The \(o\)-Malykhin property for spaces \(C_k(X)\) (Q1928425)

A space \(X\) is said to be \textit{\(o\)-Malykhin} provided for each family \((O_i)_{i\in I}\) of open subsets of \(X\) and each \(x\in \overline{\bigcup_{i\in I} O_i}\setminus \bigcup_{i\in I} \overline{O_i}\) there is an infinite \(J\subseteq I\) such that \(\{j\in J: O_j\cap V=\emptyset\}\) is finite for any neighborhood \(V\) of \(x\). The main result of the paper is a characterization of the \(o\)-Malykhin property for the space \(C_k(X)\) of real-valued continuous functions on \(X\) with the compact-open topology. It is shown that \(C_k(X)\) is \(o\)-Malykhin iff each moving off collection \(\mathcal K\) of nonempty compacts of \(X\) (i.e. such that for any compact \(L\subset X\) there is a \(K\in\mathcal K\) with \(K\cap L=\emptyset\)) has an infinite compact-finite subcollection (i.e. such that each compact in \(X\) meets only finitely many members of the subcollection). The \(o\)-Malykhin property of \(C_k(X)\) is also characterized in terms of a topological game played on \(X\), which is a modification of a game of \textit{Gruenhage} and \textit{Ma} used for characterizing Baireness of \(C_k(X)\). The relationship of the \(o\)-Malykhin property of \(C_k(X)\) to its \(\kappa\)-Fréchet-Urysohnness (i.e. when to each point \(x\) in the closure of an open set \(U\) one can find a sequence from \(U\) converging to \(x\)), and other properties is also studied.