$c$-半层空间与几乎完全正则空间一点注记

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

中文关键词:  几乎完全正则  $c$-半层空间(CSS)  半连续函数

英文关键词:almost completely regular spaces  CSS  semi-continuous functions

中文摘要:

      本文用半连续函数给出了$c$-半层空间和几乎完全正则空间的一些等价刻画. 主要结论如下: (1) 设$X$ 是一拓扑空间, 那么如下等价:(i)~~$X$ 是几乎完全正则空间.(ii)~~$X$ 中任何两个不交的紧集和闭集是完全分离的.(iii)~ 设$g,h: X \rightarrow \mathbb{I}$, 如果~$g$ 是~compact-like, $h$ 是正规下半连续的(normal lower semicontinuous), 以及满足$g \leq h$, 那么存在一连续函数~ $f:X\rightarrow \mathbb{I}$ 满足~ $g \leq f \leq h$;(2)设~ $X$ 是一拓扑空间, 那么如下等价:(a)~~$X$ 是~$c$-半层空间(CSS);(b)~~存在一算子~ $U$ 对任意递减的紧集列~ $(F_{j})_{j\in \mathbb{N}}$, 指派到一递减的开集列~$(U(n,(F_{j})))_{n\in N}$ 使得如下成立(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)~~如果两个递减的紧集列~ $(F_{j})_{j\in \mathbb{N}}$ 和~ (E_{j})_{j\in\mathbb{N}}$ 满足~ $F_{n}\subseteq E_{n}$ for each $n\in\mathbb{N}$, 那么~ $U(n,(F_{j}))\subseteq U(n,(E_{j}))$ for each $n\in\mathbb{N}$; (c)~~存在一算子$\Phi: {\rm LCL}(X,\mathbb{I})\rightarrow {\rm USC}(X,\mathbb{I})$ 满足对任意~ $h\in {\rm LCL}(X,\mathbb{I})$ 有~$0\leqslant\Phi(h)\leqslant h$, 而且当~$h(x)>0$ 时有$0<\Phi(h)(x)

英文摘要:

      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 $0<\Phi(h)(x)0$.