Coincidence of the Isbell and fine Isbell topologies
Let \(X\), \(Y\) be topological spaces and \(C(X,Y)\) denote the space of all continuous mappings from \(X\) into \(Y\). The author studies the relation between various topologies on \(C(X,Y)\) and when they coincide. The results rely on earlier work by \textit{S. Dolecki} and \textit{F. Mynard} [Topology Appl. 157, No. 8, 1370--1378 (2010; Zbl 1195.54036)]. In case \(X\) is 0-dimensional, he finds necessary and sufficient conditions on \(X\) so that (i) for all \(Y\), the Isbell and fine Isbell topologies on \(C(X, Y)\) coincide; (ii) the Isbell and fine Isbell topologies coincide on \(C(X, R)\), where \(R\) denotes the real numbers. A number of open problems are also cited.
scientific article; zbMATH DE number 5791180