The open-open topology for function spaces
Summary: Let \((X,T)\) and \((Y,T^*)\) be topological spaces and let \(F\subset Y^ X\). For each \(U\in T\), \(V\in T^*\), let \((U,V)= \{f\in F\): \(f(U)\subset V\}\). Define the set \(S_{oo}= \{(U,V)\): \(U\in T\) and \(V\in T^*\}\). Then \(S_{oo}\) is a subbasis for a topology, \(T_{oo}\) on \(F\), which is called the open-open topology. We compare \(T_{oo}\) with other topologies and discuss its properties. We also show that \(T_{oo}\), 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 \(X\).
scientific article; zbMATH DE number 130458