Box products are often discretely generated
type: Item
instance of: scholarly article
title: Box products are often discretely generated
zbMATH Open document ID: 1236.54005
author: Richard G. Wilson
author: Vladimir Tkachuk
published in: Topology and its Applications
publication date: 2011-12-06T00:00:00Z
review text: A space \(X\) is called discretely generated if for every \(A\subseteq X\) and every \(x\in cl_{X}A\), there exists a discrete set \(D\subseteq A\) such that \(x\in cl_{X}D\). This notion was introduced by \textit{A. Dow, M. G. Tkachenko, V. V. Tkachuk} and \textit{R. G. Wilson} [Glas. Mat. Ser. III 37 (1), 187--210 (2002; Zbl 1009.54005)]. Answering questions from that paper, the authors prove that (1) every dyadic compact discretely generated space is metrizable and (2) if there exists an \(\omega_1\)-Souslin line, then there exists a compact dicretely generated space which has no point-countable \(\pi\)-base. Concerning products, it is also proved that (3) every box product of monotonically normal spaces is discretely generated and (4) every finite product of Hausdorff spaces in which every point has a nested local base is discretely generated. Many open problems are raised.
reviewed by: Haruto Ohta
Mathematics Subject Classification ID: 54A20
Mathematics Subject Classification ID: 54B10
Mathematics Subject Classification ID: 54F05
Mathematics Subject Classification ID: 54D10
Mathematics Subject Classification ID: 54C25
zbMATH DE Number: 5986563
zbMATH Keywords: discretely generated space
zbMATH Keywords: monotonically normal
zbMATH Keywords: product
zbMATH Keywords: dyadic compact
zbMATH Keywords: nested local base
MaRDI profile type: Publication
full work available at URL: j.topol.2011.09.004
OpenAlex ID: W2008285705
cites work: Q3684795
cites work: Existence and applications of remote points
cites work: Applications of maximal topologies
cites work: Q3150238
cites work: Q4150804
cites work: Degree of discrete generation of compact sets
cites work: First countable spaces without point-countable π-bases
cites work: Q3938195
cites work: Q3218775
cites work: A solution to the L space problem
cites work: Embeddings and frames of function spaces
cites work: Q3221027
DOI: 10.1016/J.TOPOL.2011.09.004
description: scientific article; zbMATH DE number 5986563