Indestructibly productively Lindel\"of and Menger function spaces

TL;DR

This paper investigates the properties of certain function spaces, specifically $C_{\lambda}(X)$, focusing on their indestructible productively Lindel"of and Menger characteristics under various topological conditions.

Contribution

It introduces new results on the indestructibly productively Lindel"of and Menger properties of $C_{\lambda}(X)$ spaces, expanding understanding of their stability under forcing extensions.

Findings

Characterization of $C_{\lambda}(X)$ as indestructibly productively Lindel"of.

Conditions under which $C_{\lambda}(X)$ exhibits Menger property.

Analysis of the stability of these properties under forcing extensions.

Abstract

For a Tychonoff space $X$ and a family $\lambda$ of subsets of $X$, we denote by $C_{\lambda}(X)$ the $T_1$-space of all real-valued continuous functions on $X$ with the $\lambda$ -open topology. A topological space is productively Lindel\"of if its product with every Lindel\"of space is Lindel\"of. A space is indestructibly productively Lindel\"of if it is productively Lindel\"of in any extension by countably closed forcing. In this paper, we study indestructibly productively Lindel\"of and Menger function space $C_{\lambda}(X)$.