thesis by CANTOR_1:def 4; end; theorem Th18: for X being set, A being Subset-Family of X st A = {{},X} holds UniCl A = A & FinMeetCl A = A proof let X be set, A be Subset-Family of X such that A1: A = {{},X}; hereby let a be set; assume a in UniCl A; then consider y being Subset-Family of X such that A2: y c= A & a = union y by CANTOR_1:def 1; y = {} or y = {{}} or y = {X} or y = {{},X} by A1,A2,ZFMISC_1:42; then a = {} or a = X or a = {} \/ X & {} \/ X = X by A2,ZFMISC_1:2,31,93; hence a in A by A1,TARSKI:def 2; end; thus A c= UniCl A by CANTOR_1:1; hereby let a be set; assume a in FinMeetCl A; then consider y being Subset-Family of X such that A3: y c= A & y is finite & a = Intersect y by CANTOR_1:def 4; y = {} or y = {{}} or y = {X} or y = {{},X} by A1,A3,ZFMISC_1:42; then a = X or a = meet {{}} or a = meet {X} or a = meet {{},X} by A3,CANTOR_1:def 3; then a = X or a = {} or a = {} /\ X & {} /\ X = {} by SETFAM_1:11,12; hence a in A by A1,TARSKI:def 2; end; thus A c= FinMeetCl A by CANTOR_1:4; end; theorem Th19: for X,Y being set, A being Subset-Family of X, B being Subset-Family of Y holds (A c= B implies UniCl A c= UniCl B) & (A = B implies UniCl A = UniCl B) proof let X,Y be set, A be Subset-Family of X, B be Subset-Family of Y; A1: now let X,Y be set; let A be Subset-Family of X, B be Subset-Family of Y such that A2: A c= B; thus UniCl A c= UniCl B proof let x be set; assume x in UniCl A; then consider y being Subset-Family of X such that A3: y c= A & x = union y by CANTOR_1:def 1; A4: y c= B by A2,A3,XBOOLE_1:1; then y is Subset of bool Y by XBOOLE_1:1; then y is Subset-Family of Y by SETFAM_1:def 7; then ex y being Subset-Family of Y st y c= B & x = union y by A3,A4; hence thesis by CANTOR_1:def 1; end; end; hence A c= B implies UniCl A c= UniCl B; assume A = B; hence UniCl A c= UniCl B & UniCl B c= UniCl A by A1; end; theorem Th20: for X,Y being set, A being Subset-Family of X, B being Subset-Family of Y st A = B & X in A & X c= Y holds FinMeetCl B = {Y} \/ FinMeetCl A proof let X,Y be set, A be Subset-Family of X, B be Subset-Family of Y such that A1: A = B & X in A & X c= Y; thus FinMeetCl B c= {Y} \/ FinMeetCl A proof let x be set; assume x in FinMeetCl B; then consider y being Subset-Family of Y such that A2: y c= B & y is finite & x = Intersect y by CANTOR_1:def 4; reconsider z = y as Subset of bool X by A1,A2,XBOOLE_1:1; reconsider z as Subset-Family of X by SETFAM_1:def 7; per cases; suppose y = {}; then x = Y by A2,CANTOR_1:def 3; then x in {Y} by TARSKI:def 1; hence thesis by XBOOLE_0:def 2; suppose y <> {}; then Intersect y = meet y & Intersect z = meet y by CANTOR_1:def 3; then x in FinMeetCl A by A1,A2,CANTOR_1:def 4; hence thesis by XBOOLE_0:def 2; end; let x be set; assume A3: x in {Y} \/ FinMeetCl A; consider E being empty Subset-Family of Y; per cases by A3,XBOOLE_0:def 2; suppose x in {Y}; then x = Y & Intersect E = Y & E c= B by CANTOR_1:def 3,TARSKI:def 1, XBOOLE_1:2; hence thesis by CANTOR_1:def 4; suppose x in FinMeetCl A; then consider y being non empty finite Subset-Family of X such that A4: y c= A & x = Intersect y by A1,Th14; reconsider z = y as Subset of bool Y by A1,A4,XBOOLE_1:1; reconsider z as Subset-Family of Y by SETFAM_1:def 7; x = meet z by A4,CANTOR_1:def 3 .= Intersect z by CANTOR_1:def 3; hence thesis by A1,A4,CANTOR_1:def 4; end; theorem Th21: for X being set, A being Subset-Family of X holds UniCl FinMeetCl UniCl A = UniCl FinMeetCl A proof let X be set, A be Subset-Family of X; per cases; suppose A = {}; then FinMeetCl A = {X} & UniCl A = {{}} by Th16,Th17; then FinMeetCl UniCl A = {{},X} & UniCl FinMeetCl A = {X,{}} by Th11; hence thesis by Th18; suppose A <> {}; then reconsider A as non empty Subset-Family of X; A1: UniCl FinMeetCl UniCl A c= UniCl UniCl FinMeetCl A proof let x be set; assume x in UniCl FinMeetCl UniCl A; then consider Y being Subset-Family of X such that A2: Y c= FinMeetCl UniCl A & x = union Y by CANTOR_1:def 1; Y c= UniCl FinMeetCl A proof let y be set; assume y in Y; then consider Z being Subset-Family of X such that A3: Z c= UniCl A & Z is finite & y = Intersect Z by A2,CANTOR_1:def 4; per cases; suppose Z = {}; then y = X by A3,CANTOR_1:def 3; then y in FinMeetCl A & FinMeetCl A c= UniCl FinMeetCl A by CANTOR_1:1,8; hence thesis; suppose A4: Z <> {}; then A5: y = meet Z by A3,CANTOR_1:def 3; set G = {meet rng f where f is Element of Funcs(Z,A): for z being set st z in Z holds f.z c= z}; A6: G c= FinMeetCl A proof let a be set; assume a in G; then consider f being Element of Funcs(Z,A) such that A7: a = meet rng f & for z being set st z in Z holds f.z c= z; Funcs(Z,A) is FUNCTION_DOMAIN of Z,A by FRAENKEL:11; then f is Function of Z,A by FRAENKEL:def 2; then A8: dom f = Z & rng f c= A by FUNCT_2:def 1,RELSET_1:12; then reconsider B = rng f as Subset of bool X by XBOOLE_1:1; reconsider B as Subset-Family of X by SETFAM_1:def 7; B <> {} by A4,A8,RELAT_1:65; then Intersect B = a & B is finite by A3,A7,A8,CANTOR_1:def 3,FINSET_1:26; hence thesis by A8,CANTOR_1:def 4; end; then reconsider G as Subset of bool X by XBOOLE_1:1; reconsider G as Subset-Family of X by SETFAM_1:def 7; union G = y proof hereby let a be set; assume a in union G; then consider b being set such that A9: a in b & b in G by TARSKI:def 4; consider f being Element of Funcs(Z,A) such that A10: b = meet rng f & for z being set st z in Z holds f.z c= z by A9; Funcs(Z,A) is FUNCTION_DOMAIN of Z,A by FRAENKEL:11; then f is Function of Z,A by FRAENKEL:def 2; then A11: dom f = Z & rng f c= A by FUNCT_2:def 1,RELSET_1:12; then reconsider B = rng f as Subset of bool X by XBOOLE_1:1; reconsider B as Subset-Family of X by SETFAM_1:def 7; b c= y proof let c be set; assume A12: c in b; now let d be set; assume A13: d in Z; then f.d in B by A11,FUNCT_1:def 5; then b c= f.d by A10,SETFAM_1:4; then f.d c= d & c in f.d by A10,A12,A13; hence c in d; end; hence c in y by A4,A5,SETFAM_1:def 1; end; hence a in y by A9; end; let a be set; assume A14: a in y; defpred P[set,set] means a in $2 & $2 c= $1; A15: now let z be set; assume A16: z in Z; then A17: a in z & z in UniCl A by A3,A5,A14,SETFAM_1:def 1; consider C being Subset-Family of X such that A18: C c= A & z = union C by A3,A16,CANTOR_1:def 1; consider w being set such that A19: a in w & w in C by A17,A18,TARSKI:def 4; take w; thus w in A by A18,A19; thus P[z, w] by A18,A19,ZFMISC_1:92; end; consider f being Function such that A20: dom f = Z & rng f c= A and A21: for z being set st z in Z holds P[z, f.z] from NonUniqBoundFuncEx(A15); reconsider f as Element of Funcs(Z,A) by A20,FUNCT_2:def 2; for z being set st z in Z holds f.z c= z by A21; then A22: meet rng f in G; now thus rng f <> {} by A4,A20,RELAT_1:65; let y be set; assume y in rng f; then ex z being set st z in Z & y = f.z by A20,FUNCT_1:def 5; hence a in y by A21; end; then a in meet rng f by SETFAM_1:def 1; hence thesis by A22,TARSKI:def 4; end; hence thesis by A6,CANTOR_1:def 1; end; hence thesis by A2,CANTOR_1:def 1; end; A c= UniCl A by CANTOR_1:1; then FinMeetCl A c= FinMeetCl UniCl A by CANTOR_1:16; then UniCl FinMeetCl A c= UniCl FinMeetCl UniCl A & UniCl UniCl FinMeetCl A = UniCl FinMeetCl A by Th15,CANTOR_1:9; hence thesis by A1,XBOOLE_0:def 10; end; begin :: Bases theorem Th22: for T being TopSpace, K being Subset-Family of T holds the topology of T = UniCl K iff K is Basis of T proof let T be TopSpace, K be Subset-Family of T; thus the topology of T = UniCl K implies K is Basis of T proof assume the topology of T = UniCl K; hence K c= the topology of T & the topology of T c= UniCl K by CANTOR_1:1; end; assume K c= the topology of T; then A1: UniCl K c= UniCl the