Year: 2023
Author: Fang Liu, Xiangeng Zhou, Li Liu
Journal of Mathematical Study, Vol. 56 (2023), Iss. 2 : pp. 135–146
Abstract
Let $G_{1}, G_{2}$ be methods on topological spaces $X$ and $Y$ respectively, $f:X\rightarrow Y$ be a mapping, $\mathscr{P}$ be a cover of $X$. $f$ is said to be a $(G_{1}, G_{2})$-quotient mapping provided $f^{-1}(U)$ is $G_{1}$-open in $X$, then $U$ is $G_{2}$-open in $Y$. $\mathscr{P}$ is called a $G$-$cs'$-network of $X$ if whenever $x=\{x_n\}_{n\in\mathbb{N}}\in c_{G}(X)$ 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, $G$-$cs'$-networks and $G$-kernel covers of $X$, and study some characterizations of $(G_{1}, G_{2})$-quotient mappings, $G$-$cs'$-networks, 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 $G$-$cs'$-network, then $X$ is a meta-Lindelöf space.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jms.v56n2.23.02
Journal of Mathematical Study, Vol. 56 (2023), Iss. 2 : pp. 135–146
Published online: 2023-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 12
Keywords: $G$-methods $G$-convergence $G$-quotient mappings $G$-$cs′$ -networks $G$-kernel covers $G$-Frechet spaces.