The Vietoris hyperspace $\mathcal F(X)$ and certain generalized metric properties

Abstract

In this paper, we study several generalized metric properties of the space $\mathcal F(X)$ of finite subsets of a space $X$ endowed with the Vietoris topology. In particular, we consider such properties $(P)$ for which $\mathcal F(X)$ has $(P)$ if and only if $X$ has $(P)$. Also, we obtain some results related to the images of metric spaces under some kinds of continuous mappings.

Keywords

• hyperspace
• generalized metric properties
• $cs$-cover
• $cs^*$-cover
• $sn$-symmetric space
• (weak) Cauchy symmetric space
• (weak) Cauchy $sn$-symmetric space

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