On the topologies of a hyperspace of a metrizable topological space

The properties of the topology $\tau_{inf}$, which is the infimum of the set of all topologies generated by the Hausdorff metrics on the hyperspace $\exp X$ of a metrizable topological space $X$ are studied. As one of the main results necessary and sufficient conditions for the metrizability (with Hausdorff metric) of $\tau_{inf}$ are obtained. We also show that $\exp X$ with the topology $\tau_{inf}$ is first-countable space if and only if a space $X$ is locally compact and second-countable. Besides we investigate relations between $\tau_{inf}$ and other topologies on the $\exp X$: Vietoris topology, Fell topology and locally finite topology.