A Note on Almost Completely Regular Spaces and $c$-Semistratifiable Spaces

Abstract:

      In this paper, we give some characterizations of almost completely regular spaces and $c$-semistratifiable spaces (CSS) by semi-continuous functions. We mainly show that: (1) Let $X$ be a space. Then the following statements are equivalent: (i)~~$X$ is almost completely regular. (ii)~~Every two disjoint subsets of $X$, one of which is compact and the other is regular closed, are completely separated. (iii)~~If $g,h: X \rightarrow \mathbb{I}$, $g$ is compact-like, $h$ is normal lower semicontinuous, and $g \leq h$, then there exists a continuous function $f:X\rightarrow \mathbb{I}$ such that $g \leq f \leq h$; and (2) Let $X$ be a space. Then the following statements are equivalent: (a)~~$X$ is CSS; (b)~~There is an operator $U$ assigning to a decreasing sequence of compact sets $(F_{j})_{j\in \mathbb{N}}$, a decreasing sequence of open sets $(U(n,(F_{j})))_{n\in N}$ such that (b1)~~$F_{n}\subseteq U(n,(F_{j}))$ for each $n\in\mathbb{N}$; (b2)~~$\bigcap_{n\in\mathbb{N}}U(n,(F_{j}))=\bigcap_{n\in\mathbb{N}}F_{n}$; (b3)~~Given two decreasing sequences of compact sets $(F_{j})_{j\in \mathbb{N}}$ and $(E_{j})_{j\in\mathbb{N}}$ such that $F_{n}\subseteq E_{n}$ for each $n\in\mathbb{N}$, then $U(n,(F_{j}))\subseteq U(n,(E_{j}))$ for each $n\in\mathbb{N}$; (c)~~There is an operator $\Phi: {\rm LCL}(X,\mathbb{I})\rightarrow {\rm USC}(X,\mathbb{I})$ such that, for any $h\in {\rm LCL}(X,\mathbb{I})$, $0\leqslant\Phi(h)\leqslant h$, and $0<\Phi(h)(x)0$.