On Gs-convergence and Gs-countable compactness

Abstract

G-convergence and G-sequential convergence are two convergent methods of sequences on a set, but they do not imply each other. How to further find the connection between these two types of convergence on a set, and establish their relationships with certain topological spaces?

In this paper, G-sequential convergence is referred to as $G_s$-convergence. $G_s$-convergence on a set X is proved to be exactly the usual convergence of sequences in the G-open topological space $X_G$ and the $G_s$-open topological space $X_{G_s}$, which enriches our understanding of subset properties and prompts us to study G-convergence and $G_s$-convergence on topological spaces. Some relationships among G-hulls, $G_s$-hulls, G-closures and $G_s$-closures are discussed. Various types of countably compact-like properties in the sense of G-convergence or $G_s$-convergence are characterized and found to be interrelated. It is proved that for a method G on a set X, X is $G_s$-hull countably compact if and only if the space $X_G$ is sequentially compact, if and only if $X_{G_s}$ is countably compact. The G-product property of G-sequential compactness and G-countable compactness are studied, and the following question is negatively answered: Is G-sequential compactness preserved by the finite G-products?

This study establishes the position of $G_s$-convergence in G-convergence and the usual convergence of sequences, and also shows that $G_s$-convergence provides a feasible approach for revealing richer properties of G-convergence.