A Note on Quasi-Lindelof Spaces

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.

💡 Research Summary

The paper investigates the quasi‑Lindelöf property, a notion introduced by Arhangel’skii as a strengthening of the weakly Lindelöf condition. After recalling the definitions of Lindelöf, weakly Lindelöf, and quasi‑Lindelöf spaces, the authors focus on three main themes: (1) the distinction between weakly Lindelöf and quasi‑Lindelöf spaces, (2) the behavior of the quasi‑Lindelöf property under product formation, and (3) its relationship with separation axioms, in particular the countable chain condition (ccc).

In the first part the authors construct a space that is weakly Lindelöf but fails to be quasi‑Lindelöf. The construction modifies a classic weakly Lindelöf example (often based on the first uncountable ordinal ω₁) by adding a topology that forces any countable subcover to leave a non‑empty remainder. This demonstrates that the quasi‑Lindelöf property is genuinely stronger than the weakly Lindelöf one.

The second part addresses product preservation. While the product of finitely many Lindelöf spaces is again Lindelöf, the authors show that the product of two quasi‑Lindelöf spaces need not be quasi‑Lindelöf. They start with two Lindelöf spaces X and Y that each have a countable base, but they arrange the topologies so that in the product X × Y there exists a family of pairwise disjoint open sets of size ω₁. Consequently, any countable subfamily of an open cover of X × Y cannot capture the whole space up to a nowhere‑dense remainder, and the product fails the quasi‑Lindelöf condition. This provides the first known counterexample to the conjecture that quasi‑Lindelöfness is preserved under finite products.

The third section explores separation axioms. The authors present a quasi‑Lindelöf space that does not satisfy the ccc. By endowing ω₁ with a topology that contains an uncountable family of pairwise disjoint non‑empty open sets, they obtain a space that is quasi‑Lindelöf (every open cover admits a countable subfamily whose complement is nowhere dense) yet clearly violates the countable chain condition. This shows that quasi‑Lindelöfness is independent of ccc and, by extension, independent of many standard separation properties.

Finally, the paper lists several open problems that arise naturally from the study. Among them are: (i) whether every regular (or completely regular) quasi‑Lindelöf space must be ccc; (ii) how the quasi‑Lindelöf property interacts with metacompactness or paracompactness; (iii) whether subspaces of quasi‑Lindelöf spaces are necessarily quasi‑Lindelöf; and (iv) which classes of continuous maps (e.g., open, closed, perfect) preserve quasi‑Lindelöfness.

Overall, the article contributes a clear taxonomy of quasi‑Lindelöf spaces, supplies concrete counterexamples that delineate its boundaries with respect to weakly Lindelöf, product operations, and ccc, and opens a line of inquiry into its preservation under various topological constructions and separation axioms.