The regular open-open topology for function spaces

Abstract

The regular open-open topology, Troo, is introduced, its properties for spaces of continuous functions are discussed, and Troo is compared to Too, the open-open topology. It is then shown that Troo on H(X), the collection of all self-homeomorphisms on a topological space, (X,T), is equivalent to the topology induced on H(X) by a specific quasi-uniformity on X, when X is a semi-regular space.

Keywords

• compact-open topology
• admissible topology
• open-open topology quasi-uniformity
• regular open set
• semi-regular space
• topological group.