Volume 3, 1991

University of Bialystok

Copyright (c) 1991 Association of Mizar Users

### The abstract of the Mizar article:

### Metric Spaces as Topological Spaces --- Fundamental Concepts

**by****Agata Darmochwal, and****Yatsuka Nakamura**- Received November 21, 1991
- MML identifier: TOPMETR

- [ Mizar article, MML identifier index ]

environ vocabulary ARYTM, PRE_TOPC, SETFAM_1, TARSKI, SUBSET_1, COMPTS_1, BOOLE, RELAT_1, CONNSP_2, TOPS_1, ORDINAL2, FUNCT_1, METRIC_1, RCOMP_1, ABSVALUE, ARYTM_1, PCOMPS_1, EUCLID, FINSET_1, BORSUK_1, ARYTM_3, TOPMETR, PARTFUN1; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, ORDINAL1, NUMBERS, XCMPLX_0, XREAL_0, NAT_1, REAL_1, DOMAIN_1, RELAT_1, FUNCT_1, PARTFUN1, FUNCT_2, FINSET_1, BINOP_1, STRUCT_0, METRIC_1, PRE_TOPC, TOPS_1, TOPS_2, COMPTS_1, PCOMPS_1, CONNSP_2, RCOMP_1, ABSVALUE, BORSUK_1, EUCLID; constructors REAL_1, DOMAIN_1, TOPS_1, TOPS_2, COMPTS_1, RCOMP_1, ABSVALUE, BORSUK_1, EUCLID, MEMBERED, XBOOLE_0; clusters SUBSET_1, FUNCT_1, PRE_TOPC, PCOMPS_1, STRUCT_0, METRIC_3, METRIC_1, RELSET_1, EUCLID, TOPS_1, XREAL_0, MEMBERED, ZFMISC_1, XBOOLE_0; requirements NUMERALS, REAL, BOOLE, SUBSET, ARITHM; begin reserve r for Real, n for Nat; reserve a, b for real number; reserve T for non empty TopSpace; :: Topological spaces theorem :: TOPMETR:1 for T being TopStruct, F being Subset-Family of T holds F is_a_cover_of T iff the carrier of T c= union F; reserve A for non empty SubSpace of T; theorem :: TOPMETR:2 for p being Point of A holds p is Point of T; theorem :: TOPMETR:3 T is_T2 implies A is_T2; theorem :: TOPMETR:4 for A,B being SubSpace of T st the carrier of A c= the carrier of B holds A is SubSpace of B; reserve P,Q for Subset of T, p for Point of T; theorem :: TOPMETR:5 T|P is SubSpace of T|(P \/ Q) & T|Q is SubSpace of T|(P \/ Q); theorem :: TOPMETR:6 for P being non empty Subset of T st p in P for Q being a_neighborhood of p holds for p' being Point of T|P, Q' being Subset of T|P st Q' = Q /\ P & p'= p holds Q' is a_neighborhood of p'; theorem :: TOPMETR:7 for A,B,C being TopSpace for f being map of A,C holds f is continuous & C is SubSpace of B implies for h being map of A,B st h = f holds h is continuous; theorem :: TOPMETR:8 for A being TopSpace, B being non empty TopSpace for f being map of A,B for C being SubSpace of B holds f is continuous implies for h being map of A,C st h = f holds h is continuous; theorem :: TOPMETR:9 for A,B being TopSpace for f being map of A,B for C being Subset of B holds f is continuous implies for h being map of A,B|C st h = f holds h is continuous; theorem :: TOPMETR:10 for X being TopStruct, Y being non empty TopStruct, K0 being Subset of X, f being map of X,Y, g being map of X|K0,Y st f is continuous & g = f|K0 holds g is continuous; :: Some definitions & theorems about metrical spaces reserve M for non empty MetrSpace, p for Point of M; definition let M be MetrSpace; mode SubSpace of M -> MetrSpace means :: TOPMETR:def 1 the carrier of it c= the carrier of M & for x,y being Point of it holds (the distance of it).(x,y) = (the distance of M).