About $wcs$-covers and $wcs^*$-networks on the Vietoris hyperspace $\mathcal F(X)$

We study some generalized metric properties on the hyperspace $\mathcal F(X)$ of finite subsets of a space $X$ endowed with the Vietoris topology. We prove that $X$ has a point-star network consisting of (countable) $wcs$-covers if and only if so does $\mathcal F(X)$. Moreover, $X$ has a sequence of $wcs$-covers with property $(P)$ which is a point-star network if and only if so does $\mathcal F(X)$, where $(P)$ is one of the following properties: point-finite, point-countable, compact-finite, compact-countable, locally finite, locally countable. On the other hand, $X$ has a $wcs^*$-network with property $\sigma$-$(P)$ if and only if so does $\mathcal F(X)$. By these results, we obtain some results related to the images of metric spaces and separable metric spaces under some kinds of continuous mappings on the Vietoris hyperspace $\mathcal F(X)$.