topology of T by CANTOR_1:9; assume the topology of T c= UniCl K; hence the topology of T c= UniCl K & UniCl K c= the topology of T by A1,CANTOR_1:6; end; theorem Th23: for T being TopSpace, K being Subset-Family of T holds K is prebasis of T iff FinMeetCl K is Basis of T proof let T be TopSpace, BB be Subset-Family of T; A1: now assume T is empty; then the topology of T = {{}} & the carrier of T = {} by Th8,STRUCT_0:def 1; hence the topology of T = bool the carrier of T by ZFMISC_1:1; end; thus BB is prebasis of T implies FinMeetCl BB is Basis of T proof assume A2: BB is prebasis of T; then BB c= the topology of T by CANTOR_1:def 5; then FinMeetCl BB c= FinMeetCl the topology of T by CANTOR_1:16; then A3: FinMeetCl BB c= the topology of T by A1,CANTOR_1:5; consider A being Basis of T such that A4: A c= FinMeetCl BB by A2,CANTOR_1:def 5; the topology of T c= UniCl A & UniCl A c= UniCl FinMeetCl BB by A4,CANTOR_1:9,def 2; then the topology of T c= UniCl FinMeetCl BB by XBOOLE_1:1; hence FinMeetCl BB is Basis of T by A3,CANTOR_1:def 2; end; assume A5: FinMeetCl BB is Basis of T; then BB c= FinMeetCl BB & FinMeetCl BB c= the topology of T by CANTOR_1:4,def 2; then BB c= the topology of T by XBOOLE_1:1; hence BB is prebasis of T by A5,CANTOR_1:def 5; end; theorem Th24: for T being non empty TopSpace, B being Subset-Family of T st UniCl B is prebasis of T holds B is prebasis of T proof let T be non empty TopSpace, B be Subset-Family of T; assume UniCl B is prebasis of T; then FinMeetCl UniCl B is Basis of T by Th23; then UniCl FinMeetCl UniCl B = the topology of T by Th22; then UniCl FinMeetCl B = the topology of T by Th21; then FinMeetCl B is Basis of T by Th22; hence thesis by Th23; end; theorem Th25: for T1, T2 being TopSpace, B being Basis of T1 st the carrier of T1 = the carrier of T2 & B is Basis of T2 holds the topology of T1 = the topology of T2 proof let T1, T2 be TopSpace, B be Basis of T1; assume A1: the carrier of T1 = the carrier of T2 & B is Basis of T2; then reconsider C = B as Basis of T2; thus the topology of T1 = UniCl C by A1,Th22 .= the topology of T2 by Th22; end; theorem Th26: for T1, T2 being TopSpace, P being prebasis of T1 st the carrier of T1 = the carrier of T2 & P is prebasis of T2 holds the topology of T1 = the topology of T2 proof let T1, T2 be TopSpace, P be prebasis of T1; assume A1: the carrier of T1 = the carrier of T2 & P is prebasis of T2; then reconsider C = P as prebasis of T2; FinMeetCl P is Basis of T1 & FinMeetCl C is Basis of T2 by Th23; hence thesis by A1,Th25; end; theorem for T being TopSpace, K being Basis of T holds K is open & K is prebasis of T proof let T be TopSpace, K be Basis of T; K c= the topology of T by CANTOR_1:def 2; hence for P be Subset of T st P in K holds P is open by PRE_TOPC:def 5; thus K c= the topology of T by CANTOR_1:def 2; take K; thus K c= FinMeetCl K by CANTOR_1:4; end; theorem for T being TopSpace, K being prebasis of T holds K is open proof let T be TopSpace, K be prebasis of T; let P be Subset of T; A1: K c= the topology of T by CANTOR_1:def 5; assume P in K; hence P in the topology of T by A1; end; theorem Th29: for T being non empty TopSpace, B being prebasis of T holds B \/ {the carrier of T} is prebasis of T proof let T be non empty TopSpace, B be prebasis of T; set C = B \/ {the carrier of T}; the carrier of T in the topology of T by PRE_TOPC:def 1; then A1: B c= the topology of T & {the carrier of T} c= the topology of T by CANTOR_1:def 5,ZFMISC_1:37; then C c= the topology of T by XBOOLE_1:8; then C is Subset of bool the carrier of T by XBOOLE_1:1; then C is Subset-Family of T by SETFAM_1:def 7; then reconsider C as Subset-Family of T; C is prebasis of T proof thus C c= the topology of T by A1,XBOOLE_1:8; B c= C by XBOOLE_1:7; then FinMeetCl B c= FinMeetCl C & FinMeetCl B is Basis of T by Th23,CANTOR_1:16; hence thesis; end; hence thesis; end; theorem for T being TopSpace, B being Basis of T holds B \/ {the carrier of T} is Basis of T proof let T be TopSpace, B be Basis of T; set C = B \/ {the carrier of T}; the carrier of T in the topology of T by PRE_TOPC:def 1; then A1: B c= the topology of T & {the carrier of T} c= the topology of T by CANTOR_1:def 2,ZFMISC_1:37; then C c= the topology of T by XBOOLE_1:8; then C is Subset of bool the carrier of T by XBOOLE_1:1; then C is Subset-Family of T by SETFAM_1:def 7; then reconsider C as Subset-Family of T; C is Basis of T proof thus C c= the topology of T by A1,XBOOLE_1:8; B c= C by XBOOLE_1:7; then UniCl B c= UniCl C & the topology of T c= UniCl B by CANTOR_1:9,def 2; hence thesis by XBOOLE_1:1; end; hence thesis; end; theorem Th31: for T being TopSpace, B being Basis of T for A being Subset of T holds A is open iff for p being Point of T st p in A ex a being Subset of T st a in B & p in a & a c= A proof let T be TopSpace, K be Basis of T, A be Subset of T; hereby assume A is open; then A1: A = union {G where G is Subset of T: G in K & G c= A} by YELLOW_8: 18; let p be Point of T; assume p in A; then consider Z being set such that A2: p in Z & Z in {G where G is Subset of T: G in K & G c= A} by A1,TARSKI:def 4; ex a being Subset of T st Z = a & a in K & a c= A by A2; hence ex a being Subset of T st a in K & p in a & a c= A by A2; end; assume A3: for p being Point of T st p in A ex a being Subset of T st a in K & p in a & a c= A; set F = {G where G is Subset of T: G in K & G c= A}; F c= bool the carrier of T proof let x be set; assume x in F; then ex G being Subset of T st x = G & G in K & G c= A; hence thesis; end; then reconsider F as Subset-Family of T by SETFAM_1:def 7; reconsider F as Subset-Family of T; A4: F is open proof let x be Subset of T; assume x in F; then ex G being Subset of T st x = G & G in K & G c= A; then x in K & K c= the topology of T by CANTOR_1:def 2; hence x in the topology of T; end; A = union F proof hereby let x be set; assume A5: x in A; then reconsider p = x as Point of T; consider a being Subset of T such that A6: a in K & p in a & a c= A by A3,A5; a in F by A6; hence x in union F by A6,TARSKI:def 4; end; F c= bool A proof let x be set; assume x in F; then ex G being Subset of T st x = G & G in K & G c= A; hence thesis; end; then union F c= union bool A by ZFMISC_1:95; hence thesis by ZFMISC_1:99; end; hence thesis by A4,TOPS_2:26; end; theorem Th32: for T being TopSpace, B being Subset-Family of T st B c= the topology of T & for A being Subset of T st A is open for p being Point of T st p in A ex a being Subset of T st a in B & p in a & a c= A holds B is Basis of T proof let T be TopSpace, B be Subset-Family of T such that A1: B c= the topology of T and A2: for A being Subset of T st A is open for p being Point of T st p in A ex a being Subset of T st a in B & p in a & a c= A; thus B c= the topology of T by A1; let x be set; assume A3: x in the topology of T; then reconsider A = x as Subset of T; set Y = {V where V is Subset of T: V in B & V c= A}; Y c= bool the carrier of T proof let y be set; assume y in Y; then ex V being Subset of T st y = V & V in B & V c= A; hence thesis; end; then reconsider Y as Subset-Family of T by SETFAM_1:def 7; reconsider Y as Subset-Family of T; A4: Y c= B proof let y be set; assume y in Y; then ex V being Subset of T st y = V & V in B & V c= A; hence thesis; end; x = union Y proof hereby let p be set; assume A5: p in x; then p in A; then reconsider q = p as Point of T; A is open by A3,PRE_TOPC:def 5; then consider a being Subset of T such that A6: a in B & q in a & a c= A by A2,A5; a in Y by A6; hence p in union Y by A6,TARSKI:def 4; end; let p be set; assume p in union Y; then consider a being set such that A7: p in a & a in