Action on hyperspaces

Let X be a Tychonoff space, CL(X) the hyperspace of all non-empty closed subsets of X, H(X) the full group of self-homeomorphisms of X and e: (f,x)  H(X) x X  f(x)  X the evaluation map. It is very well-known that, if X is locally compact, the compact-open topology c.o on H(X) yields jointly continuity for the evaluation map and, at the same time, continuity for the product map. But, unfortunately, it does not yield continuity for the inverse map. In local compactness, a sufficient ( but not necessary ) condition for c.o being a topological group topology is the following one: () Any point admits a compact connected neighbourhood, [3]. We can exhibit an example of a space X that is neither locally compact nor verifying the property () for which, anyway, (H(X), c.o ) is a topological group. Again in local compactness, by using methods of non-standard analysis and action on hyperspaces endowed with the Fell topology, in [4], K.R. Wicks gave necessary and sufficient conditions for c.o being a topological group topology. We look at locally compact case only as a particular case that falls within the more general one in which the compact-open topology is substituted with (proximal) set-open topologies while the Fell topology is replaced by (proximal) hit and miss topologies.