Covering properties at a subset on the Vietoris hyperspace $\mathcal F(X)$

Abstract

In this paper, we study covering properties at a subset on the hyperspace $\mathcal F(X)$ of finite subsets of a space $X$ endowed with the Vietoris topology. We prove that $X$ has the covering property $\gamma$ at $A$ if and only if $\mathcal F(X)$ has the covering property $\gamma$ at $\langle A\rangle_{\mathcal F(X)}$ for each $A\subset X$ and some $\gamma$. By these results, we obtain some results related to the images of metric spaces under some kinds of continuous mappings at a subset on the Vietoris hyperspace $\mathcal F(X)$.