  We study the heredity of the classes of generalized metric spaces (for
example, spaces with a $\sigma$-hereditarily closure-preserving $k$-network,
spaces with a point-countable base, spaces with a base of countable order,
spaces with a point-regular base, Nagata-spaces, $c$-semi-stratifiable spaces,
$\gamma$-spaces, semi-metrizable spaces) to the hyperspaces of nonempty compact
subsets and finite subsets with the Vietoris topology.
