On products of discretely generated spaces
type: Item
instance of: scholarly article
title: On products of discretely generated spaces
zbMATH Open document ID: 1107.54006
author: Eva Murtinová
published in: Topology and its Applications
publication date: 2007-01-09T00:00:00Z
review text: A space \(X\) is said to be discretely generated if for any \(A\subseteq X\) and any \(x\in \overline{A}\), there is a discrete \(D\subset A\) such that \(x\in\overline{D}\). Two related notions are also considered in this paper: \(X\) is weakly discretely generated if any non-closed subset \(A\) contains a discrete subset that is not closed; and \(X\) is strongly discretely generated if for any subset \(A\), any \(x\in \overline{A}\) is contained the closure of a strongly discrete subset of \(A\) (i.e., a discrete subset whose points can be separated by a pairwise disjoint family of open sets). Two problems from [\textit{A. Dow, M. G. Tkachenko, V. V. Tkachuk} and \textit{R. G. Wilson}, Glas. Mat., III. Ser. 37, No. 1, 187--210 (2002; Zbl 1009.54005)] are considered: (1) If \(X\) is compact and discretely generated is also \(X\times X\) discretely generated? and (2) If \(X\) is weakly discretely generated is also \(X\times X\) weakly discretely generated? The following counterexamples are constructed: 1) Assuming CH a compact discretely generated space such that \(X\times X\) is not discretely generated. 2) Examples of strongly discretely generated spaces whose square is not weakly discretely generated are constructed under a number of different assumptions (CH or MA(countable) or the existence of a measurable cardinal). In addition it is proven that a product of two discretely generated spaces is discretely generated assuming that one of the factors is compact and strongly discretely generated. A corollary to this result is that the product of a countable family of compact strongly discretely generated spaces is discretely generated. Finally, it is shown that a countable Hausdorff space of weight less than the dominating number is strongly discretely generated.
reviewed by: Paul J. Szeptycki
Mathematics Subject Classification ID: 54A35
Mathematics Subject Classification ID: 54B10
Mathematics Subject Classification ID: 54G20
zbMATH DE Number: 5083209
zbMATH Keywords: discretely generated
zbMATH Keywords: weakly discretely generated
zbMATH Keywords: strongly discretely generated
MaRDI profile type: Publication
full work available at URL: j.topol.2005.11.014
OpenAlex ID: W2068634646
cites work: Q4844091
cites work: Spaces which are generated by discrete sets.
cites work: Q3038176
cites work: Q5186348
cites work: Q3150238
cites work: Q4289973
DOI: 10.1016/J.TOPOL.2005.11.014
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description: scientific article; zbMATH DE number 5083209