Funcation Characterizations Of Some Spaces In Which Compacta Are G_δ

In General Topology,real-valued functions are useful tools for the characterization of some topological spaces.Many important classes of spaces such as stratifiable spaces,k-semi-stratifiable spaces and perfectly normal spaces can be characterized with real-valued functions that satisfy certain conditions.In this paper,we present characterizations of some spaces in which compacta are Gδ in terms of real-valued functions,such as c-stratifiable spaces,kc-semi-stratifiable spaces and y-spaces,etc.The first chapter deals with the related background of the research,the basic notations and the main results.In the second chapter,we study the function characterizations of some spaces in which compacta are Gδ.Three types of characterizations of c-stratifiable spaces and kc-semi-stratifiable spaces are given by using semi-continuous functions.For example,we show that X is a c-stratifiable space if and only if one of the following conditions is satisfied.(a)For each K ∈ C(X),there exist fK ∈ L(X),hK ∈UKL(X)and fK ≤hk satisfying:(1)K=(?)(0)=(?)(0);(2)if K1(?)K2 then hK1≥hK2.(b)For each K∈C(X),there exists fK ∈ L(X),hK ∈ U(X)and fK≤hK satisfying:(1)K=(?)(0)=(?);(2)if K1∈ K2 then hK1≥hK2.(c)There exists a family F(?)L(X)satisfying:(1)if(?)K ∈C(X),there exists f∈F such that f(x)>0 and f(K)={0};(2)for each X∈X,F’(?)Fand ε>0,if F’(x)＝{0} then there exists an open neighborhood V of x such that F’(V)(?)[0,ε).(d)There exists a family F(?)F(X)satisfying:(1)if x(?)k∈ C(X),then there exists f∈F an open neighborhood Vof x and m∈N such thatf(v)(?)(1/m,1]and f(K)={0};(2)for each xeX,F’and ε>0,if F’(x)={0} then there exists an open neighborhood U of x such that F;(V)(?)[0,ε).(e)For each K ∈ C(X),there extist decreasing {δnK ∈L(X):n ∈N} and{ξnk∈U(X):n ∈N} satisfying:(1)(?)(2)if K1,K2∈C(X)and K1(?)K2 then for each n∈N,δnK1≤ δnK2;(3)for each K∈c(X)and n∈N,δnK ≤ξnk.In the third chapter,we give characterizations of some generalized metric spaces with real-valued functions,such as y-spaces,Nagata spaces and first countable spaces.For example,we show that X is a y-space if and only if for each K∈C(X),there exists a decreasing sequence {δnK∈L(X):n∈N} of functions satisfying:(1)(?);(2)if K1,K2∈C(X)and K1(?)K2 then for each n∈N,δnK1≤δnK2;(3)for each K∈C(X),F∈τ° with K∩F=(?),there exists m∈N such that for each x∈F,δmk(x)=0.