Remarks on quasi-Lindelöf spaces (Q2872020)

A space \(X\) is said to be weakly Lindelöf if for every open cover \(\mathcal U\) of \(X\) there exists a countable subset \(\mathcal V\) of \(\mathcal U\) such that \(\overline{\cup\{V:V\in\mathcal V\}}=X\). A space \(X\) is said to be quasi-Lindelöf if every closed subspace of \(X\) is weakly Lindelöf. \textit{P. Staynova} asked whether the product of a quasi-Lindelöf space and a compact space is quasi-Lindelöf in her papers [A comparison of Lindelöf-type covering properties of topological spaces, Rose-Hulman Undergrad. Math. J. 12(2), 163--204 (2011)] and [A Note on Quasi-Lindelöf Spaces, \url{arXiv:1212.2869v1}]. The author constructs a counterexample to Staynova's question using the Isbell-Mrówka space which shows that there exists a Tychonoff quasi-Lindelöf space \(X\) and a compact space \(Y\) such that \(X\times Y\) is not quasi-Lindelöf.