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

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.