We study some generalized metric properties on the hyperspace 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 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 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 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 F ( X ).
