Comparison of some set open topologies on \(C(X, Y)\) (Q471480)

Let \(X\) and \(Y\) be two topological spaces and \(C(X, Y)\) be the set of all continuous functions from \(X\) into \(Y\). If \(\alpha\) is a nonempty family of subsets of \(X\), then the set open topology on \(C(X, Y)\) has a subbase consisting of the sets \([A, V] =\{f\in C(X, Y) :f(A) \subseteq V\},\) where \(A\in \alpha\) and \(V\) is an open subset of \(Y\). The function space \(C(X, Y)\) endowed with this topology is denoted by \(C_{\alpha}(X, Y).\) The set open topology was first introduced by \textit{R. Arens} and \textit{J. Dugundji} [Pac. J. Math. 1, 5--31 (1951; Zbl 0044.11801)]. In this paper the authors provide a criterion for the coincidence of the set open and uniform topologies on \(C(X, Y)\) for the case when \(Y\) is an equiconnected metric space. Also, they compare the spaces \(C_{\alpha}(X, Y)\) and \(C_{\beta}(X, Y)\) for two given families \(\alpha\) and \(\beta\) of compact subsets of \(X\) and study separation axioms on \(C(X,Y)\).