Abstract: 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).
Keywords: hyperspace; generalized metric property; wcs-cover; wcs^*-network
DOI: DOI 10.14712/1213-7243.2024.011
AMS Subject Classification: 54B20 54C10 54D20 54E40