The quasi-Rothberger property of Pixley--Roy hyperspaces

Abstract

Let $\mathrm{PR}(X)$ denote the hyperspace of non-empty finite subsets of a topological space $X$ with Pixley--Roy topology. In this paper, we investigate the quasi-Rothberger property in hyperspace $\mathrm{PR}(X)$. We prove that for a space $X$, the followings are equivalent:

(1) $\mathrm{PR}(X)$ is quasi-Rothberger;

(2) $X$ satisfies $\textsf{S}_{1}(\Pi_{rcf-h},\Pi_{wrcf-h})$;

(3) $X$ is separable and each co-finite subset of $X$ satisfies $\textsf{S}_{1}(\Pi_{pcf-h},\Pi_{wpcf-h})$;

(4) $X$ is separable and $\mathrm{PR}(Y)$ is quasi-Rothberger for each co-finite subset $Y$ of $X$.\\
We also characterize the quasi-Menger property and the quasi-Hurewicz property of $\mathrm{PR}(X)$. These answer the questions posted in [8].