Topologizing homeomorphism groups of rim-compact spaces
by
Anna Di Concilio
Dipartimento di Matematica ed Informatica Universita' di Salerno Italia

Let X be a Tychonoff space , H(X) the group of all self-homeomorphisms of X with the usual composition and e: (f, x) in H(X)×X --> f(x) in X the evaluation function. Topologies on H(X) providing continuity of the evaluation function are called admissible. Topologies on H(X) compatible with the group operations are called topological group structures. R. Arens proved, when X is locally compact T₂, the existence of the minimum among all admissible topological group structures on H(X) which can be described simply as a set-open topology, further agreeing with the compact-open topology when X is locally connected. We show the same extra local compactness in two essentially different cases of rim-compactness. The former one, when X is rim-compact T₂ and locally connected. The latter one, when X agrees with the rational number space Q equipped with the euclidean topology. In the first case when X is more separable metric then the minimum is separable and completely metrizable too. When X is further finite union of disjoint connected subspaces, then the minimum is closely linked to its Freudenthal compactification and it is the closed-open topology determined from all closed sets with compact boundaries. In the rational case again the minimum is linked to the Freudenthal compactification of Q and it is just the closed-open topology. The Freudenthal compactification in rim-compactness plays a key role as one-point compactification in local compactness. In the rational case we investigate as the fine or Whitney topology on H(Q) induces an admissible topological group structure on H(Q) stronger than the closed-open topology.

Date received: June 12, 2002