Y by TARSKI:def 4; ex V being Subset of T st a = V & V in B & V c= A by A7; hence thesis by A7; end; hence thesis by A4,CANTOR_1:def 1; end; theorem Th33: for S being TopSpace, T being non empty TopSpace, K being Basis of T for f being map of S,T holds f is continuous iff for A being Subset of T st A in K holds f"A` is closed proof let S be TopSpace, T be non empty TopSpace, BB be Basis of T, f be map of S,T; A1: BB c= the topology of T by CANTOR_1:def 2; hereby assume A2: f is continuous; let A be Subset of T; assume A in BB; then A is open by A1,PRE_TOPC:def 5; then A` is closed by TOPS_1:30; hence f"A` is closed by A2,PRE_TOPC:def 12; end; assume A3:for A being Subset of T st A in BB holds f"A` is closed; let A be Subset of T; assume A is closed; then A` is open by TOPS_1:29; then A4: A` = union {G where G is Subset of T: G in BB & G c= A`} by YELLOW_8: 18; set F = {f"G where G is Subset of T: G in BB & G c= A`}; F c= bool the carrier of S proof let a be set; assume a in F; then ex G being Subset of T st a = f"G & G in BB & G c= A`; hence thesis; end; then reconsider F as Subset-Family of S by SETFAM_1:def 7; reconsider F as Subset-Family of S; F c= the topology of S proof let a be set; assume a in F; then consider G being Subset of T such that A5: a = f"G & G in BB & G c= A`; f"G` is closed & (f"G)` = f"G` by A3,A5,Th2; then f"G is open by TOPS_1:30; hence thesis by A5,PRE_TOPC:def 5; end; then A6: union F in the topology of S by PRE_TOPC:def 1; defpred P[Subset of T] means $1 in BB & $1 c= A`; deffunc F(Subset of T) = $1; f"union {F(G) where G is Subset of T: P[G]} = union {f"F(G) where G is Subset of T: P[G]} from ABC; then f"A` is open by A4,A6,PRE_TOPC:def 5; then (f"A)` is open by Th2; hence f"A is closed by TOPS_1:29; end; theorem for S being TopSpace,T being non empty TopSpace, K being Basis of T for f being map of S,T holds f is continuous iff for A being Subset of T st A in K holds f"A is open proof let S be TopSpace,T be non empty TopSpace, K be Basis of T; let f be map of S,T; hereby assume A1: f is continuous; let A be Subset of T; assume A in K; then f"A` is closed by A1,Th33; then (f"A)` is closed by Th2; hence f"A is open by TOPS_1:30; end; assume A2: for A being Subset of T st A in K holds f"A is open; now let A be Subset of T; assume A in K; then f"A is open by A2; then (f"A)` is closed by TOPS_1:30; hence f"A` is closed by Th2; end; hence thesis by Th33; end; theorem Th35: for S being TopSpace,T being non empty TopSpace, K being prebasis of T for f being map of S,T holds f is continuous iff for A being Subset of T st A in K holds f"A` is closed proof let S be TopSpace,T be non empty TopSpace, BB be prebasis of T, f be map of S,T; A1: BB c= the topology of T by CANTOR_1:def 5; hereby assume A2: f is continuous; let A be Subset of T; assume A in BB; then A is open by A1,PRE_TOPC:def 5; then A` is closed by TOPS_1:30; hence f"A` is closed by A2,PRE_TOPC:def 12; end; assume A3:for A being Subset of T st A in BB holds f"A` is closed; reconsider C = FinMeetCl BB as Basis of T by Th23; now let A be Subset of T; assume A in C; then consider Y being Subset-Family of T such that A4: Y c= BB & Y is finite & A = Intersect Y by CANTOR_1:def 4; reconsider Y as Subset-Family of T; reconsider CY = COMPLEMENT Y as Subset-Family of T; defpred P[set] means $1 in Y; deffunc F(Subset of T) = $1`; A5: f"A` = f"union CY by A4,YELLOW_8:16 .= f"union {F(a) where a is Subset of T: P[a]} by Th3 .= union {f"F(y) where y is Subset of T: P[y]} from ABC; set X = {f"y` where y is Subset of T: y in Y}; X c= bool the carrier of S proof let x be set; assume x in X; then ex y being Subset of T st x = f"y` & y in Y; hence thesis; end; then reconsider X as Subset-Family of S by SETFAM_1:def 7; reconsider X as Subset-Family of S; A6: X is closed proof let a be Subset of S; assume a in X; then ex y being Subset of T st a = f"y` & y in Y; hence a is closed by A3,A4; end; A7: Y is finite by A4; deffunc F(Subset of T) = f"$1`; A8: {F(y) where y is Subset of T: y in Y} is finite from FraenkelFin(A7); X is finite by A8; hence f"A` is closed by A5,A6,TOPS_2:28; end; hence thesis by Th33; end; theorem for S being TopSpace,T being non empty TopSpace, K being prebasis of T for f being map of S,T holds f is continuous iff for A being Subset of T st A in K holds f"A is open proof let S be TopSpace,T be non empty TopSpace, K be prebasis of T; let f be map of S,T; hereby assume A1: f is continuous; let A be Subset of T; assume A in K; then f"A` is closed by A1,Th35; then (f"A)` is closed by Th2; hence f"A is open by TOPS_1:30; end; assume A2: for A being Subset of T st A in K holds f"A is open; now let A be Subset of T; assume A in K; then f"A is open by A2; then (f"A)` is closed by TOPS_1:30; hence f"A` is closed by Th2; end; hence thesis by Th35; end; theorem Th37: for T being non empty TopSpace, x being Point of T, X being Subset of T for K being Basis of T st for A being Subset of T st A in K & x in A holds A meets X holds x in Cl X proof let T be non empty TopSpace, x be Point of T, X be Subset of T; let BB be Basis of T such that A1: for A being Subset of T st A in BB & x in A holds A meets X; now let G be a_neighborhood of x; A2: Int G = union {A where A is Subset of T: A in BB & A c= G} by YELLOW_8:20; x in Int G by CONNSP_2:def 1; then consider Z being set such that A3: x in Z & Z in {A where A is Subset of T: A in BB & A c= G} by A2,TARSKI:def 4; consider A being Subset of T such that A4: Z = A & A in BB & A c= G by A3; A meets X by A1,A3,A4; hence G meets X by A4,XBOOLE_1:63; end; hence x in Cl X by YELLOW_6:6; end; theorem Th38: for T being non empty TopSpace, x being Point of T, X being Subset of T for K being prebasis of T st for Z being finite Subset-Family of T st Z c= K & x in Intersect Z holds Intersect Z meets X holds x in Cl X proof let T be non empty TopSpace, x be Point of T, X be Subset of T; let BB be prebasis of T such that A1: for Z being finite Subset-Family of T st Z c= BB & x in Intersect Z holds Intersect Z meets X; reconsider BB' = FinMeetCl BB as Basis of T by Th23; now let A be Subset of T; assume A in BB'; then consider Y being Subset-Family of T such that A2: Y c= BB & Y is finite & A = Intersect Y by CANTOR_1:def 4; reconsider Y as finite Subset-Family of T by A2; assume x in A; then Intersect Y meets X by A1,A2; hence A meets X by A2; end; hence thesis by Th37; end; theorem for T being non empty TopSpace, K being prebasis of T, x being Point of T for N being net of T st for A being Subset of T st A in K & x in A holds N is_eventually_in A for S being Subset of T st rng netmap(N,T) c= S holds x in Cl S proof let T be non empty TopSpace, BB be prebasis of T, x be Point of T, N be net of T such that A1: for A being Subset of T st A in BB & x in A holds N is_eventually_in A; let S be Subset of T such that A2: rng netmap(N,T) c= S; A3: the carrier of N = [#]N & [#]N is directed by PRE_TOPC:12,WAYBEL_0:def 6; A4: the mapping of N = netmap(N,T) by WAYBEL_0:def 7; now let Z be finite Subset-Family of T; assume A5: Z c= BB & x in Intersect Z; defpred P[set,set] means for i,j being Element of N st $2 = i & i <= j holds N.j in $1; A6: now let a be set; assume A7: a in Z; then reconsider A = a as Subset of T; A in BB & x in A by A5,A7,CANTOR_1:10; then N is_eventually_in A by A1; then consider i being Element of N such that A8: for j being Element of N st i <= j holds N.j in A by WAYBEL_0:def 11; reconsider b = i as set; take b; thus b in the carrier of N & P[a, b] by A8; end; consider f being Function such that A9: dom f = Z & rng f c= the carrier of N & for a being set st a in Z holds P[a, f.a] from