Volume 10, 1998

University of Bialystok

Copyright (c) 1998 Association of Mizar Users

### The abstract of the Mizar article:

### On \Tone\ Reflex of Topological Space

**by****Adam Naumowicz, and****Mariusz \Lapinski**- Received March 7, 1998
- MML identifier: T_1TOPSP

- [ Mizar article, MML identifier index ]

environ vocabulary FUNCT_1, RELAT_1, PRE_TOPC, EQREL_1, BORSUK_1, TARSKI, BOOLE, SUBSET_1, SETFAM_1, PROB_1, URYSOHN1, ORDINAL2, T_1TOPSP; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, RELAT_1, FUNCT_1, FUNCT_2, SETFAM_1, STRUCT_0, PRE_TOPC, EQREL_1, TOPS_2, URYSOHN1, BORSUK_1; constructors URYSOHN1, TOPS_2; clusters PRE_TOPC, BORSUK_1, STRUCT_0, FSM_1, FUNCT_1, RELSET_1, SUBSET_1, XBOOLE_0, EQREL_1, PARTFUN1; requirements BOOLE, SUBSET; begin reserve X for non empty set, x for Element of X, y,v,w for set; canceled; theorem :: T_1TOPSP:2 for T being non empty TopSpace, A being non empty a_partition of the carrier of T, y being Subset of space A holds (Proj(A))"y = union y; theorem :: T_1TOPSP:3 for X being non empty set, S being a_partition of X, A being Subset of S holds (union S) \ (union A) = union (S \ A); theorem :: T_1TOPSP:4 for X being non empty set,A being Subset of X, S being a_partition of X holds A in S implies union(S \ {A}) = X \ A; theorem :: T_1TOPSP:5 for T being non empty TopSpace, S being non empty a_partition of the carrier of T, A being Subset of space S, B being Subset of T holds B = union A implies (A is closed iff B is closed); :: Classes of partitions of a set definition let X be non empty set,x be Element of X,S1 be a_partition of X; func EqClass(x,S1) -> Subset of X means :: T_1TOPSP:def 1 x in it & it in S1; end; theorem :: T_1TOPSP:6 for S1,S2 being a_partition of X st (for x being Element of X holds EqClass(x,S1) = EqClass(x,S2)) holds S1=S2; theorem :: T_1TOPSP:7 for X being non empty set holds {X} is a_partition of X; definition let X be set; mode Family-Class of X means :: T_1TOPSP:def 2 it c= bool bool X; end; definition let X be set; let F be Family-Class of X; attr F is partition-membered means :: T_1TOPSP:def 3 for S being set st S in F holds S is a_partition of X; end; definition let X be set; cluster partition-membered Family-Class of X; end; definition let X be set; mode Part-Family of X is partition-membered Family-Class of X; end; reserve F for Part-Family of X; definition let X be non empty set; cluster non empty a_partition of X; end; theorem :: T_1TOPSP:8 for X being set, p being a_partition of X holds {p} is Part-Family of X; definition let X be set; cluster non empty Part-Family of X; end; theorem :: T_1TOPSP:9 for S1 being a_partition of X, x,y being Element of X holds EqClass(x,S1) meets EqClass(y,S1) implies EqClass(x,S1) = EqClass(y,S1); theorem :: T_1TOPSP:10 for A being set,X being non empty set,S being a_partition of X holds A in S implies ex x being Element of X st A = EqClass(x,S); definition let X be non empty set,F be non empty Part-Family of X; func Intersection F -> non empty a_partition of X means :: T_1TOPSP:def 4 for x being Element of X holds EqClass(x,it) = meet{EqClass(x,S) where S is a_partition of X : S in F}; end; :: Families of partitions of topological spaces reserve T for non empty TopSpace; theorem :: T_1TOPSP:11 { A where A is a_partition of the carrier of T : A is closed } is Part-Family of the carrier of T; definition let T; func Closed_Partitions T -> non empty Part-Family of the carrier of T equals :: T_1TOPSP:def 5 { A where A is a_partition of the carrier of T : A is closed }; end; :: T_1 reflex of a topological space definition let T be non empty TopSpace; func T_1-reflex T -> TopSpace equals :: T_1TOPSP:def 6 space (Intersection Closed_Partitions T); end; definition let T; cluster T_1-reflex T -> strict non empty; end; theorem :: T_1TOPSP:12 for T being non empty TopSpace holds T_1-reflex T is being_T1; definition let T; cluster T_1-reflex T -> being_T1; end; definition let T be non empty TopSpace; func T_1-reflect T -> continuous map of T,T_1-reflex T equals :: T_1TOPSP:def 7 Proj Intersection Closed_Partitions T; end; theorem :: T_1TOPSP:13 for T,T1 being non empty TopSpace,f being continuous map of T,T1 holds T1 is being_T1 implies ({f"{z} where z is Element of T1 : z in rng f} is a_partition of the carrier of T) & (for A being Subset of T st A in {f"{z} where z is Element of T1 : z in rng f} holds A is closed); theorem :: T_1TOPSP:14 for T,T1 being non empty TopSpace,f being continuous map of T,T1 holds T1 is being_T1 implies for w for x being Element of T holds w = EqClass(x,(Intersection Closed_Partitions T)) implies w c= f"{f.x}; theorem :: T_1TOPSP:15 for T,T1 being non empty TopSpace,f being continuous map of T,T1 holds T1 is being_T1 implies for w st w in the carrier of T_1-reflex T ex z being Element of T1 st z in rng f & w c= f"{z}; :: The theorem on factorization theorem :: T_1TOPSP:16 for T,T1 being non empty TopSpace,f being continuous map of T,T1 holds T1 is being_T1 implies ex h being continuous map of T_1-reflex T, T1 st f = h*T_1-reflect T; definition let T,S be non empty TopSpace; let f be continuous map of T,S; func T_1-reflex f -> continuous map of T_1-reflex T, T_1-reflex S means :: T_1TOPSP:def 8 (T_1-reflect S) * f = it * (T_1-reflect T); end;

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