Compact productivity of Lindelöf-type properties
type: Item
instance of: scholarly article
title: Compact productivity of Lindelöf-type properties
author: L. E. Gutiérrez-Dominguez
author: Ofelia T. Alas
author: Richard G. Wilson
published in: Acta Mathematica Hungarica
publication date: 2022-11-09T00:00:00Z
review text: All hypothesized spaces are Hausdorff. For a space \(X\) and topological property \(P\), the space \(X\) is said to be \textit{almost cellular-}\(P\) (resp., \textit{strongly almost cellular}-\(P\)) if whenever \(\mathcal{U}\) is a pairwise disjoint family of nonempty open sets of \(X\) there is a subspace (resp., closed subspace) of \(X\) which has property \(P\) and meets \(\vert \mathcal{U}\vert \) elements of \(\mathcal{U}\). \par A few of the many very nice results the authors establish, some of which they use to answer previously published questions or raise new ones, are the following. A generalized Isbell-Mrówka \(\Psi\)-space extension of the discrete space \(\omega_1\) (Example 2.6) is a Tychonoff strongly almost cellular-Lindelöf space, but is neither cellular-Lindelöf nor weakly Lindelöf. The Tychonoff space due to \textit{M. Bell} et al. [Pac. J. Math. 79, 37--45 (1979; Zbl 0367.54003)], shown by them to be weakly Lindelöf but not Lindelöf, is also not almost cellular-Lindelöf. A feebly Lindelöf space with a rank 3 diagonal has cardinality at most \(\mathfrak c\). Thus, no first countable almost cellular-Lindelöf space has cardinality \(>\mathfrak c\). For a space \(X\), \(X\times K\) is almost cellular-Lindelöf for each compact space \(K\) iff \(X\times A_\kappa\) is almost cellular-Lindelöf for each infinite cardinal \(\kappa\) (where \(A_\kappa\) denotes the one-point compactification of a discrete space of cardinality \(\kappa\)). For a space \(X\), \(X\times K\) is almost cellular-Lindelöf (resp., almost strongly cellular-Lindelöf) whenever \(K\) is compact iff for every set \(\mathcal{U}\) of nonempty open subsets of \(X\), there is a Lindelöf subspace (resp., a closed Lindelöf subspace) of \(X\) which meets \(\vert \mathcal{U}\vert \)-many elements of \(\mathcal{U}\). Applying this characterization theorem to the space in Example 2.6, they present a proof providing a negative answer to the natural question: If \(X\) is a space such that \(X\times…
Mathematics Subject Classification ID: 54D45
Mathematics Subject Classification ID: 54D99
zbMATH DE Number: 7614061
zbMATH Keywords: Lindelöf space
zbMATH Keywords: feebly Lindelöf space
zbMATH Keywords: cellular-Lindelöf space
zbMATH Keywords: strongly cellular-Lindelöf space
zbMATH Keywords: almost cellular-Lindelöf space
zbMATH Keywords: weakly Lindelöf space
zbMATH Keywords: weakly linearly Lindelöf space
zbMATH Keywords: \(G_{\delta}\)-diagonal
reviewed by: Robert M. Stephenson Jr.
MaRDI profile type: Publication
full work available at URL: s10474-022-01254-x
OpenAlex ID: W4296087258
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DOI: 10.1007/S10474-022-01254-X
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description: scientific article; zbMATH DE number 7614061