NonUniqBoundFuncEx(A6); consider k being Element of N; per cases by A9,RELAT_1:65; suppose Z = {}; then A10: Intersect Z = the carrier of T by CANTOR_1:def 3; N.k = netmap(N,T).k & the carrier of T <> {} by A4,WAYBEL_0:def 8; then N.k in rng netmap(N,T) by FUNCT_2:6; hence Intersect Z meets S by A2,A10,XBOOLE_0:3; suppose rng f <> {}; then reconsider Y = rng f as finite non empty Subset of N by A9,FINSET_1:26; consider i being Element of N such that A11: i in the carrier of N & i is_>=_than Y by A3,WAYBEL_0:1; now let y be set; assume A12: y in Z; then A13: f.y in Y by A9,FUNCT_1:def 5; then reconsider j = f.y as Element of N; j <= i by A11,A13,LATTICE3:def 9; hence N.i in y by A9,A12; end; then A14: N.i in Intersect Z by CANTOR_1:10; N.i = netmap(N,T).i & the carrier of T <> {} by A4,WAYBEL_0:def 8; then N.i in rng netmap(N,T) by FUNCT_2:6; hence Intersect Z meets S by A2,A14,XBOOLE_0:3; end; hence x in Cl S by Th38; end; begin :: Product topology theorem Th40: for T1,T2 being non empty TopSpace for B1 being Basis of T1, B2 being Basis of T2 holds {[:a,b:] where a is Subset of T1, b is Subset of T2: a in B1 & b in B2} is Basis of [:T1,T2:] proof let T1,T2 be non empty TopSpace; let B1 be Basis of T1, B2 be Basis of T2; set BB = {[:a,b:] where a is Subset of T1, b is Subset of T2: a in B1 & b in B2}; set T = [:T1,T2:]; A1: the carrier of T = [:the carrier of T1, the carrier of T2:] by BORSUK_1:def 5; BB c= bool the carrier of T proof let x be set; assume x in BB; then ex a being Subset of T1, b being Subset of T2 st x = [:a,b:] & a in B1 & b in B2; hence thesis; end; then reconsider BB as Subset-Family of T by SETFAM_1:def 7; reconsider BB as Subset-Family of T; A2: B1 c= the topology of T1 & B2 c= the topology of T2 by CANTOR_1:def 2; BB is Basis of T proof hereby let x be set; assume x in BB; then consider a being Subset of T1, b being Subset of T2 such that A3: x = [:a,b:] & a in B1 & b in B2; a is open & b is open by A2,A3,PRE_TOPC:def 5; then [:a,b:] is open by BORSUK_1:46; hence x in the topology of T by A3,PRE_TOPC:def 5; end; let x be set; assume A4: x in the topology of T; then reconsider X = x as Subset of T; X is open by A4,PRE_TOPC:def 5; then A5: X = union Base-Appr X by BORSUK_1:54; set Y = BB /\ Base-Appr X; A6: Y c= BB & Y c= Base-Appr X by XBOOLE_1:17; reconsider Y as Subset-Family of T; union Y = X proof thus union Y c= X by A5,A6,ZFMISC_1:95; let z be set; assume A7: z in X; then reconsider p = z as Point of T; consider z1,z2 being set such that A8: z1 in the carrier of T1 & z2 in the carrier of T2 & p = [z1,z2] by A1,ZFMISC_1:def 2; reconsider z1 as Point of T1 by A8; reconsider z2 as Point of T2 by A8; consider Z being set such that A9: z in Z & Z in Base-Appr X by A5,A7,TARSKI:def 4; A10: Base-Appr X = {[:a,b:] where a is Subset of T1, b is Subset of T2: [:a,b:] c= X & a is open & b is open} by BORSUK_1:def 6; then consider a being Subset of T1, b being Subset of T2 such that A11: Z = [:a,b:] & [:a,b:] c= X & a is open & b is open by A9; A12: z1 in a & z2 in b by A8,A9,A11,ZFMISC_1:106; then consider a' being Subset of T1 such that A13: a' in B1 & z1 in a' & a' c= a by A11,Th31; consider b' being Subset of T2 such that A14: b' in B2 & z2 in b' & b' c= b by A11,A12,Th31; [:a',b':] c= [:a,b:] by A13,A14,ZFMISC_1:119; then [:a',b':] c= X & a' is open & b' is open by A2,A11,A13,A14,PRE_TOPC:def 5,XBOOLE_1:1; then [:a',b':] in Base-Appr X & [:a',b':] in BB by A10,A13,A14; then p in [:a',b':] & [:a',b':] in Y by A8,A13,A14,XBOOLE_0:def 3,ZFMISC_1:106; hence thesis by TARSKI:def 4; end; hence x in UniCl BB by A6,CANTOR_1:def 1; end; hence thesis; end; theorem Th41: for T1,T2 being non empty TopSpace for B1 being prebasis of T1, B2 being prebasis of T2 holds {[:the carrier of T1, b:] where b is Subset of T2: b in B2} \/ {[:a, the carrier of T2:] where a is Subset of T1: a in B1} is prebasis of [:T1,T2:] proof let T1,T2 be non empty TopSpace; set T = [:T1,T2:]; let B1 be prebasis of T1, B2 be prebasis of T2; set C2 = {[:the carrier of T1, b:] where b is Subset of T2: b in B2}; set C1 = {[:a, the carrier of T2:] where a is Subset of T1: a in B1}; reconsider D1 = FinMeetCl B1 as Basis of T1 by Th23; reconsider D2 = FinMeetCl B2 as Basis of T2 by Th23; reconsider D = {[:a,b:] where a is Subset of T1, b is Subset of T2: a in D1 & b in D2} as Basis of T by Th40; the carrier of T1 c= the carrier of T1; then reconsider cT1 = the carrier of T1 as Subset of T1; the carrier of T2 c= the carrier of T2; then reconsider cT2 = the carrier of T2 as Subset of T2; A1: cT1 in the topology of T1 & cT2 in the topology of T2 by PRE_TOPC:def 1; A2: B1 c= the topology of T1 & B2 c= the topology of T2 by CANTOR_1:def 5; C1 c= bool the carrier of T proof let x be set; assume x in C1; then ex a being Subset of T1 st x = [:a, cT2:] & a in B1; hence thesis; end; then reconsider C1 as Subset-Family of T by SETFAM_1:def 7; reconsider C1 as Subset-Family of T; C2 c= bool the carrier of T proof let x be set; assume x in C2; then ex a being Subset of T2 st x = [:cT1, a:] & a in B2; hence thesis; end; then reconsider C2 as Subset-Family of T by SETFAM_1:def 7; reconsider C2 as Subset-Family of T; reconsider C = C2 \/ C1 as Subset-Family of T; C is prebasis of T proof hereby let x be set; assume x in C; then x in C1 or x in C2 by XBOOLE_0:def 2; then (ex a being Subset of T1 st x = [:a, cT2:] & a in B1) or (ex a being Subset of T2 st x = [:cT1, a:] & a in B2); then consider a being Subset of T1, b being Subset of T2 such that A3: x = [:a,b:] & a in the topology of T1 & b in the topology of T2 by A1,A2 ; a is open & b is open by A3,PRE_TOPC:def 5; then [:a,b:] is open by BORSUK_1:46; hence x in the topology of T by A3,PRE_TOPC:def 5; end; take D; let d be set; assume d in D; then consider a being Subset of T1, b being Subset of T2 such that A4: d = [:a,b:] & a in D1 & b in D2; consider aY being Subset-Family of T1 such that A5: aY c= B1 and A6: aY is finite and A7: a = Intersect aY by A4,CANTOR_1:def 4; consider bY being Subset-Family of T2 such that A8: bY c= B2 and A9: bY is finite and A10: b = Intersect bY by A4,CANTOR_1:def 4; deffunc F(Subset of T1) = [:$1, cT2:]; A11: {F(s) where s is Subset of T1: s in aY} is finite from FraenkelFin(A6); deffunc G(Subset of T2) = [:cT1, $1:]; A12: {G(s) where s is Subset of T2: s in bY} is finite from FraenkelFin(A9); set Y1 = {[:s, cT2:] where s is Subset of T1: s in aY}; set Y2 = {[:cT1, s:] where s is Subset of T2: s in bY}; A13: Y1 c= C proof let x be set; assume x in Y1; then consider s being Subset of T1 such that A14: x = [:s, cT2:] & s in aY; s in B1 & s is Subset of T1 by A5,A14; then x in C1 by A14; hence thesis by XBOOLE_0:def 2; end; A15: Y2 c= C proof let x be set; assume x in Y2; then consider s being Subset of T2 such that A16: x = [:cT1, s:] & s in bY; s in B2 & s is Subset of T2 by A8,A16; then x in C2 by A16; hence thesis by XBOOLE_0:def 2; end; set Y = Y1 \/ Y2; A17: Y c= C by A13,A15,XBOOLE_1:8; then Y is Subset of bool the carrier of T by XBOOLE_1:1; then Y is Subset-Family of T by SETFAM_1:def 7; then reconsider Y as Subset-Family of T; A18: Y is finite by A11,A12,FINSET_1:14; Intersect Y = d proof hereby let p be set; assume A19: p in Intersect Y; then consider p1 being Point of T1, p2 being Point of T2 such that A20: p = [p1,p2] by BORSUK_1:50; now let z be set; assume A21: z in aY; then reconsider z' = z as Subset of T1; [:z', cT2:] in Y1 by A21; then [:z', cT2:] in Y by XBOOLE_0:def 2; then p in [:z', cT2:] by A19,CANTOR_1:10; hence p1 in z by A20,ZFMISC_1:106; end; then A22: p1 in a by A7,CANTOR_1:10; now let z be set; assume A23: z in bY; then reconsider z' = z as Subset of