The degree of weakly discretely generated spaces
type: Item
instance of: scholarly article
title: The degree of weakly discretely generated spaces
zbMATH Open document ID: 1324.54045
author: Lucia R. Junqueira
author: Richard G. Wilson
author: Ofelia T. Alas
published in: Acta Mathematica Hungarica
publication date: 2014-08-14T00:00:00Z
review text: A space \(X\) is discretely generated if for every subset \(A\) of \(X\) and every \(x\in \text{cl}\, A\), there is a discrete subset \(D\) of \(A\) such that \(x\in\text{cl}\, D\). \(X\) is weakly discretely generated if for every non-closed subset \(A\) of \(X\) there is a discrete subset \(D\) of \(A\) such that \(\text{cl}\, D\setminus A\neq\emptyset\). For every subset \(A\) of \(X\) and every ordinal \(\alpha\), \([A]_0=A\), \([A]_\alpha=\bigcup\{[A]_\beta:\beta< \alpha\}\) if \(\alpha\) is a limit ordinal and \([A]_\alpha= \bigcup \{\text{cl}\, D: D\) a discrete subset of \([A]_\gamma\}\) if \(\alpha= \gamma +1\). Finally, the degree of discrete generation of a weakly discretely generated space \(X\) is defined by \[ idc(X) =\min \{\alpha : [A]_\alpha = \text{cl}\, A \text{ for all subsets } A \text{ of } X\}. \] This paper contains several new results concerning these notions, particularly, in the classes of pseudocompact, locally compact and Čech-complete spaces.
Mathematics Subject Classification ID: 54D99
Mathematics Subject Classification ID: 54G10
Mathematics Subject Classification ID: 54G20
zbMATH DE Number: 6330522
zbMATH Keywords: discretely generated space
zbMATH Keywords: weakly discretely generated space
zbMATH Keywords: one point compactification
zbMATH Keywords: Stone-Čech compactification
zbMATH Keywords: Katetov extension
zbMATH Keywords: ordinal function idc
MaRDI profile type: Publication
full work available at URL: s10474-014-0417-1
OpenAlex ID: W2080706286
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DOI: 10.1007/S10474-014-0417-1
description: scientific article; zbMATH DE number 6330522