# A note on k-c-semistratifiable spaces and strong $\beta$-spaces

Mathematica Bohemica (2011)

• Volume: 136, Issue: 3, page 287-299
• ISSN: 0862-7959

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## Abstract

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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 $x\in X$, $\left\{x\right\}=\bigcap \left\{g\left(x,n\right):n\in ℕ\right\}$ and $g\left(x,n+1\right)\subseteq g\left(x,n\right)$ for each $n\in ℕ$. (2) If a sequence ${\left\{{x}_{n}\right\}}_{n\in ℕ}$ of $X$ converges to a point $x\in X$ and ${y}_{n}\in g\left({x}_{n},n\right)$ for each $n\in ℕ$, then for any convergent subsequence ${\left\{{y}_{{n}_{k}}\right\}}_{k\in ℕ}$ of ${\left\{{y}_{n}\right\}}_{n\in ℕ}$ we have that ${\left\{{y}_{{n}_{k}}\right\}}_{k\in ℕ}$ 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 $X=\bigcup \left\{\mathrm{Int}\left({X}_{n}\right):n\in ℕ\right\}$ 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 $X=\bigcup \left\{{X}_{n}:n\in ℕ\right\}$ and ${X}_{n}$ is a closed strong $\beta$-space for each $n\in ℕ$, then $X$ is a strong $\beta$-space.

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