This paper surveys topologies, called admissible group topologies, of the full group of self-homeomorphisms of a Tychonoff space , 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 on.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 of all admissible group topologies on admits a least Hayward SP1090 Skimmer Parts 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 Fridge Hinge Insert 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 is not a necessary condition for the compact-open topology to be an admissible group topology of.