Covering compacta by discrete and other separated sets
type: Item
instance of: scholarly article
title: Covering compacta by discrete and other separated sets
zbMATH Open document ID: 1171.54002
published in: Topology and its Applications
publication date: 2009-04-23T00:00:00Z
review text: A space is said to be crowded if it has no isolated points. In [Topology Appl. 154, No.~2, 283--286 (2007; Zbl 1107.54004)], \textit{I. Juhász} and \textit{J. van Mill} asked: If \(X\) is a compact Hausdorff crowded space, is it true that \(X\) cannot be covered by fewer than \(\mathfrak c\)-many discrete subspaces? The author proves: (2.2) If \(\kappa\) is an infinite cardinal, and \(X\) is the union of \(\kappa\)-many discrete subspaces, then so is any perfect image of \(X\). As a corollary to 2.2, he obtains an affirmative answer to the preceding question. He also notes that for \(\kappa=\omega\), 2.2 is due to [\textit{D. Burke} and \textit{R. Hansell}, Ann. N. Y. Acad. Sci. 788, 54--56 (1996; Zbl 0914.54029)]. In the final section the author studies coverings by right-separated and left-separated subspaces. Recall that a space \(Y\) is right (resp., left)-separated if there is a well-order on \(Y\) such that each point \(y\in Y\) has a neighborhood missing its successors (resp., predecessors). Two theorems he obtains are the following. If the continuum \(\mathfrak c\) is regular, then no crowded first-countable compact Hausdorff space can be covered by fewer than \(\mathfrak c\)-many right-separated subspaces. If the continuum \(\mathfrak c\) is regular, then no crowded first-countable compact Hausdorff space can be covered by fewer than \(\mathfrak c\)-many left-separated subspaces. He remarks that it is unsolved whether or not either first countability or the restriction on \(\mathfrak c\) is a necessary assumption.
Mathematics Subject Classification ID: 54A25
zbMATH DE Number: 5546618
zbMATH Keywords: compact spaces
zbMATH Keywords: discrete subspaces
zbMATH Keywords: left-separated
zbMATH Keywords: right-separated
author: Gary Gruenhage
reviewed by: Robert M. Stephenson Jr.
MaRDI profile type: Publication
full work available at URL: j.topol.2008.12.021
OpenAlex ID: W2078834567
cites work: Q4229007
cites work: Q3417758
cites work: Covering compacta by discrete subspaces
cites work: A strengthening of the Čech-Pospišil theorem
DOI: 10.1016/J.TOPOL.2008.12.021
Recommended article: A strengthening of the Čech-Pospišil theorem
Recommended article: Q5496133
Recommended article: Q4424985
Recommended article: The cardinality of countably compact Hausdorff spaces
Recommended article: A note on discrete sets
Recommended article: On the cardinality of certain Hausdorff spaces
Recommended article: Covering compacta by discrete subspaces
Recommended article: On partition of topological spaces
Recommended article: A countable widely connected Hausdorff space
Recommended article: Closures of discrete sets in compact spaces
description: scientific article; zbMATH DE number 5546618