More on cellular-Lindelof spaces

摘要

The class of cellular-Lindelof spaces was introduced by A. Bella and S. Spadaro (2017) [5]. Recall that a topological space X is cellular-Lindelof if for every family u of pairwise disjoint non-empty open sets of X there is a Lindelof subspace L subset of X such that U boolean AND L not equal empty set, for every U is an element of u. Cellular-Lindelof spaces generalize both Lindelof spaces and spaces with the countable chain condition. In this paper, we first discuss some basic properties of cellular-Lindelof spaces such as the behavior with respect to products and subspaces. We also establish cardinal inequalities for cellular-Lindelof quasitopological groups by using Erdos-Rado's theorem. Finally, we introduce and study the class of cellular-compact (cellular-sigma-compact) spaces. In particular, we prove that every cellular-sigma-compact Hausdorff space having either a rank 2-diagonal or a regular G(delta)-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. (C) 2019 Elsevier B.V. All rights reserved.