Topologizing homeomorphism groups of rim-compact spaces

Abstract

Let X be a Tychonoff space, H ( X ) the group of all self-homeomorphisms of X with the usual composition and e : ( f , x ) ∈ H ( X ) × X → f ( x ) ∈ 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 group topologies. Whenever X is locally compact T 2 , there is the minimum among all admissible group topologies on H ( X ) . That can be described simply as a set-open topology, further agreeing with the compact-open topology if X is also locally connected. We show the same result in two essentially different cases of rim-compactness. The former one, where X is rim-compact T 2 and locally connected. The latter one, where X agrees with the rational number space Q equipped with the euclidean topology. In the first case the minimal admissible group topology on H ( X ) is the closed-open topology determined by all closed sets with compact boundaries contained in some component of X. Moreover, whenever X is also separable metric, it is Polish. In the rational case the minimal admissible group topology on H ( Q ) is just the closed-open topology. In both cases the minimal admissible group topology on H ( X ) is closely linked to the Freudenthal compactification of X. The Freudenthal compactification in rim-compactness plays a key role as the one-point compactification does in local compactness. In the rational case we investigate whether the fine or Whitney topology on H ( Q ) induces an admissible group topology on H ( Q ) stronger than the closed-open topology.