A regular non-weakly discretely generated \(P\)-space
type: Item
instance of: scholarly article
title: A regular non-weakly discretely generated \(P\)-space
zbMATH Open document ID: 1499.54041
author: Yanyan Li
published in: Acta Mathematica Hungarica
publication date: 2022-03-16T00:00:00Z
review text: A space is discretely generated if the topology is generated by its discrete subsets in the sense that if \(x\in\overline A\) then \(x\in\overline D\) for some (relatively) discrete subset \(D\) of \(A\). It is \textit{weakly} discretely generated if every non-closed set \(A\) contains a (relatively) discrete subset \(D\) such that \(\overline D\setminus A\neq\emptyset\).\par After surveying what is known about these properties the authors construct, under the assumption that \(2^{\aleph_0}=\aleph_1\) and \(2^{\aleph_1}=\aleph_2\), a regular \(P\)-space that is not weakly discretely generated. This is related to the result that compact Hausdorff spaces are weakly discretely generated; since Lindelöf \(P\)-spaces behave much like compact Hausdorff spaces the question whether they are also weakly discretely generated is a natural one. The present example is the first regular \(P\)-space that is not weakly discretely generated; but it is not Lindelöf. Thus the original question remains open as well as the question for a regular example in ZFC.
reviewed by: K. P. Hart
Mathematics Subject Classification ID: 54A25
Mathematics Subject Classification ID: 54G10
zbMATH DE Number: 7491525
zbMATH Keywords: discretely generated
zbMATH Keywords: \(P\)-space
zbMATH Keywords: convergence property
MaRDI profile type: Publication
full work available at URL: s10474-022-01214-5
OpenAlex ID: W4210857364
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DOI: 10.1007/S10474-022-01214-5
description: scientific article; zbMATH DE number 7491525