T2; [:cT1, z':] in Y2 by A23; then [:cT1, z':] in Y by XBOOLE_0:def 2; then p in [:cT1, z':] by A19,CANTOR_1:10; hence p2 in z by A20,ZFMISC_1:106; end; then p2 in b by A10,CANTOR_1:10; hence p in d by A4,A20,A22,ZFMISC_1:106; end; let p be set; assume p in d; then consider p1,p2 being set such that A24: p1 in a & p2 in b & [p1,p2] = p by A4,ZFMISC_1:def 2; reconsider p1 as Point of T1 by A24; reconsider p2 as Point of T2 by A24; now let z be set; assume A25: z in Y; per cases by A25,XBOOLE_0:def 2; suppose z in Y1; then consider s being Subset of T1 such that A26: z = [:s, cT2:] & s in aY; p1 in s by A7,A24,A26,CANTOR_1:10; hence [p1,p2] in z by A26,ZFMISC_1:106; suppose z in Y2; then consider s being Subset of T2 such that A27: z = [:cT1, s:] & s in bY; p2 in s by A10,A24,A27,CANTOR_1:10; hence [p1,p2] in z by A27,ZFMISC_1:106; end; hence p in Intersect Y by A24,CANTOR_1:10; end; hence d in FinMeetCl C by A17,A18,CANTOR_1:def 4; end; hence thesis; end; theorem for X1,X2 being set, A being Subset-Family of [:X1,X2:] for A1 being non empty Subset-Family of X1 for A2 being non empty Subset-Family of X2 st A = {[:a,b:] where a is Subset of X1, b is Subset of X2: a in A1 & b in A2} holds Intersect A = [:Intersect A1, Intersect A2:] proof let X1,X2 be set, A be Subset-Family of [:X1,X2:]; let A1 be non empty Subset-Family of X1, A2 be non empty Subset-Family of X2 such that A1: A = {[:a,b:] where a is Subset of X1, b is Subset of X2: a in A1 & b in A2}; hereby let x be set; assume A2: x in Intersect A; then consider x1,x2 being set such that A3: x1 in X1 & x2 in X2 & [x1,x2] = x by ZFMISC_1:def 2; consider a1 being Element of A1, a2 being Element of A2; reconsider a1 as Subset of X1; reconsider a2 as Subset of X2; now let a be set; assume a in A1; then [:a,a2:] in A by A1; then x in [:a,a2:] by A2,CANTOR_1:10; hence x1 in a by A3,ZFMISC_1:106; end; then A4: x1 in Intersect A1 by A3,CANTOR_1:10; now let a be set; assume a in A2; then [:a1,a:] in A by A1; then x in [:a1,a:] by A2,CANTOR_1:10; hence x2 in a by A3,ZFMISC_1:106; end; then x2 in Intersect A2 by A3,CANTOR_1:10; hence x in [:Intersect A1, Intersect A2:] by A3,A4,ZFMISC_1:106; end; let x be set; assume A5: x in [:Intersect A1, Intersect A2:]; then consider x1,x2 being set such that A6: x1 in Intersect A1 & x2 in Intersect A2 & [x1,x2] = x by ZFMISC_1:def 2; now let c be set; assume c in A; then consider a being Subset of X1, b being Subset of X2 such that A7: c = [:a,b:] & a in A1 & b in A2 by A1; x1 in a & x2 in b by A6,A7,CANTOR_1:10; hence x in c by A6,A7,ZFMISC_1:106; end; hence x in Intersect A by A5,CANTOR_1:10; end; theorem for T1,T2 being non empty TopSpace for B1 being prebasis of T1, B2 being prebasis of T2 st union B1 = the carrier of T1 & union B2 = the carrier of T2 holds {[:a,b:] where a is Subset of T1, b is Subset of T2: a in B1 & b in B2} is prebasis of [:T1,T2:] proof let T1,T2 be non empty TopSpace; let B1 be prebasis of T1, B2 be prebasis of T2 such that A1: union B1 = the carrier of T1 & union B2 = the carrier of T2; set cT1 = the carrier of T1, cT2 = the carrier of T2; set BB1 = {[:the carrier of T1, b:] where b is Subset of T2: b in B2}, BB2 = {[:a, the carrier of T2:] where a is Subset of T1: a in B1}; set CC = {[:a,b:] where a is Subset of T1, b is Subset of T2: a in B1 & b in B2}; set T = [:T1,T2:]; reconsider BB=BB1 \/ BB2 as prebasis of [:T1,T2:] by Th41; A2: FinMeetCl BB is Basis of T by Th23; CC c= bool the carrier of [:T1,T2:] proof let x be set; assume x in CC; then ex a being Subset of T1, b being Subset of T2 st x = [:a,b:] & a in B1 & b in B2; hence thesis; end; then reconsider CC as Subset-Family of [:T1,T2:] by SETFAM_1:def 7; reconsider CC as Subset-Family of [:T1,T2:]; CC c= the topology of T proof let x be set; assume x in CC; then consider a being Subset of T1, b being Subset of T2 such that A3: x = [:a,b:] & a in B1 & b in B2; B1 c= the topology of T1 & B2 c= the topology of T2 by CANTOR_1:def 5; then a is open & b is open by A3,PRE_TOPC:def 5; then [:a,b:] is open by BORSUK_1:46; hence thesis by A3,PRE_TOPC:def 5; end; then UniCl CC c= UniCl the topology of T by CANTOR_1:9; then A4: UniCl CC c= the topology of T by CANTOR_1:6; BB c= UniCl CC proof let x be set; assume A5: x in BB; per cases by A5,XBOOLE_0:def 2; suppose x in BB1; then consider b being Subset of T2 such that A6: x = [:cT1,b:] & b in B2; set Y = {[:a,b:] where a is Subset of T1: a in B1}; Y c= bool the carrier of T proof let y be set; assume y in Y; then ex a being Subset of T1 st y = [:a,b:] & a in B1; hence thesis; end; then reconsider Y as Subset-Family of T by SETFAM_1:def 7; reconsider Y as Subset-Family of T; A7: Y c= CC proof let y be set; assume y in Y; then ex a being Subset of T1 st y = [:a,b:] & a in B1; hence thesis by A6; end; x = union Y proof hereby let z be set; assume z in x; then consider p1, p2 being set such that A8: p1 in cT1 & p2 in b & [p1,p2] = z by A6,ZFMISC_1:def 2; consider a being set such that A9: p1 in a & a in B1 by A1,A8,TARSKI:def 4; reconsider a as Subset of T1 by A9; [:a,b:] in Y & z in [:a,b:] by A8,A9,ZFMISC_1:def 2; hence z in union Y by TARSKI:def 4; end; let z be set; assume z in union Y; then consider y being set such that A10: z in y & y in Y by TARSKI:def 4; ex a being Subset of T1 st y = [:a,b:] & a in B1 by A10; then y c= x by A6,ZFMISC_1:118; hence thesis by A10; end; hence thesis by A7,CANTOR_1:def 1; suppose x in BB2; then consider a being Subset of T1 such that A11: x = [:a,cT2:] & a in B1; set Y = {[:a,b:] where b is Subset of T2: b in B2}; Y c= bool the carrier of T proof let y be set; assume y in Y; then ex b being Subset of T2 st y = [:a,b:] & b in B2; hence thesis; end; then reconsider Y as Subset-Family of T by SETFAM_1:def 7; reconsider Y as Subset-Family of T; A12: Y c= CC proof let y be set; assume y in Y; then ex b being Subset of T2 st y = [:a,b:] & b in B2; hence thesis by A11; end; x = union Y proof hereby let z be set; assume z in x; then consider p1, p2 being set such that A13: p1 in a & p2 in cT2 & [p1,p2] = z by A11,ZFMISC_1:def 2; consider b being set such that A14: p2 in b & b in B2 by A1,A13,TARSKI:def 4; reconsider b as Subset of T2 by A14; [:a,b:] in Y & z in [:a,b:] by A13,A14,ZFMISC_1:def 2; hence z in union Y by TARSKI:def 4; end; let z be set; assume z in union Y; then consider y being set such that A15: z in y & y in Y by TARSKI:def 4; ex b being Subset of T2 st y = [:a,b:] & b in B2 by A15; then y c= x by A11,ZFMISC_1:118; hence thesis by A15; end; hence thesis by A12,CANTOR_1:def 1; end; then FinMeetCl BB c= FinMeetCl UniCl CC by CANTOR_1:16; then UniCl CC is prebasis of T by A2,A4,CANTOR_1:def 5; hence thesis by Th24; end; begin :: Topological augmentations definition let R be RelStr; mode TopAugmentation of R -> TopRelStr means: Def4: the RelStr of it = the RelStr of R; existence proof consider O being Subset-Family of R; take TopRelStr(#the carrier of R, the InternalRel of R, O#); thus thesis; end; end; definition let R be RelStr; let T be TopAugmentation of R; redefine attr T is TopSpace-like; synonym T is correct; end; definition let R be RelStr; cluster correct discrete strict TopAugmentation of R; existence proof reconsider BB = bool the carrier of R as Subset-Family of R by SETFAM_1:def 7; set T = TopRelStr(#the carrier of R, the InternalRel of R, BB#); the RelStr of R = the RelStr of T; then reconsider T as TopAugmentation of R by Def4; take T; T is discrete TopStruct by TDLAT_3:def 1; hence thesis; end; end; theorem for T being TopRelStr holds T is TopAugmentation of T proof let T be TopRelStr; thus the RelStr of T = the RelStr of T; end; theorem for S being TopRelStr, T being TopAugmentation of S holds S is TopAugmentation of T proof let S be TopRelStr, T be TopAugmentation of S; thus the RelStr of S = the RelStr of T by Def4; end; theorem for R being RelStr, T1 being TopAugmentation of R for T2 being TopAugmentation of T1 holds T2 is TopAugmentation of R proof let R be RelStr, T1 be TopAugmentation of R; let T2 be TopAugmentation of T1; thus the RelStr of T2 = the RelStr of T1 by Def4 .