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

Abstract

For a Tychonoff space X and a family λ of subsets of X, we denote by Cλ(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 Cλ(X). Our main results state that Cλ(X) is Menger if and only if Cλ(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. © 2019 Mathematical Institute Slovak Academy of Sciences.