Remarks on $R$-separability 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, motivated by [4], we introduced $cf$-covers and $rcf$-covers of $X$ to establish the $R$-selective separability and the $M$-selective separability in $\mathrm{PR}(X)$ under the Pixley--Roy topology. We proved that the following statements are equivalent for a space $X$:\\
(1) $\mathrm{PR}(X)$ is $R$-separable\ (resp., $M$-separable);\\
(2) $X$ satisfies $\textsf{S}_{1}(\mathcal{C}_{rcf},\mathcal{C}_{rcf})\ (resp., \textsf{S}_{\textsf{fin}}(\mathcal{C}_{rcf},\mathcal{C}_{rcf}))$;\\
(3) $X$ is countable and each co-finite subset of $X$ satisfies $\textsf{S}_{1}(\mathcal{C}_{cf},\mathcal{C}_{cf})\ (resp., \textsf{S}_{\textsf{fin}}(\mathcal{C}_{cf},\mathcal{C}_{cf}))$;\\
(4) $X$ is countable and $\mathrm{PR}(X)$ has countable strong fan tightness\ (resp., $\mathrm{PR}(X)$ has countable fan tightness).