A Comparison of Lindelof-Type Covering Properties of Topological Spaces

TL;DR

This paper compares Lindel"of, almost Lindel"of, and weakly Lindel"of spaces, presenting analogous theorems, counterexamples, and exploring their relationships with other topological properties.

Contribution

It provides a systematic comparison of these covering properties, proves new analogous theorems, and introduces novel counterexamples illustrating their interrelations.

Findings

Analogous theorems for weakly and almost Lindel"of spaces are established.

Counterexamples demonstrate differences between these properties and other topological notions.

New features of examples highlight subtle distinctions among covering properties.

Abstract

Lindel\"of spaces are studied in any basic Topology course. However, there are other interesting covering properties with similar behaviour, such as almost Lindel\"of, weakly Lindel\"of, and quasi-Lindel\"of, that have been considered in various research papers. Here we present a comparison between the standard results on Lindel\"of spaces and analogous results for weakly and almost Lindel\"of spaces. Some theorems, similar to the published ones, will be proved. We also consider counterexamples, most of which have not been included in the standard Topological textbooks, that show the interrelations between those properties and various basic topological notions, such as separability, separation axioms, first countability, and others. Some new features of those examples will be noted in view of the present comparison. We also pose several open questions.