Characterizations of $\delta$-stratifiable spaces

Abstract

In this paper, we give some characterizations of$\delta$-stratifiable spaces by means of $g$-functions and semi-continuousfunctions. It is established that:ıtem{(1)} A topological space $X$ in which every point is a regular$G_\delta$-set is $\delta$-stratifiable if and only if thereexists a $g$-function $g:N\times X\rightarrow \tau $ satisfiesthat if $Fın RG(X)$ and $y\notin F$, then there is an $mın N$such that $y\notin \overline{g(m,F)}$;ıtem{(2)} If there is an order preserving map $\varphi:USC(X)\rightarrow LSC(X) $ suchthat for any $hın USC(X),0\leq \varphi(h)\leq h$ and$0<\varphi(h)(x)<h(x)$ whenever $h(x)>0$, then $X$ is $\delta$-stratifiable space.