Volume 7, 1995

University of Bialystok

Copyright (c) 1995 Association of Mizar Users

### The abstract of the Mizar article:

### Dyadic Numbers and T$_4$ Topological Spaces

**by****Jozef Bialas, and****Yatsuka Nakamura**- Received July 29, 1995
- MML identifier: URYSOHN1

- [ Mizar article, MML identifier index ]

environ vocabulary GROUP_1, ARYTM_3, PRALG_2, BOOLE, PRE_TOPC, FUNCT_1, FINSEQ_1, RELAT_1, ARYTM_1, CONNSP_2, TOPS_1, SUBSET_1, SETFAM_1, TARSKI, TOPMETR, COMPTS_1, URYSOHN1; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, NUMBERS, XREAL_0, DOMAIN_1, FUNCT_1, FUNCT_2, REAL_1, FINSEQ_1, NAT_1, TOPMETR, COMPTS_1, STRUCT_0, PRE_TOPC, CARD_4, TOPS_1, CONNSP_2; constructors DOMAIN_1, REAL_1, NAT_1, TOPMETR, COMPTS_1, CARD_4, TOPS_1, MEMBERED; clusters SUBSET_1, STRUCT_0, PRE_TOPC, RELSET_1, TOPS_1, ARYTM_3, XREAL_0, MEMBERED, ZFMISC_1; requirements REAL, NUMERALS, BOOLE, SUBSET, ARITHM; begin definition func R<0 ->Subset of REAL means :: URYSOHN1:def 1 for x being Real holds x in it iff x < 0; end; definition func R>1 ->Subset of REAL means :: URYSOHN1:def 2 for x being Real holds x in it iff 1 < x; end; definition let n be Nat; func dyadic(n) -> Subset of REAL means :: URYSOHN1:def 3 for x being Real holds x in it iff ex i being Nat st (0 <= i & i <= (2|^n) & x = i/(2|^n)); end; definition func DYADIC -> Subset of REAL means :: URYSOHN1:def 4 for a being Real holds a in it iff ex n being Nat st a in dyadic(n); end; definition func DOM -> Subset of REAL equals :: URYSOHN1:def 5 R<0 \/ DYADIC \/ R>1; end; definition let T be TopSpace; let A be non empty Subset of REAL; let F be Function of A,bool the carrier of T; let r be Element of A; redefine func F.r -> Subset of T; end; canceled 4; theorem :: URYSOHN1:5 for n being Nat holds for x being Real st x in dyadic(n) holds 0 <= x & x <= 1; theorem :: URYSOHN1:6 dyadic(0) = {0,1}; theorem :: URYSOHN1:7 dyadic(1) = {0,1/2,1}; definition let n be Nat; cluster dyadic(n) -> non empty; end; theorem :: URYSOHN1:8 for x, n being Nat holds x|^n is Nat; theorem :: URYSOHN1:9 for n being Nat holds ex FS being FinSequence st (dom FS = Seg ((2|^n)+1) & for i being Nat st i in dom FS holds FS.i = (i-1)/(2|^n)); definition let n be Nat; func dyad(n) -> FinSequence means :: URYSOHN1:def 6 dom it = Seg((2|^n)+1) & for i being Nat st i in dom it holds it.i = (i-1)/(2|^n); end; theorem :: URYSOHN1:10 for n being Nat holds dom dyad(n) = Seg((2|^n)+1) & rng dyad(n) = dyadic(n); definition cluster DYADIC -> non empty; end; definition cluster DOM -> non empty; end; theorem :: URYSOHN1:11 for n being Nat holds dyadic(n) c= dyadic(n+1); theorem :: URYSOHN1:12 for n being Nat holds 0 in dyadic(n) & 1 in dyadic(n); theorem :: URYSOHN1:13 for n,i being Nat st 0 < i & i <= 2|^n holds (i*2-1)/(2|^(n+1)) in dyadic(n+1) \ dyadic(n); theorem :: URYSOHN1:14 for n,i being Nat st 0 <= i & i < 2|^n holds (i*2+1)/(2|^(n+1)) in dyadic(n+1) \ dyadic(n); theorem :: URYSOHN1:15 for n being Nat holds 1/(2|^(n+1)) in dyadic(n+1) \ dyadic(n); definition let n be Nat; let x be Element of dyadic(n); func axis(x,n) -> Nat means :: URYSOHN1:def 7 x = it/(2|^n); end; theorem :: URYSOHN1:16 for n being Nat holds for x being Element of dyadic(n) holds x = axis(x,n)/(2|^n) & 0 <= axis(x,n) & axis(x,n) <= (2|^n); theorem :: URYSOHN1:17 for n being Nat holds for x being Element of dyadic(n) holds (axis(x,n)-1)/(2|^n) < x & x < (axis(x,n)+1)/(2|^n); theorem :: URYSOHN1:18 for n being Nat holds for x being Element of