Productively Lindelof and Indestructibly Lindelof Spaces

TL;DR

This paper explores the properties and relationships of productively Lindelof and indestructibly Lindelof spaces, examining their behavior under various conditions and their connections with topological games and selection principles.

Contribution

It provides new insights into the properties of these spaces and their interrelations, extending previous work on their behavior and connections.

Findings

Productively Lindelof spaces maintain Lindelof property under product with any Lindelof space.

Indestructible Lindelof spaces remain Lindelof in all countably closed forcing extensions.

New connections between these spaces, topological games, and selection principles are established.

Abstract

There has recently been considerable interest in productively Lindelof spaces, i.e. spaces such that their product with every Lindelof space is Lindelof. Here we make several related remarks about such spaces. Indestructible Lindelof spaces, i.e. spaces that remain Lindelof in every countably closed forcing extension, were introduced by Tall in 1995. Their connection with topological games and selection principles was explored by Scheepers and Tall in 2010. We find further connections here.