Topologizing Homeomorphism Groups

Abstract EN

This paper surveys topologies, called admissible group topologies, of the full group of self-homeomorphisms ℋ(X) of a Tychonoff space X, which yield continuity of both the group operations and at the same time provide continuity of the evaluation function or, in other words, make the evaluation function a group action of ℋ(X) on X.

By means of a compact extension procedure, beyond local compactness and in two essentially different cases of rim-compactness, we show that the complete upper-semilattice ℒH(X) of all admissible group topologies on ℋ(X) admits a least element, that can be described simply as a set-open topology and contemporaneously as a uniform topology.

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.

Finally, we give necessary and sufficient conditions for the topology of uniform convergence on the bounded sets of a local proximity space to be an admissible group topology.

Also, we cite that local compactness of X is not a necessary condition for the compact-open topology to be an admissible group topology of ℋ(X).