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

Wang, Li-Xia, and Peng, Liang-Xue. "A note on k-c-semistratifiable spaces and strong $\beta $-spaces." Mathematica Bohemica 136.3 (2011): 287-299. <http://eudml.org/doc/197240>.

@article{Wang2011,
abstract = {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 $x\in X$, $\lbrace x\rbrace =\bigcap \lbrace g(x, n)\colon n\in \mathbb \{N\}\rbrace $ and $ g(x, n+1)\subseteq g(x, n)$ for each $n\in \mathbb \{N\}$. (2) If a sequence $\lbrace x_n\rbrace _\{n\in \mathbb \{N\}\}$ 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 $\lbrace y_\{n_k\}\rbrace _\{k\in \mathbb \{N\}\}$ of $\lbrace y_n\rbrace _\{n\in \mathbb \{N\}\}$ we have that $\lbrace y_\{n_k\}\rbrace _\{k\in \mathbb \{N\}\}$ 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 \lbrace \{\rm Int\}(X_n)\colon n\in \mathbb \{N\}\rbrace $ 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 \lbrace X_n\colon n\in \mathbb \{N\}\rbrace $ and $X_n$ is a closed strong $\beta $-space for each $n\in \mathbb \{N\}$, then $X$ is a strong $\beta $-space.},
author = {Wang, Li-Xia, Peng, Liang-Xue},
journal = {Mathematica Bohemica},
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 -space},
language = {eng},
number = {3},
pages = {287-299},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A note on k-c-semistratifiable spaces and strong $\beta $-spaces},
url = {http://eudml.org/doc/197240},
volume = {136},
year = {2011},
}

TY - JOUR
AU - Wang, Li-Xia
AU - Peng, Liang-Xue
TI - A note on k-c-semistratifiable spaces and strong $\beta $-spaces
JO - Mathematica Bohemica
PY - 2011
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 136
IS - 3
SP - 287
EP - 299
AB - 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 $x\in X$, $\lbrace x\rbrace =\bigcap \lbrace g(x, n)\colon n\in \mathbb {N}\rbrace $ and $ g(x, n+1)\subseteq g(x, n)$ for each $n\in \mathbb {N}$. (2) If a sequence $\lbrace x_n\rbrace _{n\in \mathbb {N}}$ 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 $\lbrace y_{n_k}\rbrace _{k\in \mathbb {N}}$ of $\lbrace y_n\rbrace _{n\in \mathbb {N}}$ we have that $\lbrace y_{n_k}\rbrace _{k\in \mathbb {N}}$ 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 \lbrace {\rm Int}(X_n)\colon n\in \mathbb {N}\rbrace $ 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 \lbrace X_n\colon n\in \mathbb {N}\rbrace $ and $X_n$ is a closed strong $\beta $-space for each $n\in \mathbb {N}$, then $X$ is a strong $\beta $-space.
LA - eng
KW - 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 -space
UR - http://eudml.org/doc/197240
ER -