More on cellular-Lindelöf spaces.

The class of cellular-Lindelöf spaces was introduced by A. Bella and S. Spadaro (2017) [5]. Recall that a topological space X is cellular-Lindelöf if for every family U of pairwise disjoint non-empty open sets of X there is a Lindelöf subspace L ⊂ X such that U ∩ L ≠ ∅ , for every U ∈ U. Cellular-Lindelöf spaces generalize both Lindelöf spaces and spaces with the countable chain condition. In this paper, we first discuss some basic properties of cellular-Lindelöf spaces such as the behavior with respect to products and subspaces. We also establish cardinal inequalities for cellular-Lindelöf quasitopological groups by using Erdös-Radó's theorem. Finally, we introduce and study the class of cellular-compact (cellular- σ -compact) spaces. In particular, we prove that every cellular- σ -compact Hausdorff space having either a rank 2-diagonal or a regular G δ -diagonal has cardinality at most c , which partially answers Question 8 and Question 9 of S. Spadaro and A. Bella (2018) [6]. Some new questions are also posed. [ABSTRACT FROM AUTHOR]