Volume 10, 1998

University of Bialystok

Copyright (c) 1998 Association of Mizar Users

### The abstract of the Mizar article:

### Bases and Refinements of Topologies

**by****Grzegorz Bancerek**- Received March 9, 1998
- MML identifier: YELLOW_9

- [ Mizar article, 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; 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 Y() = {F(a) where a is Element of A(): a in X()} and for a being Element of A() holds F(F(a)) = a; 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 X() = {F(a) where a is Element of A(): a in Y()}; 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 X() = {F(a)` where a is Element of A(): a in Y()}; theorem :: YELLOW_9:1 for R being non empty RelStr, x,y being Element of R holds y in (uparrow x)` iff not x <= y; theorem :: YELLOW_9:2 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`; theorem :: YELLOW_9:3 for T being 1-sorted, F being Subset-Family of T holds COMPLEMENT F = {a` where a is Subset of T: a in F}; theorem :: YELLOW_9:4 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}; theorem :: YELLOW_9:5 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)`; definition cluster strict trivial reflexive non empty discrete finite TopRelStr; end; definition cluster strict complete trivial TopLattice; end; definition let S be non empty RelStr, T be upper-bounded non empty reflexive antisymmetric RelStr; cluster infs-preserving map of S,T; end; definition let S be non empty RelStr, T be lower-bounded non empty reflexive antisymmetric RelStr; cluster sups-preserving map of S,T; end; definition let R,S be 1-sorted; assume the carrier of S c= the carrier of R; func incl(S,R) -> map of S,R equals :: YELLOW_9:def 1 id the carrier of S; end; definition let R be non empty RelStr; let S be non empty SubRelStr of R; cluster incl(S,R) -> monotone; 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 :: YELLOW_9:def 2 (incl(subrelstr X, R))*((subrelstr X)+id); func X opp+id -> strict non empty NetStr over R equals :: YELLOW_9:def 3 (incl(subrelstr X, R))*((subrelstr X)opp+id); end; theorem :: YELLOW_9:6 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; theorem :: YELLOW_9:7 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; definition let R be non empty reflexive RelStr; let X be non empty Subset of R; cluster X +id -> reflexive; cluster X opp+id -> reflexive; end; definition let R be non empty transitive RelStr; let X be non empty Subset of R; cluster X +id -> transitive; cluster X opp+id -> transitive; end; definition let R be non empty reflexive RelStr; let I be directed Subset of R; cluster subrelstr I -> directed; end; definition let R be non empty reflexive RelStr; let I be directed non empty Subset of R; cluster I+id -> directed; end; definition let R be non empty reflexive RelStr; let F be filtered non empty Subset of R; cluster (subrelstr F) opp+id -> directed; end; definition let R be non empty reflexive RelStr; let F be filtered non empty Subset of R; cluster F opp+id -> directed; end; begin :: Operations on families of open sets theorem :: YELLOW_9:8 for T being TopSpace st T is empty holds the topology of T = {{}}; theorem :: YELLOW_9:9 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}}; theorem :: YELLOW_9:10 for T being trivial non empty TopSpace holds {the carrier of T} is Basis of T & {} is prebasis of T & {{}} is prebasis of T; theorem :: YELLOW_9:11 for X,Y being set, A being Subset-Family of X st A = {Y} holds FinMeetCl A = {Y,X} & UniCl A = {Y,{}}; theorem :: YELLOW_9:12 for X being set, A,B being Subset-Family of X st A = B \/ {X} or B = A \ {X} holds Intersect A = Intersect B; theorem :: YELLOW_9:13 for X being set, A,B being Subset-Family of X st A = B \/ {X} or B = A \ {X} holds FinMeetCl A = FinMeetCl B; theorem :: YELLOW_9:14 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; theorem :: YELLOW_9:15 for X being set, A being Subset-Family of X holds UniCl UniCl A = UniCl A; theorem :: YELLOW_9:16 for X being set, A being empty Subset-Family of X holds UniCl A = {{}}; theorem :: YELLOW_9:17 for X being set, A being empty Subset-Family of X holds FinMeetCl A = {X}; theorem :: YELLOW_9:18 for X being set, A being Subset-Family of X st A = {{},X} holds UniCl A = A & FinMeetCl A = A; theorem :: YELLOW_9:19 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); theorem :: YELLOW_9:20 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; theorem :: YELLOW_9:21 for X being set, A being Subset-Family of X holds UniCl FinMeetCl UniCl A = UniCl FinMeetCl A; begin :: Bases theorem :: YELLOW_9:22 for T being TopSpace, K being Subset-Family of T holds the topology of T = UniCl K iff K is Basis of T; theorem :: YELLOW_9:23 for T being TopSpace, K being Subset-Family of T holds K is prebasis of T iff FinMeetCl K is Basis of T; theorem :: YELLOW_9:24 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; theorem :: YELLOW_9:25 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; theorem :: YELLOW_9:26 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; theorem :: YELLOW_9:27 for T being TopSpace, K being Basis of T holds K is open & K is prebasis of T; theorem :: YELLOW_9:28 for T being TopSpace, K being prebasis of T holds K is open; theorem :: YELLOW_9:29 for T being non empty TopSpace, B being prebasis of T holds B \/ {the carrier of T} is prebasis of T; theorem :: YELLOW_9:30 for T being TopSpace, B being Basis of T holds B \/ {the carrier of T} is Basis of T; theorem :: YELLOW_9:31 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; theorem :: YELLOW_9:32 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; theorem :: YELLOW_9:33 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; theorem :: YELLOW_9:34 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; theorem :: YELLOW_9:35 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; theorem :: YELLOW_9:36 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; theorem :: YELLOW_9:37 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; theorem :: YELLOW_9:38 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; theorem :: YELLOW_9:39 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; begin :: Product topology theorem :: YELLOW_9:40 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:]; theorem :: YELLOW_9:41 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:]; theorem :: YELLOW_9:42 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:]; theorem :: YELLOW_9:43 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:]; begin :: Topological augmentations definition let R be RelStr; mode TopAugmentation of R -> TopRelStr means :: YELLOW_9:def 4 the RelStr of it = the RelStr of R; 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; end; theorem :: YELLOW_9:44 for T being TopRelStr holds T is TopAugmentation of T; theorem :: YELLOW_9:45 for S being TopRelStr, T being TopAugmentation of S holds S is TopAugmentation of T; theorem :: YELLOW_9:46 for R being RelStr, T1 being TopAugmentation of R for T2 being TopAugmentation of T1 holds T2 is TopAugmentation of R; definition let R be non empty RelStr; cluster -> non empty TopAugmentation of R; end; definition let R be reflexive RelStr; cluster -> reflexive TopAugmentation of R; end; definition let R be transitive RelStr; cluster -> transitive TopAugmentation of R; end; definition let R be antisymmetric RelStr; cluster -> antisymmetric TopAugmentation of R; end; definition let R be complete (non empty RelStr); cluster -> complete TopAugmentation of R; end; theorem :: YELLOW_9:47 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; theorem :: YELLOW_9:48 for S being non empty reflexive RelStr, T being TopAugmentation of S st the topology of T = sigma S holds T is correct; theorem :: YELLOW_9:49 for S being complete LATTICE, T being TopAugmentation of S st the topology of T = sigma S holds T is Scott; definition let R be complete LATTICE; cluster Scott strict correct TopAugmentation of R; end; theorem :: YELLOW_9:50 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; theorem :: YELLOW_9:51 for S being complete LATTICE, T being Scott TopAugmentation of S holds the topology of T = sigma S; theorem :: YELLOW_9:52 for S,T being complete LATTICE st the RelStr of S = the RelStr of T holds sigma S = sigma T; definition let R be complete LATTICE; cluster Scott -> correct TopAugmentation of R; end; begin :: Refinements definition let T be TopStruct; mode TopExtension of T -> TopSpace means :: YELLOW_9:def 5 the carrier of T = the carrier of it & the topology of T c= the topology of it; end; theorem :: YELLOW_9:53 for S being TopStruct ex T being TopExtension of S st T is strict & the topology of S is prebasis of T; definition let T be TopStruct; cluster strict discrete TopExtension of T; end; definition let T1,T2 be TopStruct; mode Refinement of T1,T2 -> TopSpace means :: YELLOW_9:def 6 the carrier of it = (the carrier of T1) \/ (the carrier of T2) & (the topology of T1) \/ (the topology of T2) is prebasis of it; end; definition let T1 be non empty TopStruct, T2 be TopStruct; cluster -> non empty Refinement of T1,T2; cluster -> non empty Refinement of T2,T1; end; theorem :: YELLOW_9:54 for T1,T2 being TopStruct, T, T' being Refinement of T1,T2 holds the TopStruct of T = the TopStruct of T'; theorem :: YELLOW_9:55 for T1,T2 being TopStruct, T being Refinement of T1,T2 holds T is Refinement of T2,T1; theorem :: YELLOW_9:56 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; theorem :: YELLOW_9:57 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; theorem :: YELLOW_9:58 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; theorem :: YELLOW_9:59 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; theorem :: YELLOW_9:60 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; theorem :: YELLOW_9:61 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;

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