The open-open topology for function spaces

Abstract

Let (X,T) and (Y,T*) be topological spaces and let F⊂YX. For each U∈T, V∈T*, let (U,V)={f∈F:f(U)⊂V}. Define the set S∘∘={(U,V):U∈T and V∈T*}. Then S∘∘ is a subbasis for a topology, T∘∘ on F, which is called the open-open topology. We compare T∘∘ with other topologies and discuss its properties. We also show that T∘∘, on H(X), the collection of all self-homeomorphisms on X, is equivalent to the topology induced on H(X) by the Pervin quasi-uniformity on X.