Topologizing homeomorphism groups

This paper is a survey of results relative to a special kind of function spaces like the full group of homeomorphisms of a Tychonoff space and its subgroups. The area has initially evolved relaxing compactness condition by passing from the class of compact metric spaces to the class of $T_2$ locally compact spaces and then beyond local compactness. Let $X$ be a Tychonoff space, ${\cal H}(X)$ the full group of self-homeomorphisms of $X$ with the usual composition and $ e: (f,x) \in {\cal H}(X) \times X \rightarrow f(x) \in X$ the evaluation function. Topologies on ${\cal H}(X)$ which yield continuity of both the group operations, product and inverse function, and at same time provide continuity of the evaluation function are called admissible group topologies. Denote by ${\cal L}_H(X)$ the upper-semilattice of all admissible group topologies on ${\cal H}(X)$ ordered by the usual inclusion. Whenever $X$ is locally compact $T_2, \ {\cal L}_H(X)$ admits a least element. That can be described simply as a set-open topology and contemporaneously as a uniform topology, further agreeing with the compact-open topology if $X$ is also locally connected. Beyond local compactness and by means of a compact extension procedure we perform 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, in a first step where $X$ is the rational number space $\mathbb{Q}$ equipped with the Euclidean topology and, next, where $X$ is a product of $T_2$ zero-dimensional spaces each satisfying the property: {\it any two non-empty clopen subspaces are homeomorphic} and their products. In both cases the least admissible group topology on ${\cal H}(X)$ 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. But, then, carrying on another efficient way to produce admissible group topologies in substitution of, or in parallel with, the compact extension procedure, we show that rim-compactness is not a necessary condition for the existence of the least admissible group topology. More precisely, we show that the full group of self-homeomorphisms of the product space $\mathbb{R}\times \mathbb{Q}$, where $\mathbb{R}$ and $\mathbb{Q}$ are the sets of the real and rational numbers respectively, both carrying the Euclidean topology, admits a least admissible group topology even though notoriously $\mathbb{R} \times \mathbb{Q}$ is not rim-compact. Finally, by making to fall the compact case in the more general one of local proximity space, we give necessary and sufficient conditions for a topology of uniform convergence on the bounded sets of a local proximity space is an admissible group topology. We cite also that recently we have proven that local compactness is not necessary condition for ${\cal H}(X)$ equipped with the compact-open topology being a topological group acting continuously on $X$ via the evaluation map.