Let X be a metrizable space. A Vietoris-type topology, called the locally finite topology, is defined on the hyperspace \(2^ X\) of all closed, nonempty subsets of X. We show that the locally finite topology coincides with the supremum of all Hausdorff metric topologies corresponding to equivalent metrics on X. We also investigate when the locally finite topology coincides with the more usual topologies on \(2^ X\) and when the locally finite topology is metrizable.