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

Abstract

Abstract For a Tychonoff space X and a family λ of subsets of X, we denote by C λ(X) the space of all real-valued continuous functions on X with the set-open topology. A Menger space is a topological space in which for every sequence of open covers 𝓤1, 𝓤2, … of the space there are finite sets 𝓕1 ⊂ 𝓤1, 𝓕2 ⊂ 𝓤2, … such that family 𝓕1 ∪ 𝓕2 ∪ … covers the space. In this paper, we study the Menger and projective Menger properties of a Hausdorff space C λ(X). Our main results state that C λ(X) is Menger if and only if C λ(X) is σ-compact; Cp (Y | X) is projective Menger if and only if Cp (Y | X) is σ-pseudocompact where Y is a dense subset of X.