= the RelStr of R by Def4; end; definition let R be non empty RelStr; cluster -> non empty TopAugmentation of R; coherence proof let T be TopAugmentation of R; the RelStr of T = the RelStr of R by Def4; hence the carrier of T is non empty; end; end; definition let R be reflexive RelStr; cluster -> reflexive TopAugmentation of R; coherence proof let T be TopAugmentation of R; the RelStr of T = the RelStr of R by Def4; hence the InternalRel of T is_reflexive_in the carrier of T by ORDERS_1:def 4; end; end; definition let R be transitive RelStr; cluster -> transitive TopAugmentation of R; coherence proof let T be TopAugmentation of R; the RelStr of T = the RelStr of R by Def4; then the InternalRel of T is_transitive_in the carrier of T by ORDERS_1:def 5; hence thesis by ORDERS_1:def 5; end; end; definition let R be antisymmetric RelStr; cluster -> antisymmetric TopAugmentation of R; coherence proof let T be TopAugmentation of R; the RelStr of T = the RelStr of R by Def4; then the InternalRel of T is_antisymmetric_in the carrier of T by ORDERS_1:def 6; hence thesis by ORDERS_1:def 6; end; end; definition let R be complete (non empty RelStr); cluster -> complete TopAugmentation of R; coherence proof let T be TopAugmentation of R; the RelStr of T = the RelStr of R by Def4; hence thesis by YELLOW_0:3; end; end; theorem Th47: for S being up-complete antisymmetric (non empty reflexive RelStr), T being non empty reflexive RelStr st the RelStr of S = the RelStr of T for A being Subset of S, C being Subset of T st A = C & A is inaccessible holds C is inaccessible proof let S be up-complete antisymmetric (non empty reflexive RelStr), T be non empty reflexive RelStr such that A1: the RelStr of S = the RelStr of T; let A be Subset of S, C be Subset of T such that A2: A = C and A3: for D being non empty directed Subset of S st sup D in A holds D meets A; let D be non empty directed Subset of T such that A4: sup D in C; reconsider E = D as non empty directed Subset of S by A1,WAYBEL_0:3; ex_sup_of E,S by WAYBEL_0:75; then sup D = sup E by A1,YELLOW_0:26; hence D meets C by A2,A3,A4; end; theorem Th48: for S being non empty reflexive RelStr, T being TopAugmentation of S st the topology of T = sigma S holds T is correct proof let R be non empty reflexive RelStr; let T be TopAugmentation of R such that A1: the topology of T = sigma R; A2: the RelStr of T = the RelStr of R by Def4; set IT = ConvergenceSpace Scott-Convergence R; the carrier of IT = the carrier of R & sigma R = the topology of IT by WAYBEL11:def 12,YELLOW_6:def 27; then the carrier of T in sigma R & (for a being Subset-Family of T st a c= the topology of T holds union a in the topology of T) & (for a,b being Subset of T st a in the topology of T & b in the topology of T holds a /\ b in the topology of T) by A1,A2,PRE_TOPC:def 1; hence thesis by A1,PRE_TOPC:def 1; end; theorem Th49: for S being complete LATTICE, T being TopAugmentation of S st the topology of T = sigma S holds T is Scott proof let R be complete LATTICE; let T be TopAugmentation of R such that A1: the topology of T = sigma R; A2: the RelStr of T = the RelStr of R by Def4; A3: sigma R = the topology of ConvergenceSpace Scott-Convergence R by WAYBEL11:def 12; T is Scott proof let S be Subset of T; reconsider A = S as Subset of R by A2; thus S is open implies S is inaccessible upper proof assume S in the topology of T; then A is inaccessible upper by A1,A3,WAYBEL11:31; hence thesis by A2,Th47,WAYBEL_0:25; end; assume S is inaccessible upper; then A is inaccessible upper by A2,Th47,WAYBEL_0:25; hence S in the topology of T by A1,A3,WAYBEL11:31; end; hence thesis; end; definition let R be complete LATTICE; cluster Scott strict correct TopAugmentation of R; existence proof set T = TopRelStr(#the carrier of R, the InternalRel of R, sigma R#); the RelStr of T = the RelStr of R; then reconsider T as TopAugmentation of R by Def4; take T; thus thesis by Th48,Th49; end; end; theorem Th50: for S,T being complete Scott (non empty reflexive transitive antisymmetric TopRelStr) st the RelStr of S = the RelStr of T for F being Subset of S, G being Subset of T st F = G holds F is open implies G is open proof let S,T be complete Scott (non empty reflexive transitive antisymmetric TopRelStr) such that A1: the RelStr of S = the RelStr of T; let F be Subset of S, G be Subset of T; assume A2: F = G & F is open; then F is upper inaccessible by WAYBEL11:def 4; then G is upper inaccessible by A1,A2,Th47,WAYBEL_0:25; hence G is open by WAYBEL11:def 4; end; theorem Th51: for S being complete LATTICE, T being Scott TopAugmentation of S holds the topology of T = sigma S proof let S be complete LATTICE; let T be Scott TopAugmentation of S; set R = TopRelStr(#the carrier of S, the InternalRel of S, sigma S#); the RelStr of R = the RelStr of S; then reconsider R as TopAugmentation of S by Def4; reconsider R as correct Scott TopAugmentation of S by Th48,Th49; A1: the RelStr of T = the RelStr of R by Def4; thus the topology of T c= sigma S proof let x be set; assume A2: x in the topology of T; then reconsider A = x as Subset of T; reconsider C = A as Subset of R by A1; A is open by A2,PRE_TOPC:def 5; then C is open by A1,Th50; hence thesis by PRE_TOPC:def 5; end; let x be set; assume A3: x in sigma S; then reconsider A = x as Subset of R; reconsider C = A as Subset of T by A1; A is open by A3,PRE_TOPC:def 5; then C is open by A1,Th50; hence thesis by PRE_TOPC:def 5; end; theorem for S,T being complete LATTICE st the RelStr of S = the RelStr of T holds sigma S = sigma T proof let S,T be complete LATTICE such that A1: the RelStr of S = the RelStr of T; consider A being Scott correct TopAugmentation of S; the RelStr of A = the RelStr of S by Def4; then A is Scott correct TopAugmentation of T by A1,Def4; then the topology of A = sigma T by Th51; hence thesis by Th51; end; definition let R be complete LATTICE; cluster Scott -> correct TopAugmentation of R; coherence proof let T be TopAugmentation of R; assume T is Scott; then the topology of T = sigma R by Th51; hence thesis by Th48; end; end; Lm2: for S being TopStruct ex T being strict TopSpace st the carrier of T = the carrier of S & the topology of S is prebasis of T proof let S be TopStruct; set T = TopStruct(#the carrier of S, UniCl FinMeetCl the topology of S#); A1: {{},{}} = {{}} by ENUMSET1:69; now assume A2: the carrier of S = {}; then the topology of S = {} or the topology of S = {{}} by ZFMISC_1:1,39 ; then FinMeetCl the topology of S = {{}} by A1,A2,Th11,Th17; then UniCl FinMeetCl the topology of S = {{}} by A1,Th11; hence T is discrete TopStruct by A2,TDLAT_3:def 1,ZFMISC_1:1; end; then reconsider T as strict TopSpace by CANTOR_1:17; take T; the topology of S c= FinMeetCl the topology of S & FinMeetCl the topology of S c= the topology of T by CANTOR_1:1,4; then A3: the topology of S c= the topology of T by XBOOLE_1:1; FinMeetCl the topology of S is Basis of T by Th22; hence thesis by A3,CANTOR_1:def 5; end; begin :: Refinements definition let T be TopStruct; mode TopExtension of T -> TopSpace means: Def5: the carrier of T = the carrier of it & the topology of T c= the topology of it; existence proof consider R being strict TopSpace such that A1: