Cardinal invariants of cellular-Lindelöf spaces (Q2317608)

In [Monatsh. Math, 186, 345--353 (2018; Zbl 1398.54008)] the authors of this article introduced cellular-Lindelöf spaces: a space \(X\) is said to be cellular-Lindelöf if for every cellular family \(\mathcal U\) in \(X\) there is a Lindelöf subspace \(L\) of \(X\) such that \(L\cap U \neq \emptyset\) for every \(U \in\mathcal U\). All ccc and all Lindelöf spaces are cellular-Lindelöf. They asked if a first-countable cellular-Lindelöf space \(X\) has cardinality \(\le \mathfrak{c}\) (which would be a common generalization of two famous cardinal inequalities by Arhangel'skii and by Hajnal-Juhász). The authors prove that every normal cellular-Lindelöf space with a \(G_{\delta}\) diagonal of rank 2 has cardinality \(\le \mathfrak{c}\), and that under \(2^{<\mathfrak{c}} = \mathfrak{c}\), every normal cellular-Lindelöf first countable space has cardinality \(\le \mathfrak{c}\). It is also proved that in the class of monotonically normal spaces cellular-Lindelöfness and Lindelöfness coincide. The paper ends with nine open problems.