Characterizations of δ-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 $g:N\times X\to \tau$ satisfiesthat if $FınRG(X)$ and $y\notin F$, then there is an $mınN$such that $y\notin \overline{g(m,F)}$;ıtem{(2)} If there is an order preserving map $\phi :USC(X)\to LSC(X)$ suchthat for any $hınUSC(X),0\le \phi (h)\le h$ and$0<\phi (h)(x)<h(x)$ whenever $h(x)>0$, then $X$ is $\delta$-stratifiable space.