the carrier of R = the carrier of T & the topology of T is prebasis of R by Lm2; take R; thus thesis by A1,CANTOR_1:def 5; end; end; theorem Th53: for S being TopStruct ex T being TopExtension of S st T is strict & the topology of S is prebasis of T proof let S be TopStruct; consider T being strict TopSpace such that A1: the carrier of T = the carrier of S & the topology of S is prebasis of T by Lm2; the topology of S c= the topology of T by A1,CANTOR_1:def 5; then reconsider T as TopExtension of S by A1,Def5; take T; thus thesis by A1; end; definition let T be TopStruct; cluster strict discrete TopExtension of T; existence proof reconsider S = bool the carrier of T as Subset-Family of T by SETFAM_1:def 7; reconsider S as Subset-Family of T; set R = TopStruct(#the carrier of T, S#); A1: R is discrete TopStruct by TDLAT_3:def 1; the topology of T c= S; then R is TopExtension of T by A1,Def5; hence thesis by A1; end; end; definition let T1,T2 be TopStruct; mode Refinement of T1,T2 -> TopSpace means: Def6: the carrier of it = (the carrier of T1) \/ (the carrier of T2) & (the topology of T1) \/ (the topology of T2) is prebasis of it; existence proof set c1 = the carrier of T1, c2 = the carrier of T2; set t1 = the topology of T1, t2 = the topology of T2; c1 c= c1 \/ c2 & c2 c= c1 \/ c2 by XBOOLE_1:7; then bool c1 c= bool (c1 \/ c2) & bool c2 c= bool (c1 \/ c2) by ZFMISC_1:79; then t1 c= bool (c1 \/ c2) & t2 c= bool (c1 \/ c2) by XBOOLE_1:1; then reconsider t = t1 \/ t2 as Subset of bool (c1 \/ c2) by XBOOLE_1:8; reconsider t as Subset-Family of c1 \/ c2 by SETFAM_1:def 7; set S = TopStruct(#c1 \/ c2, t#); consider T being TopExtension of S such that A1: T is strict & t is prebasis of T by Th53; reconsider T as strict TopExtension of S by A1; take T; thus thesis by A1,Def5; end; end; definition let T1 be non empty TopStruct, T2 be TopStruct; cluster -> non empty Refinement of T1,T2; coherence proof let T be Refinement of T1,T2; the carrier of T = (the carrier of T2) \/ (the carrier of T1) by Def6; hence the carrier of T is non empty; end; cluster -> non empty Refinement of T2,T1; coherence proof let T be Refinement of T2,T1; the carrier of T = (the carrier of T2) \/ (the carrier of T1) by Def6; hence the carrier of T is non empty; end; end; theorem for T1,T2 being TopStruct, T, T' being Refinement of T1,T2 holds the TopStruct of T = the TopStruct of T' proof let T1, T2 be TopStruct, T,T' be Refinement of T1,T2; the carrier of T = (the carrier of T1) \/ (the carrier of T2) & (the topology of T1) \/ (the topology of T2) is prebasis of T & the carrier of T' = (the carrier of T1) \/ (the carrier of T2) & (the topology of T1) \/ (the topology of T2) is prebasis of T' by Def6; hence thesis by Th26; end; theorem for T1,T2 being TopStruct, T being Refinement of T1,T2 holds T is Refinement of T2,T1 proof let T1, T2 be TopStruct, T be Refinement of T1,T2; the carrier of T = (the carrier of T1) \/ (the carrier of T2) & (the topology of T1) \/ (the topology of T2) is prebasis of T by Def6; hence thesis by Def6; end; theorem for T1,T2 being TopStruct, T being Refinement of T1,T2 for X being Subset-Family of T st X = (the topology of T1) \/ (the topology of T2) holds the topology of T = UniCl FinMeetCl X proof let T1,T2 be TopStruct, T be Refinement of T1,T2; let X be Subset-Family of T; assume X = (the topology of T1) \/ (the topology of T2); then X is prebasis of T by Def6; then FinMeetCl X is Basis of T by Th23; hence thesis by Th22; end; theorem for T1, T2 being TopStruct st the carrier of T1 = the carrier of T2 for T being Refinement of T1, T2 holds T is TopExtension of T1 & T is TopExtension of T2 proof let T1, T2 be TopStruct such that A1: the carrier of T1 = the carrier of T2; let T be Refinement of T1, T2; A2: the carrier of T = (the carrier of T1) \/ the carrier of T2 by Def6 .= the carrier of T1 by A1; (the topology of T1) \/ the topology of T2 is prebasis of T by Def6; then the topology of T1 c= (the topology of T1) \/ the topology of T2 & the topology of T2 c= (the topology of T1) \/ the topology of T2 & (the topology of T1) \/ the topology of T2 c= the topology of T by CANTOR_1:def 5,XBOOLE_1:7; then the topology of T1 c= the topology of T & the topology of T2 c= the topology of T by XBOOLE_1:1; hence thesis by A1,A2,Def5; end; theorem for T1,T2 being non empty TopSpace, T be Refinement of T1, T2 for B1 being prebasis of T1, B2 being prebasis of T2 holds B1 \/ B2 \/ {the carrier of T1, the carrier of T2} is prebasis of T proof let T1,T2 be non empty TopSpace, T be Refinement of T1,T2; let B1 be prebasis of T1, B2 be prebasis of T2; reconsider B = (the topology of T1) \/ (the topology of T2) as prebasis of T by Def6; set cT1 = the carrier of T1, cT2 = the carrier of T2; reconsider C1 = B1 \/ {cT1} as prebasis of T1 by Th29; reconsider C2 = B2 \/ {cT2} as prebasis of T2 by Th29; A1: B1 c= the topology of T1 & C1 c= the topology of T1 & B2 c= the topology of T2 & C2 c= the topology of T2 & cT1 in the topology of T1 & cT2 in the topology of T2 by CANTOR_1:def 5,PRE_TOPC:def 1; then A2: B1 \/ B2 c= B & B c= the topology of T & cT1 in B & cT2 in B by CANTOR_1:def 5,XBOOLE_0:def 2,XBOOLE_1:13; then A3: B1 \/ B2 c= the topology of T & {cT1,cT2} c= the topology of T & {cT1,cT2} c= B by XBOOLE_1:1,ZFMISC_1:38; then B1 \/ B2 \/ {cT1,cT2} c= the topology of T by XBOOLE_1:8; then B1 \/ B2 \/ {cT1, cT2} is Subset of bool the carrier of T by XBOOLE_1:1; then B1 \/ B2 \/ {cT1, cT2} is Subset-Family of T by SETFAM_1: def 7; then reconsider BB = B1 \/ B2 \/ {cT1, cT2} as Subset-Family of T; the topology of T1 c= B & the topology of T2 c= B by XBOOLE_1:7; then C1 c= B & C2 c= B by A1,XBOOLE_1:1; then reconsider D1 = C1, D2 = C2 as Subset of bool the carrier of T by XBOOLE_1:1; reconsider D1, D2 as Subset-Family of T by SETFAM_1:def 7; reconsider D1, D2 as Subset-Family of T; A4: UniCl FinMeetCl BB = UniCl FinMeetCl FinMeetCl BB by CANTOR_1:13 .= UniCl FinMeetCl UniCl FinMeetCl BB by Th21; FinMeetCl B is Basis of T & FinMeetCl C1 is Basis of T1 & FinMeetCl C2 is Basis of T2 by Th23; then A5: UniCl FinMeetCl B = the topology of T & UniCl FinMeetCl C1 = the topology of T1 & UniCl FinMeetCl C2 = the topology of T2 by Th22; B1 c= B1 \/ B2 & B2 c= B1 \/ B2 & {cT1} c= {cT1,cT2} & {cT2} c= {cT1,cT2} by XBOOLE_1:7,ZFMISC_1:12; then D1 c= BB & D2 c= BB & BB c= B by A2,A3,XBOOLE_1:8,13; then A6: FinMeetCl BB c= FinMeetCl B & FinMeetCl D1 c= FinMeetCl BB & FinMeetCl D2 c= FinMeetCl BB by CANTOR_1:16; then A7: UniCl FinMeetCl BB c= the topology of T & UniCl FinMeetCl D1 c= UniCl FinMeetCl BB & UniCl FinMeetCl D2 c= UniCl FinMeetCl BB by A5,CANTOR_1:9; cT1 in {cT1} & cT2 in {cT2} by TARSKI:def 1; then cT1 in C1 & cT2 in C2 by XBOOLE_0:def 2; then FinMeetCl D1 = {the carrier of T} \/ FinMeetCl C1 & FinMeetCl D2 = {the carrier of T} \/ FinMeetCl C2 by Th20; then FinMeetCl C1 c= FinMeetCl D1 & FinMeetCl C2 c= FinMeetCl D2 by XBOOLE_1:7; then FinMeetCl C1 c= FinMeetCl BB & FinMeetCl C2 c= FinMeetCl BB by A6,XBOOLE_1:1; then the topology of T1 c= UniCl FinMeetCl BB & the topology of T2 c= UniCl FinMeetCl BB by A5,Th19; then B c= UniCl FinMeetCl BB by XBOOLE_1:8; then FinMeetCl B c= FinMeetCl UniCl FinMeetCl BB by CANTOR_1:16; then the topology of T c= UniCl FinMeetCl BB by A4,A5,CANTOR_1:9; then the topology of T = UniCl FinMeetCl BB by A7,XBOOLE_0:def 10; then FinMeetCl BB is Basis of T by Th22; hence thesis by Th23; end; theorem Th59: for T1,T2 being non empty TopSpace for B1 being Basis of T1, B2 being Basis of T2 for T being Refinement of T1, T2 holds B1 \/ B2 \/ INTERSECTION(B1,B2) is Basis of T proof let T1,T2 be non empty TopSpace; let B1 be