A Note on Quasi-Lindelof Spaces

TL;DR

This paper investigates the properties and behaviors of quasi-Lindel"of spaces, providing examples and exploring their relationships with other topological properties, while highlighting gaps in current understanding.

Contribution

It offers new examples and insights into quasi-Lindel"of spaces, examining their behavior under topological operations and their relation to separation axioms.

Findings

A weakly Lindel"of space that is not quasi-Lindel"of

A product of Lindel"of spaces that is not quasi-Lindel"of

A quasi-Lindel"of space that is not ccc

Abstract

The quasi-Lindel\"of property was first introduced by Arhangelski in \cite{Arc}, as a strengthening of the weakly Lindel\"of property. However, unlike Lindel\"of and weakly Lindel\"of spaces, very little is known about how quasi-Lindel\"of spaces behave under the main topological operations, and how the property relates to separation axioms. In the present paper, we look at several properties of quasi-Lindel\"of spaces. We consider several examples: a weakly Lindel\"of space which is not quasi-Lindel\"of, a product of Lindel\"of spaces which is not even quasi-Lindel\"of, and a quasi-Lindel\"of space which is not ccc. At the end, we pose some open questions.