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Keywords:
c-semistratifiable space; k-c-semistratifiable space; submesocompact space; $g$ function; strong $\beta $-space
c-semistratifiable space; k-c-semistratifiable space; submesocompact space; $g$ function; strong $\beta $-space
Summary:
Recall that a space $X$ is a c-semistratifiable (CSS) space, if the compact sets of $X$ are $G_\delta $-sets in a uniform way. In this note, we introduce another class of spaces, denoting it by k-c-semistratifiable (k-CSS), which generalizes the concept of c-semistratifiable. We discuss some properties of k-c-semistratifiable spaces. We prove that a $T_2$-space $X$ is a k-c-semistratifiable space if and only if $X$ has a $g$ function which satisfies the following conditions: (1) For each $x\in X$, $\{ x\}=\bigcap \{g(x, n)\colon n\in \mathbb {N}\}$ and $ g(x, n+1)\subseteq g(x, n)$ for each $n\in \mathbb {N}$. (2) If a sequence $\{x_n\}_{n\in \mathbb {N}}$ of $X$ converges to a point $x\in X$ and $y_n\in g(x_n, n)$ for each $n\in \mathbb {N}$, then for any convergent subsequence $\{y_{n_k}\}_{k\in \mathbb {N}}$ of $\{y_n\}_{n\in \mathbb {N}}$ we have that $\{y_{n_k}\}_{k\in \mathbb {N}}$ converges to $x$. By the above characterization, we show that if $X$ is a submesocompact locally k-c-semistratifiable space, then $X$ is a k-c-semistratifible space, and the countable product of k-c-semistratifiable spaces is a k-c-semistratifiable space. If $X=\bigcup \{{\rm Int}(X_n)\colon n\in \mathbb {N}\}$ and $X_n$ is a closed k-c-semistratifiable space for each $n$, then $X$ is a k-c-semistratifiable space. In the last part of this note, we show that if $X=\bigcup \{X_n\colon n\in \mathbb {N}\}$ and $X_n$ is a closed strong $\beta $-space for each $n\in \mathbb {N}$, then $X$ is a strong $\beta $-space.
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