| Title:
|
A note on k-c-semistratifiable spaces and strong $\beta $-spaces (English) |
| Author:
|
Wang, Li-Xia |
| Author:
|
Peng, Liang-Xue |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
136 |
| Issue:
|
3 |
| Year:
|
2011 |
| Pages:
|
287-299 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
c-semistratifiable space |
| Keyword:
|
k-c-semistratifiable space |
| Keyword:
|
submesocompact space |
| Keyword:
|
$g$ function |
| Keyword:
|
strong $\beta $-space |
| MSC:
|
54D20 |
| MSC:
|
54E20 |
| idZBL:
|
Zbl 1249.54063 |
| idMR:
|
MR2893977 |
| DOI:
|
10.21136/MB.2011.141650 |
| . |
| Date available:
|
2011-09-22T14:58:43Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141650 |
| . |
| Reference:
|
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