The Rothberger property on \(C_p(X,2)\) (Q891252)

A space \(Z\) has the Rothberger property if for each sequence \((\mathcal U_n:n\in \mathbb N)\) of open covers of \(Z\) there are sets \(U_n\in\mathcal U_n\), \(n\in\mathbb N\), such that \(\{U_n:n\in\mathbb N\}\) is an open cover of \(Z\). The Rothberger property of function spaces \(C_p(X,2)\) is investigated. It is proved that \(C_p(X,2)\) is Rothberger implies that \(X\) is pseudocompact. If \(\Psi(\mathcal A)\) is the Mrówka-Isbell space obtained from a so-called Mrówka maximal almost disjoint family \(\mathcal A\), then \(C_p(\Psi(\mathcal A),2)\) is Rothberger if and only if it is Lindelöf. Several results on the Rothberger property of all finite powers of \(C_p(X,2)\) are obtained. For a Corson compact \(X\), \(C_p(X,2)\) is Rothberger in all finite powers. If \(X\) is a Sokolov space, then \(C_p(X,2)\) is Rothberger in all finite powers if and only if \(X\) is pseudocompact and all finite powers of \(X\) have countable tightness. For a metrizable space \(X\), \(C_p(X,2)\) is Rothberger in all finite powers if and only if \(X\) is compact.