(x,y); end; definition let M be MetrSpace; cluster strict SubSpace of M; end; definition let M be non empty MetrSpace; cluster strict non empty SubSpace of M; end; reserve A for non empty SubSpace of M; canceled; theorem :: TOPMETR:12 for p being Point of A holds p is Point of M; theorem :: TOPMETR:13 for r being real number for M being MetrSpace, A being SubSpace of M for x being Point of M, x' being Point of A st x = x' holds Ball(x',r) = Ball(x,r) /\ the carrier of A; definition let M be non empty MetrSpace, A be non empty Subset of M; func M|A -> strict SubSpace of M means :: TOPMETR:def 2 the carrier of it = A; end; definition let M be non empty MetrSpace, A be non empty Subset of M; cluster M|A -> non empty; end; definition let a,b be real number; assume a <= b; func Closed-Interval-MSpace(a,b) -> strict non empty SubSpace of RealSpace means :: TOPMETR:def 3 for P being non empty Subset of RealSpace st P = [. a,b .] holds it = RealSpace | P; end; theorem :: TOPMETR:14 a <= b implies the carrier of Closed-Interval-MSpace(a,b) = [. a,b .]; reserve F,G for Subset-Family of M; definition let M be MetrStruct, F be Subset-Family of M; attr F is being_ball-family means :: TOPMETR:def 4 for P being set holds P in F implies ex p being Point of M, r st P = Ball(p,r); synonym F is_ball-family; pred F is_a_cover_of M means :: TOPMETR:def 5 the carrier of M c= union F; end; theorem :: TOPMETR:15 for p,q being Point of RealSpace, x,y being real number holds x=p & y=q implies dist(p,q) = abs(x-y); :: Metric spaces and topology theorem :: TOPMETR:16 for M being MetrStruct holds the carrier of M = the carrier of TopSpaceMetr M & the topology of TopSpaceMetr M = Family_open_set M; canceled 2; theorem :: TOPMETR:19 TopSpaceMetr(A) is SubSpace of TopSpaceMetr(M); theorem :: TOPMETR:20 for P being Subset of TOP-REAL n, Q being non empty Subset of Euclid n holds P = Q implies (TOP-REAL n)|P = TopSpaceMetr((Euclid n)|Q); theorem :: TOPMETR:21 for r being real number for M being triangle MetrStruct, p being Point of M for P being Subset of TopSpaceMetr(M) st P = Ball(p,r) holds P is open; theorem :: TOPMETR:22 for P being Subset of TopSpaceMetr(M) holds P is open iff for p being Point of M st p in P ex r being real number st r>0 & Ball(p,r) c= P; definition let M be MetrStruct; attr M is compact means :: TOPMETR:def 6 TopSpaceMetr(M) is compact; end; theorem :: TOPMETR:23 M is compact iff for F st F is_ball-family & F is_a_cover_of M ex G st G c= F & G is_a_cover_of M & G is finite; :: REAL as topological space definition func R^1 -> strict TopSpace equals :: TOPMETR:def 7 TopSpaceMetr(RealSpace); end; definition cluster R^1 -> non empty; end; theorem :: TOPMETR:24 the carrier of R^1 = REAL; definition let C be set, f be PartFunc of C, the carrier of R^1, x be set; cluster f.x -> real; end; definition let a,b be real number; func Closed-Interval-TSpace(a,b) -> strict non empty SubSpace of R^1 equals :: TOPMETR:def 8 TopSpaceMetr(Closed-Interval-MSpace(a,b)); end; theorem :: TOPMETR:25 a <= b implies the carrier of Closed-Interval-TSpace(a,b) = [. a,b .]; theorem :: TOPMETR:26 a <= b implies for P being Subset of R^1 st P = [. a,b .] holds Closed-Interval-TSpace(a,b) = R^1|P; theorem :: TOPMETR:27 Closed-Interval-TSpace(0,1) = I[01]; definition redefine func I[01] -> strict SubSpace of R^1; end; theorem :: TOPMETR:28 for f being map of R^1,R^1 st ex a,b being Real st for x being Real holds f.x = a*x + b holds f is continuous;

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