TY - JOUR
T1 - On $G$-Quotient Mappings and Networks Defined by $G$-Convergence
AU - Liu , Fang
AU - Zhou , Xiangeng
AU - Liu , Li
JO - Journal of Mathematical Study
VL - 2
SP - 135
EP - 146
PY - 2023
DA - 2023/07
SN - 56
DO - http://doi.org/10.4208/jms.v56n2.23.02
UR - https://global-sci.org/intro/article_detail/jms/21840.html
KW - $G$-methods, $G$-convergence, $G$-quotient mappings, $G$-$cs′$
-networks, $G$-kernel covers, $G$-Frechet spaces.
AB - <p style="text-align: justify;"><span style="text-align: justify;">Let $G_{1}, G_{2}$ be methods on topological spaces $X$ and $Y$ respectively,&nbsp; $f:X\rightarrow Y$ be a mapping, $\mathscr{P}$ be a cover of $X$. $f$ is said to be a $(G_{1}, G_{2})$-quotient mapping&nbsp; provided $f^{-1}(U)$ is $G_{1}$-open in $X$, then $U$ is $G_{2}$-open in $Y$. $\mathscr{P}$ is called a $G$-$cs&#39;$-network of $X$ if&nbsp; whenever $x=\{x_n\}_{n\in\mathbb{N}}\in c_{G}(X)$&nbsp; and $G(x)=x\in U$ with $U$ open in $X$, then there exists some $n_{0}\in \mathbb{N}$ such that $\{x ,x_{n_{0}}\}\subset P \subset U$ for some $P\in \mathscr{P}$. $\mathscr{P}$ is called a $G$-kernel cover of $X$ if $\{(U)_{G}:U\in \mathscr{P}\}$ is a cover of $X$. In this paper, we introduce the concepts of $(G_{1}, G_{2})$-quotient mappings, $G$-$cs&#39;$-networks and $G$-kernel covers of $X$, and study some characterizations of $(G_{1}, G_{2})$-quotient mappings,&nbsp; $G$-$cs&#39;$-networks,&nbsp; and $G$-kernel covers of $X$. In particular, we obtain that if $G$ is a subsequential method and $X$ is a $G$-Fréchet space with a point-countable $G$-$cs&#39;$-network, then $X$ is a meta-Lindelöf space.</span></p>