The weakly Rothberger property of Pixley--Roy hyperspaces

Abstract

Let $\mathrm{PR}(X)$ denote the hyperspace of nonempty finite subsets of a topological space $X$ with Pixley--Roy topology.
In this paper, by introducing closed-miss-finite networks and using principle ultrafilters, we proved that the following statements are equivalent for a space $X$:\\
(1) $\mathrm{PR}(X)$ is weakly Rothberger;\\
(2) $X$ satisfies $\textsf{S}_{1}(\Pi_{rcf},\Pi_{wrcf})$;\\
(3) $X$ is separable and $X-\{x\}$ satisfies $\textsf{S}_{1}(\Pi_{cf},\Pi_{wcf})$ for each $x\in X$;\\
(4) $X$ is separable and each principle ultrafilter $\mathcal{F}[x]$ in $\mathrm{PR}(X)$ is weakly Rothberger in $\mathrm{PR}(X)$.\\
We also characterize the weakly Menger property and the weakly Hurewicz property of $\mathrm{PR}(X)$.