On the \(\sigma\)-countable compactness of spaces of continuous functions with the set-open topology (Q471010)

Let \(X\) be a Tychonoff space, \(\lambda\) be a family of subsets of \(X\) and \(C_{\lambda}(X)\) be the set of all continuous real-valued functions on \(X\) endowed with the set-open topology. For a subset \(Y\subseteq X\), \(C_p(Y|X)\) denotes the subspace of \(C_p(Y)\) consisting of functions \(f|_{Y}\), where \(f\in C(X)\). Denote by \(X(P)\) the set of all \(P\)-points of \(X\). In this work, the authors provide a criterion for the \(\sigma\)-countable compactness of \(C_{\lambda}(X)\). They prove that \(C_{\lambda}(X)\) is \(\sigma\)-countably compact if and only if \(X\) is pseudocompact, \(X(P)\) is dense in \(X\), the family \(\lambda\) consists of finite subsets of the set \(X(P)\), and the space \(C_p(X(P)|X)\) is \(\sigma\)-countably compact.