Topologizing homeomorphism groups of rim-compact spaces

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 T2, 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 T2 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