Topologizing Homeomorphism Groups

Topologizing Homeomorphism Groups

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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 read more topologies on 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 princess polly dresses long sleeve 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.