Some properties of set-open topologies (Q941981)

Let \(C(X)\) denote the set of all real-valued continuous functions on a completely regular space \(X\). If \({\mathcal F}\subseteq X\) is a subset of \(X\) and \(U\subseteq\mathbb{R}\) an open subset in \(\mathbb{R}\), then \(\langle{\mathcal F},U\rangle\) stands for the set of all functions \(f\in C(X)\) for which \(f({\mathcal F})\subseteq U\). If \(\lambda\) is a nonempty family of subsets of \(X\), then \(C^\lambda(X)\) denotes the set \(C(X)\) topologized by the topology having as a subbase the sets of the form \(\langle{\mathcal F},U\rangle\), where \({\mathcal F}\in\lambda\) and \(U\) is open in \(\mathbb{R}\). This topology on \(C(X)\) is called set-open topology and the author studies various properties of the space \(C^\lambda(X)\) depending on the choice of the family \(\lambda\). Some results concerning separation axioms in \(C^\lambda(X)\), local compactness and \(\sigma\)-compactness of \(C^\lambda(X)\) are given.