Abstract
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 Gδ-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.
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