Generalized metric properties on hyperspaces with the Vietoris topology

Abstract

We present interrelations between several topological properties of, for example, spaces with a point-countable base, spaces with a $\sigma$-hereditarily closure-preserving $k$-network, spaces with a base of countable order, spaces with a point-regular base, Nagata-spaces, $c$-semi-stratifiable spaces, $\gamma$-spaces, or semi-metrizable spaces of $X$ and the hyperspaces of nonempty compact subsets and finite subsets with the Vietoris topology.