A Comparison of Lindelof-Type Covering Properties of Topological Spaces

Abstract

Lindelöf spaces are studied in any basic Topology course. However, there are other interesting covering properties with similar behaviour, such as almost Lindelöf, weakly Lindelöf, and quasi-Lindelöf, that have been considered in various research papers. Here we present a comparison between the standard results on Lindelöf spaces and analogous results for weakly and almost Lindelöf 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. Historical Overview and MotivationOne of the basic theorems in Real Analysis, the Heine-Borel Theorem, states (in modern terminology) that every closed interval on the real line is compact. Later it was discovered that a similar property holds in more general metric spaces: every closed and bounded subset turns out to be compact and conversely, every compact subset is closed and bounded. It turned out that compactness is in fact a covering property: the modern description of compactness via open covers emerged from the work of P. S. Alexandrov and P. S. Urysohn in their famous "Memoire sur les espaces topologiques compacts" [AU29]. Compact spaces in many ways resemble finite sets. For example, the fact that in Hausdorff topological spaces two different points can be separated by disjoint open sets easily generalizes to the fact that in such spaces two disjoint compact sets can also be separated by disjoint open sets. Any finite set is compact in any topology and the fact that compact spaces should be accompanied by some kind of separation axiom comes from the fact that any set with the co-finite topology is compact (but fails to be Hausdorff).However, our favourite "real" objects such as the real line, real plane etc. fail to be compact. Ernst Lindelöf was able to identify the first compactness-like covering property (which was later given his name): the property that from every open cover, one can choose a countable subcover. The Lindelöf theorem, stating that every second countable space is Lindelöf, was proved by him for Euclidean spaces as early as 1903 in [Lin03]. Many facts that held for compact spaces, such as that every closed subspace of a compact space is also compact, remain true in Lindelöf spaces. In metric spaces the Lindelöf property was proved to be equivalent to separability, the existence of a countable basis and the countable chain condition -all of which hold on the real line. But many other properties, such as preservation under products, fail even in the finite case for Lindelöf spaces.Another generalization that is much closer to compactness is the notion of an H-closed space: this is a Hausdorff space in which from every open cover we can choose a finit...