dyadic(n) holds ((axis(x,n)-1)/(2|^n) < (axis(x,n)+1)/(2|^n)); canceled; theorem :: URYSOHN1:20 for n being Nat holds for x being Element of dyadic(n+1) holds (not x in dyadic(n) implies ((axis(x,n+1)-1)/(2|^(n+1)) in dyadic(n) & (axis(x,n+1)+1)/(2|^(n+1)) in dyadic(n))); theorem :: URYSOHN1:21 for n being Nat holds for x1,x2 being Element of dyadic(n) st x1 < x2 holds axis(x1,n) < axis(x2,n); theorem :: URYSOHN1:22 for n being Nat holds for x1,x2 being Element of dyadic(n) st x1 < x2 holds x1 <= (axis(x2,n)-1)/(2|^n) & (axis(x1,n)+1)/(2|^n) <= x2; theorem :: URYSOHN1:23 for n being Nat holds for x1,x2 being Element of dyadic(n+1) st x1 < x2 & not x1 in dyadic(n) & not x2 in dyadic(n) holds (axis(x1,n+1)+1)/(2|^(n+1)) <= (axis(x2,n+1)-1)/(2|^(n+1)); begin :: :: Normal TopSpaces :: definition let T be non empty TopSpace; let x be Point of T; redefine mode a_neighborhood of x means :: URYSOHN1:def 8 ex A being Subset of T st A is open & x in A & A c= it; synonym Nbhd of x,T; end; theorem :: URYSOHN1:24 for T being non empty TopSpace for A being Subset of T holds A is open iff (for x being Point of T holds x in A implies ex B being Nbhd of x,T st B c= A); canceled; theorem :: URYSOHN1:26 for T being non empty TopSpace holds for A being Subset of T holds (for x being Point of T st x in A holds A is Nbhd of x,T) implies A is open; definition let T be TopStruct; attr T is being_T1 means :: URYSOHN1:def 9 for p,q being Point of T st not p = q ex W,V being Subset of T st W is open & V is open & p in W & not q in W & q in V & not p in V; synonym T is_T1; end; theorem :: URYSOHN1:27 for T being non empty TopSpace holds T is_T1 iff for p being Point of T holds {p} is closed; definition let T be non empty TopSpace; let F be map of T,R^1; let x be Point of T; redefine func F.x -> Real; end; theorem :: URYSOHN1:28 for T being non empty TopSpace st T is_T4 holds for A,B being open Subset of T st A <> {} & Cl(A) c=B holds ex C being Subset of T st C <> {} & C is open & Cl(A) c= C & Cl(C) c= B; theorem :: URYSOHN1:29 for T being non empty TopSpace holds T is_T3 iff for A being open Subset of T for p being Point of T st p in A holds ex B being open Subset of T st p in B & Cl(B) c= A; theorem :: URYSOHN1:30 for T being non empty TopSpace st T is_T4 & T is_T1 holds for A being open Subset of T st A <> {} holds ex B being Subset of T st B <> {} & Cl(B) c= A; theorem :: URYSOHN1:31 for T being non empty TopSpace st T is_T4 for A,B being Subset of T st A is open & B is closed & B <> {} & B c= A ex C being Subset of T st C is open & B c= C & Cl(C) c= A; begin :: :: Some increasing family of sets in normal space :: definition let T be non empty TopSpace; let A,B be Subset of T; assume T is_T4 & A <> {} & A is open & B is open & Cl(A) c= B; mode Between of A,B -> Subset of T means :: URYSOHN1:def 10 it <> {} & it is open & Cl(A) c= it & Cl(it) c= B; end; theorem :: URYSOHN1:32 for T being non empty TopSpace st T is_T4 for A,B being closed Subset of T st A <> {} & A misses B for n being Nat for G being Function of dyadic(n),bool the carrier of T st A c= G.0 & B = [#](T) \ G.1 & for r1,r2 being Element of dyadic(n) st r1 < r2 holds G.r1 is open & G.r2 is open & Cl(G.r1) c= G.r2 ex F being Function of dyadic(n+1),bool the carrier of T st A c= F.0 & B = [#](T) \ F.1 & for r1,r2,r being Element of dyadic(n+1) st r1 < r2 holds F.r1 is open & F.r2 is open & Cl(F.r1) c= F.r2 & (r in dyadic(n) implies F.r = G.r);

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