Action on hyperspaces (Q2850621)

Let \(X\) be a Tychonoff space, \(HomX\) the group of self-homeomorphisms of \(X\) and let \(CLX\) be the hyperspace of all nonempty closed subsets of \(X\). In the present paper, the author proves that local compactness is not a necessary condition for \(HomX\) endowed with the compact-open topology to be a topological group acting continuously on \(X\) by the evaluation map \(e : HomX \times X \rightarrow X\). Furthermore, she gives necessary and sufficient conditions for \(HomX\) endowed with the set-open topology on a Urysohn family to be a topological group acting continuously on \(CLX\) by the evaluation map \(E : HomX \times CLX \rightarrow CLX\). Also the same is done when \(HomX\) is equipped with the following two topologies: (1) the topology of uniform convergence on a uniformly Urysohn family; (2) the proximal set-open topology relative to a proximity and a boundedness giving a local proximity space.