Beer, Himmelberg, Prikry and Van Vleck showed that if $X$ is a metrizable space, then the locally finite topology on $2^X$ is the sup of all Hausdorff metric topologies on $2^X$ generated by compatible metrics on $X$. The second author and Sharma generalized this result to normal spaces. In this paper it is shown that the locally finite topology on $2^X$ is the Hausdorff generalized uniform topology corresponding to a generalized uniformity on $X$ which is Mozzocchi as well as local.