Characterizations of $ẹlta$-stratifiable spaces

Kedian Li

In this paper, we give some characterizations of $\delta$-stratifiable spaces by means of $g$-functions and semi-continuous functions. 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 there exists a $g$-function $g:N\times X\rightarrow \tau $ satisfies that if $F\in RG(X)$ and $y\notin F$, then there is an $m\in N$ such that $y\notin \overline{g(m,F)}$; ıtem{(2)} If there is an order preserving map $\varphi:USC(X)\rightarrow LSC(X) $ such that for any $h\in USC(X),0\leq \varphi(h)\leq h$ and $0<\varphi(h)(x)0$, then $X$ is $\delta$-stratifiable space.