Necessary and sufficient conditions for admissible set open topologies

Abstract

Let X and Y be topological spaces and Cα(X,Y) be the set of all continuous functions from X to Y endowed with the set open topology where α is a nonempty family of subsets of X. We obtain criteria in order that set open topology on C(X,Y) to be admissible. We prove that the topology of Cα(X,Y) is admissible if and only if α is a regular family, when Y is an equiconnected topological space. In the case when X is a topological monoid, we study the relationship between the admissibility of the set open topology on C(X,Y) and the continuity of an action of X on C(X,Y). We prove that the set open topology on C(X,Y) is admissible if and only if Cα(X,Y) with the action s.f=(t↦f(st)) is the cofree X-space over Y.