The Menger and projective Menger properties of function spaces with the set-open topology

Abstract

For a Tychonoff space $X$ and a family $\lambda$ of subsets of $X$, we denote by $C_{\lambda}(X)$ the space of all real-valued continuous functions on $X$ with the set-open topology. In this paper, we study the Menger and projective Menger properties of a Hausdorff space $C_{\lambda}(X)$. Our main results state that if $\lambda$ is a $\pi$-network of $X$ then (1) $C_{\lambda}(X)$ is Menger space if and only if it is $\sigma$-compact, if $Y$ is a dense subset of $X$ then (2) $C_{p}(Y\vert X)$ is projective Menger space if and only if it is $\sigma$-pseudocompact.