| Real-valued functions are useful tools for the characterization of some topological spaces.Many classes of spaces can be characterized with real-valued functions that satisfy certain conditions,such as topological spaces,normal spaces,semi-stratifiable spaces,Perfectly normal spaces,etc.For example,consider the following characterization of semi-stratifiable spaces:A space X is k-semi-stratifiable spaces if and only if there is an function d:X×τc →[0,1]such that:(a)for each x∈X,d(x,F)=0 if and only if x∈F;(b)if F,H∈τand F(?)H,then d(x,F)≥d(x,H);(c)for each F∈τc,d(x,F)is upper Semi-continuous and k-Lower Semi-continuous on x.In this paper,we present some characterizations of perfectly normal spaces,and stratifiable spaces in terms of real-valued functions.The paper is divided into three parts.The first chapter deals with the basic notions and notations and the related background of the research.In chapter 2,some characterizations of Perfectly normal spaces are given and The main results are:For a space X,the following are equivalent:(a)Xis Perfectly normal spaces;(b)there exist a map:Φ:L(X)→C(X),such that for each h ∈ L(X),Φ(h)≤ h,and if h(x)>0,there is an open neighborhood Ox and r>0 for each x’∈Ox,Φ>(h)(x’)≥r;there exist a map:Φ:L(X))C(X),such thatτfor each h ∈ L(X),Φ(h)≤h,and if h(x)>0,then Φ(h)(x)>0.In chapter 3,we present some characterization of stratifiable spaces.The main results is:For a space X,the following are equivalent:(x)Xis a stratifiable space;(h)there exist an order-preserving map:Ψ ’:L(X)→(X),Φ:L(X)→UKL(X)such that for each h ∈ L(X),then Ψ(h)≤Φ(h)≤ h,and if h(x)>0,then Ψ(h)(x)>0.(c)there exist an order-preserving map:Ψ:L(X)→L(X),Φ:L(X)→U(X),such that for each h∈L(X),then Ψ(h)≤Φ(h)≤h and if h(x)>0,then Ψ(h)(x)>0... |
