Action on hyperspaces

Let X be a Tychonoff space, HomX the full group of self-homeomorphisms of X and CLX the hyperspace of all nonempty closed subsets of X: First, by constructing a not locally compact model of topologistfs comb, we show that local compactness is not a necessary condition for HomX equipped with the compactopen topology being a topological group acting continuously on X by the evaluation map e : (f, x) ∈ HomX × X → f(x) ∈ X: Then, generalizing the compact case, we give necessary and sufficient conditions for HomX equipped with a set-open topology based on a Urysohn family being a topological group acting continuously on CLX by the evaluation map E : (f,C)∈ HomX×CLX → f(C) ∈ CLX: Furthermore, when X is a uniform space, we give necessary and sufficient conditions for HomX equipped with the topology of uniform convergence on a uniformly Urysohn family being a topological group acting continuously on CLX: Besides, we do the same in the proximal case. © 2012 Topology Proceedings.