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

TL;DR

This paper investigates the Menger and projective Menger properties of function spaces with the set-open topology, establishing conditions under which these spaces are Menger or projective Menger, based on properties like $\sigma$-compactness.

Contribution

It provides new characterizations of Menger and projective Menger properties for function spaces with the set-open topology, linking these properties to $\sigma$-compactness and $\sigma$-pseudocompactness.

Findings

$C_{ extstyle{ ext{lambda}}}(X)$ is Menger iff it is $\sigma$-compact when $ extstyle{ ext{lambda}}$ is a $ extstyle{ ext{pi}}$-network.

$C_{p}(Y|X)$ is projective Menger iff it is $\sigma$-pseudocompact for dense $Y$ in $X$.

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.