Weak covering properties and selection principles

TL;DR

This paper investigates the relationships between various covering properties in topology, focusing on when spaces with certain productive properties also satisfy stronger or related properties, expanding understanding of productively Lindelof spaces.

Contribution

It explores conditions under which productively X spaces are also productively Y, especially relating to Alster's property and weakly Lindelof spaces, offering new insights into their interrelations.

Findings

Spaces with Alster's property are also productively weakly Lindelof.

Productively X spaces may also be productively Y under certain conditions.

The paper clarifies when stronger covering properties imply weaker ones in product spaces.

Abstract

No convenient internal characterization of spaces that are productively Lindelof is known. Perhaps the best general result known is Alster's internal characterization, under the Continuum Hypothesis, of productively Lindelof spaces which have a basis of cardinality at most $\aleph_1$. It turns out that topological spaces having Alster's property are also productively weakly Lindelof. The weakly Lindelof spaces form a much larger class of spaces than the Lindelof spaces. In many instances spaces having Alster's property satisfy a seemingly stronger version of Alster's property and consequently are productively X, where X is a covering property stronger than the Lindelof property. This paper examines the question: When is it the case that a space that is productively X is also productively Y, where X and Y are covering properties related to the Lindelof property.