Volume 4, 1992

University of Bialystok

Copyright (c) 1992 Association of Mizar Users

### The abstract of the Mizar article:

### Finite Topological Spaces

**by****Hiroshi Imura, and****Masayoshi Eguchi**- Received November 27, 1992
- MML identifier: FIN_TOPO

- [ Mizar article, MML identifier index ]

environ vocabulary FINSEQ_1, FINSET_1, BOOLE, ZFMISC_1, FUNCT_1, RELAT_1, ABSVALUE, ARYTM_1, CAT_1, FUNCT_2, SUBSET_1, PRE_TOPC, TARSKI, RELAT_2, SETFAM_1, CARD_1, FIN_TOPO, HAHNBAN, FINSEQ_4; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, ORDINAL1, NUMBERS, XCMPLX_0, XREAL_0, FINSET_1, NAT_1, SETFAM_1, STRUCT_0, RELSET_1, RELAT_1, FUNCT_1, FUNCT_2, CQC_LANG, DOMAIN_1, GROUP_1, FINSEQ_1, FINSEQ_4, CARD_1; constructors NAT_1, CQC_LANG, DOMAIN_1, GROUP_1, FINSEQ_4, MEMBERED, XBOOLE_0; clusters SUBSET_1, INT_1, RELSET_1, STRUCT_0, CQC_LANG, FINSEQ_1, NAT_1, XREAL_0, MEMBERED, ZFMISC_1, XBOOLE_0, NUMBERS, ORDINAL2; requirements REAL, NUMERALS, SUBSET, BOOLE, ARITHM; begin theorem :: FIN_TOPO:1 for A being set, f being FinSequence of bool A st (for i being Nat st 1 <= i & i < len f holds f/.i c= f/.(i+1)) for i, j being Nat st i <= j & 1 <= i & j <= len f holds f/.i c= f/.j; theorem :: FIN_TOPO:2 for A being set, f being FinSequence of bool A st (for i being Nat st 1 <= i & i < len f holds f/.i c= f/.(i+1)) for i, j being Nat st i < j & 1 <= i & j <= len f & f/.j c= f/.i for k being Nat st i <= k & k <= j holds f/.j = f/.k; theorem :: FIN_TOPO:3 for F being set st F is finite & F <> {} & F is c=-linear ex m being set st m in F & for C being set st C in F holds C c= m; canceled; theorem :: FIN_TOPO:5 for f being Function st (for i being Nat holds f.i c= f.(i+1)) for i, j being Nat st i <= j holds f.i c= f.j; scheme MaxFinSeqEx {X() -> non empty set, A() -> Subset of X(), B() -> Subset of X(), F(Subset of X()) -> Subset of X()}: ex f being FinSequence of bool X() st len f > 0 & f/.1=B() & (for i being Nat st i > 0 & i < len f holds f/.(i+1)=F(f/.i)) & F(f/.len f)=f/.len f & (for i, j being Nat st i > 0 & i < j & j <= len f holds f/.i c= A() & f/.i c< f/.j) provided A() is finite and B() c= A() and for A being Subset of X() st A c= A() holds A c= F(A) & F(A) c= A(); definition struct ( 1-sorted ) FT_Space_Str (# carrier -> set, Nbd -> Function of the carrier, bool the carrier #); end; definition cluster non empty strict FT_Space_Str; end; reserve FT for non empty FT_Space_Str; reserve x, y, z for Element of FT; definition let FT be non empty FT_Space_Str; let x be Element of FT; func U_FT x -> Subset of FT equals :: FIN_TOPO:def 1 ( the Nbd of FT ).x; end; definition let x be set, y be Subset of {x}; redefine func x.-->y -> Function of {x}, bool {x}; end; definition func FT{0} -> strict FT_Space_Str equals :: FIN_TOPO:def 2 FT_Space_Str (#{0},0.-->[#]{0}#); end; definition cluster FT{0} -> non empty; end; definition let IT be non empty FT_Space_Str; attr IT is filled means :: FIN_TOPO:def 3 for x being Element of IT holds x in U_FT x; end; canceled; theorem :: FIN_TOPO:7 FT{0} is filled; theorem :: FIN_TOPO:8 FT{0} is finite; definition cluster finite filled strict (non empty FT_Space_Str); end; definition let T be 1-sorted, F be set; canceled; pred F is_a_cover_of T means :: FIN_TOPO:def 5 the carrier of T c= union F; end; theorem :: FIN_TOPO:9 for FT being filled (non empty FT_Space_Str) holds {U_FT x where x is Element of FT : not contradiction} is_a_cover_of FT; reserve A for Subset of FT; definition let FT; let A be Subset of FT; func A^delta -> Subset of FT equals :: FIN_TOPO:def 6 {x:U_FT x meets A & U_FT x meets A` }; end; theorem :: FIN_TOPO:10 x in A^delta iff U_FT x meets A & U_FT x meets A`; definition let FT; let A be Subset of FT; func A^deltai -> Subset of FT equals :: FIN_TOPO:def 7 A /\ (A^delta); func A^deltao -> Subset of FT equals :: FIN_TOPO:def 8 A` /\ (A^delta); end; theorem :: FIN_TOPO:11 A^delta = A^deltai \/ A^deltao; definition let FT; let A be Subset of FT; func A^i -> Subset of FT equals :: FIN_TOPO:def 9 {x:U_FT x c= A}; func A^b -> Subset of FT equals :: FIN_TOPO:def 10 {x:U_FT x meets A}; func A^s -> Subset of FT equals :: FIN_TOPO:def 11 {x:x in A & U_FT x \ {x} misses A }; end; definition let FT; let A be Subset of FT; func A^n -> Subset of FT equals :: FIN_TOPO:def 12 A \ A^s; func A^f -> Subset of FT equals :: FIN_TOPO:def 13 {x:ex y st y in A & x in U_FT y}; end; definition let IT be non empty FT_Space_Str; attr IT is symmetric means :: FIN_TOPO:def 14 for x, y being Element of IT holds y in U_FT x implies x in U_FT y; end; theorem :: FIN_TOPO:12 x in A^i iff U_FT x c= A; theorem :: FIN_TOPO:13 x in A^b iff U_FT x meets A; theorem :: FIN_TOPO:14 x in A^s iff x in A & U_FT x \ {x} misses A; theorem :: FIN_TOPO:15 x in A^n iff x in A & U_FT x \ {x} meets A; theorem :: FIN_TOPO:16 x in A^f iff ex y st y in A & x in U_FT y; theorem :: FIN_TOPO:17 FT is symmetric iff for A holds A^b = A^f; reserve F for Subset of FT; definition let FT; let IT be Subset of FT; attr IT is open means :: FIN_TOPO:def 15 IT = IT^i; attr IT is closed means :: FIN_TOPO:def 16 IT = IT^b; attr IT is connected means :: FIN_TOPO:def 17 for B,C being Subset of FT st IT = B \/ C & B <> {} & C <> {} & B misses C holds B^b meets C; end; definition let FT; let A be Subset of FT; func A^fb -> Subset of FT equals :: FIN_TOPO:def 18 meet{F:A c= F & F is closed}; func A^fi -> Subset of FT equals :: FIN_TOPO:def 19 union{F:A c= F & F is open}; end; theorem :: FIN_TOPO:18 for FT being filled (non empty FT_Space_Str), A being Subset of FT holds A c= A^b; theorem :: FIN_TOPO:19 for FT being non empty FT_Space_Str, A, B being Subset of FT holds A c= B implies A^b c= B^b; theorem :: FIN_TOPO:20 for FT being filled finite (non empty FT_Space_Str), A being Subset of FT holds A is connected iff for x being Element of FT st x in A ex S being FinSequence of bool the carrier of FT st len S > 0 & S/.1 = {x} & (for i being Nat st i > 0 & i < len S holds S/.(i+1) = (S/.i)^b /\ A) & A c= S/.len S; theorem :: FIN_TOPO:21 for E being non empty set, A being Subset of E, x being Element of E holds x in A` iff not x in A; theorem :: FIN_TOPO:22 ((A`)^i)` = A^b; theorem :: FIN_TOPO:23 ((A`)^b)` = A^i; theorem :: FIN_TOPO:24 A^delta = (A^b) /\ ((A`)^b); theorem :: FIN_TOPO:25 (A`)^delta = A^delta; theorem :: FIN_TOPO:26 x in A^s implies not x in (A \ {x})^b; theorem :: FIN_TOPO:27 A^s <> {} & Card A <> 1 implies A is not connected; theorem :: FIN_TOPO:28 for FT being filled (non empty FT_Space_Str), A being Subset of FT holds A^i c= A; theorem :: FIN_TOPO:29 for E being set, A, B being Subset of E holds A = B iff A` = B`; theorem :: FIN_TOPO:30 A is open implies A` is closed; theorem :: FIN_TOPO:31 A is closed implies A` is open;

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