Basis of T1, B2 be Basis of T2; let T be Refinement of T1,T2; set BB = B1 \/ B2 \/ INTERSECTION(B1,B2); reconsider B = (the topology of T1) \/ the topology of T2 as prebasis of T by Def6; A1: FinMeetCl B is Basis of T by Th23; A2: (the carrier of T1) \/ the carrier of T2 = the carrier of T by Def6; A3: B1 c= the topology of T1 & B2 c= the topology of T2 & B c= the topology of T by CANTOR_1:def 2,def 5; the topology of T1 c= B & the topology of T2 c= B by XBOOLE_1:7; then A4: B1 c= B & B2 c= B & B c= FinMeetCl B by A3,CANTOR_1:4,XBOOLE_1:1; then B1 c= FinMeetCl B & B2 c= FinMeetCl B & B1 \/ B2 c= B by A3,XBOOLE_1:1,13; then INTERSECTION(B1,B2) c= FinMeetCl B & B1 \/ B2 c= FinMeetCl B by A4,CANTOR_1:15,XBOOLE_1:1; then A5: BB c= FinMeetCl B & FinMeetCl B c= the topology of T by A1,CANTOR_1:def 2,XBOOLE_1:8; then A6: BB c= the topology of T by XBOOLE_1:1; BB is Subset of bool the carrier of T by A5,XBOOLE_1:1; then BB is Subset-Family of T by SETFAM_1:def 7; then reconsider BB as Subset-Family of T; now let A be Subset of T such that A7: A is open; let p be Point of T; assume p in A; then consider a being Subset of T such that A8: a in FinMeetCl B & p in a & a c= A by A1,A7,Th31; consider Y being Subset-Family of T such that A9: Y c= B & Y is finite & a = Intersect Y by A8,CANTOR_1:def 4; Y /\ the topology of T1 c= the topology of T1 by XBOOLE_1:17; then Y /\ the topology of T1 is Subset of bool the carrier of T1 by XBOOLE_1:1; then Y /\ the topology of T1 is Subset-Family of T1 by SETFAM_1:def 7; then reconsider Y1 = Y /\ the topology of T1 as Subset-Family of T1 ; Y /\ the topology of T2 c= the topology of T2 by XBOOLE_1:17; then Y /\ the topology of T2 is Subset of bool the carrier of T2 by XBOOLE_1:1; then Y /\ the topology of T2 is Subset-Family of T2 by SETFAM_1:def 7; then reconsider Y2 = Y /\ the topology of T2 as Subset-Family of T2 ; A10: Y = B /\ Y by A9,XBOOLE_1:28 .= Y1 \/ Y2 by XBOOLE_1:23; the carrier of T1 c= the carrier of T1; then reconsider cT1 = the carrier of T1 as Subset of T1; the carrier of T2 c= the carrier of T2; then reconsider cT2 = the carrier of T2 as Subset of T2; cT1 in the topology of T1 & cT2 in the topology of T2 by PRE_TOPC:def 1; then A11: cT1 is open & cT2 is open & cT1 in B & cT2 in B by PRE_TOPC:def 5,XBOOLE_0:def 2; thus ex a being Subset of T st a in BB & p in a & a c= A proof per cases by A2,XBOOLE_0:def 2; suppose A12: Y1 \/ Y2 = {} & p in cT1; then consider b1 being Subset of T1 such that A13: b1 in B1 & p in b1 & b1 c= cT1 by A11,Th31; b1 in B1 \/ B2 & b1 in B & a = the carrier of T by A3,A9,A10,A12,A13,CANTOR_1:def 3,XBOOLE_0:def 2; then b1 in BB & b1 is Subset of T & b1 c= A by A8,XBOOLE_0:def 2,XBOOLE_1:1; hence thesis by A13; suppose A14: Y1 \/ Y2 = {} & p in cT2; then consider b2 being Subset of T2 such that A15: b2 in B2 & p in b2 & b2 c= cT2 by A11,Th31; b2 in B1 \/ B2 & b2 in B & a = the carrier of T by A3,A9,A10,A14,A15,CANTOR_1:def 3,XBOOLE_0:def 2; then b2 in BB & b2 is Subset of T & b2 c= A by A8,XBOOLE_0:def 2,XBOOLE_1:1; hence thesis by A15; suppose A16: Y1 = {} & Y2 <> {} & Y <> {}; then A17: a = meet Y2 by A9,A10,CANTOR_1:def 3 .= Intersect Y2 by A16,CANTOR_1:def 3; Y2 c= Y by A10,XBOOLE_1:7; then Y2 is finite & Y2 c= the topology of T2 by A9,FINSET_1:13,XBOOLE_1:17; then a in FinMeetCl the topology of T2 by A17,CANTOR_1:def 4; then a in the topology of T2 & the topology of T2 = UniCl B2 by Th22,CANTOR_1:5; then consider Z being Subset-Family of T2 such that A18: Z c= B2 & a = union Z by CANTOR_1:def 1; consider z being set such that A19: p in z & z in Z by A8,A18,TARSKI:def 4; z in B1 \/ B2 by A18,A19,XBOOLE_0:def 2; then A20: z in BB & z c= a by A18,A19,XBOOLE_0:def 2,ZFMISC_1:92; then z is Subset of T & z c= A by A8,XBOOLE_1:1; hence thesis by A19,A20; suppose A21: Y1 <> {} & Y2 = {} & Y <> {}; then A22: a = meet Y1 by A9,A10,CANTOR_1:def 3 .= Intersect Y1 by A21,CANTOR_1:def 3; Y1 c= Y by A10,XBOOLE_1:7; then Y1 is finite & Y1 c= the topology of T1 by A9,FINSET_1:13,XBOOLE_1:17; then a in FinMeetCl the topology of T1 by A22,CANTOR_1:def 4; then a in the topology of T1 & the topology of T1 = UniCl B1 by Th22,CANTOR_1:5; then consider Z being Subset-Family of T1 such that A23: Z c= B1 & a = union Z by CANTOR_1:def 1; consider z being set such that A24: p in z & z in Z by A8,A23,TARSKI:def 4; z in B1 \/ B2 by A23,A24,XBOOLE_0:def 2; then A25: z in BB & z c= a by A23,A24,XBOOLE_0:def 2,ZFMISC_1:92; then z is Subset of T & z c= A by A8,XBOOLE_1:1; hence thesis by A24,A25; thus thesis; suppose A26: Y1 <> {} & Y2 <> {} & Y <> {}; then A27: a = meet Y by A9,CANTOR_1:def 3 .= (meet Y1)/\meet Y2 by A10,A26,SETFAM_1:10 .= (meet Y1)/\Intersect Y2 by A26,CANTOR_1:def 3 .= (Intersect Y1)/\Intersect Y2 by A26,CANTOR_1:def 3; Y1 c= Y & Y2 c= Y by A10,XBOOLE_1:7; then Y1 is finite & Y2 is finite & Y1 c= the topology of T1 & Y2 c= the topology of T2 by A9,FINSET_1:13,XBOOLE_1:17; then Intersect Y1 in FinMeetCl the topology of T1 & Intersect Y2 in FinMeetCl the topology of T2 by CANTOR_1:def 4; then A28: Intersect Y1 in the topology of T1 & the topology of T1 = UniCl B1 & Intersect Y2 in the topology of T2 & the topology of T2 = UniCl B2 by Th22,CANTOR_1:5; then consider Z1 being Subset-Family of T1 such that A29: Z1 c= B1 & Intersect Y1 = union Z1 by CANTOR_1:def 1; p in Intersect Y1 by A8,A27,XBOOLE_0:def 3; then consider z1 being set such that A30: p in z1 & z1 in Z1 by A29,TARSKI:def 4; consider Z2 being Subset-Family of T2 such that A31: Z2 c= B2 & Intersect Y2 = union Z2 by A28,CANTOR_1:def 1; p in Intersect Y2 by A8,A27,XBOOLE_0:def 3; then consider z2 being set such that A32: p in z2 & z2 in Z2 by A31,TARSKI:def 4; z1 /\ z2 in INTERSECTION(B1,B2) & z1 c= union Z1 & z2 c= union Z2 by A29,A30,A31,A32,SETFAM_1:def 5,ZFMISC_1:92; then A33: z1 /\ z2 in BB & z1 /\ z2 c= a by A27,A29,A31,XBOOLE_0:def 2, XBOOLE_1:27; then z1 /\ z2 is Subset of T & z1 /\ z2 c= A & p in z1 /\ z2 by A8,A30,A32,XBOOLE_0:def 3,XBOOLE_1:1; hence thesis by A33; end; end; hence thesis by A6,Th32; end; theorem Th60: for T1,T2 being non empty TopSpace st the carrier of T1 = the carrier of T2 for T being Refinement of T1, T2 holds INTERSECTION(the topology of T1, the topology of T2) is Basis of T proof let T1,T2 be non empty TopSpace such that A1: the carrier of T1 = the carrier of T2; let T be Refinement of T1, T2; set B1 = the topology of T1, B2 = the topology of T2; UniCl B1 = B1 by CANTOR_1:6; then reconsider B1 as Basis of T1 by Th22; UniCl B2 = B2 by CANTOR_1:6; then reconsider B2 as Basis of T2 by Th22; A2: B1 \/ B2 \/ INTERSECTION(B1,B2) is Basis of T by Th59; A3: the carrier of T1 in B1 & the carrier of T2 in B2 by PRE_TOPC:def 1; A4: B1 c= INTERSECTION(B1,B2) proof let a be set; assume a in B1; then a /\ the carrier of T1 in INTERSECTION(B1,B2) & a c= the carrier of T1 by A1,A3,SETFAM_1:def 5; hence thesis by XBOOLE_1:28; end; B2 c= INTERSECTION(B1,B2) proof let a be set; assume a in B2; then a /\ the carrier of T2 in INTERSECTION(B1,B2) & a c= the carrier of T2 by A1,A3,SETFAM_1:def 5; hence thesis by XBOOLE_1:28; end; then B1 \/ B2 c= INTERSECTION(B1,B2) by A4,XBOOLE_1:8; hence INTERSECTION(the topology of T1, the topology of T2) is Basis of T by A2,XBOOLE_1:12; end; theorem for L being non empty RelStr for T1,T2 being correct TopAugmentation of L for T be Refinement of T1, T2 holds INTERSECTION(the topology of T1, the topology of T2) is Basis of T proof let L be non empty RelStr; let T1,T2 be correct TopAugmentation of L; the RelStr of T1 = the RelStr of L & the RelStr of T2 = the RelStr of L by Def4; hence thesis by Th60; end;

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