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Productivity of cellular-Lindelöf spaces. (English) Zbl 1462.54031

A. Bella and S. Spadaro [Monatsh. Math. 186, No. 2, 345–353 (2018; Zbl 1398.54008)] defined a space \(X\) to be cellular-Lindelöf if for every family \(\mathcal{U}\) of pairwise disjoint nonempty open subsets of \(X\) there is a Lindelöf subspace \(L\subseteq X\) such that \(U\cap L \ne \emptyset\) for every \(U\in\mathcal{U}\). Cellular-compact spaces are defined in the same way but the subspace \(L\) is required to be compact, see [V. V. Tkachuk and R. G. Wilson, Acta Math. Hung. 159, No. 2, 674–688 (2019; Zbl 1449.54033)].
W.-F. Xuan and Y.-K. Song [Topology Appl. 266, Article ID 106861 12 p. (2019; Zbl 1431.54013)] studied cellular-Lindelöf spaces and in their Theorem 3.12 they stated that the product of a cellular-Lindelöf space with a space of countable spread is cellular-Lindelöf. In the same paper the authors also asked if the product of a compact space and a cellular-Lindelöf space is cellular-Lindelöf.
In the paper under review, in Section 2, by constructing appropriate counterexamples, the authors provide a negative answer to the above question and to the question whether or not the property cellular-compact is productive. But the main goal of the authors of this paper is to show that the assertion in the above-mentioned Theorem 3.12 is not true. For that end, in Section 4, using a Souslin tree, they construct consistent counterexamples, and in Section 5, using Moore’s \(L\)-space [J. T. Moore, J. Am. Math. Soc. 19, No. 3, 717–736 (2006; Zbl 1107.03056)], the authors construct ZFC counterexamples of the claim in Xuan and Song’s Theorem 3.12.

MSC:

54G20 Counterexamples in general topology
54B10 Product spaces in general topology
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54F65 Topological characterizations of particular spaces
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References:

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