A measurable cardinal associated with an epireflective subcategory of Hausdorff spaces
type: Item
instance of: scholarly article
title: A measurable cardinal associated with an epireflective subcategory of Hausdorff spaces
zbMATH Open document ID: 1093.54002
author: Anthony W. Hager
author: Richard MacKenzie
published in: Topology and its Applications
publication date: 2005-03-08T00:00:00Z
review text: The authors associate with every non-trivial epireflective subcategory \(\mathbf{A}\) of the category \textbf{Haus} of Hausdorff spaces and continuous maps the cardinal \(\sigma (\mathbf{A}) = \sup \{ | X| \;\mid X\) is a discrete space belonging to \(\mathbf{A}\}\), if the latter set is bounded, and \(\sigma (\mathbf{A})= \infty\) otherwise. Generalizing known results about \(k\)-compact and topologically complete spaces, they prove in particular that (1) \(\sigma (\mathbf{A})\) is non-measurable or equals \(\infty\). (2) If \(k\) is non-measurable or equals \(\infty\) and \(\mathbf{A}\) is the epireflective hull of all discrete spaces \(X\) with \(| X| < k\), then \(\sigma (\mathbf{A}) = k\). (3) A discrete space \(X\) belongs to \(\mathbf{A}\) iff \(| X| < \sigma (\mathbf{A})\). (4) \(\mathbf{A}\) is closed under the formation of topological sums ( = coproducts with at most \(k\) summands) iff \(k < \sigma (\mathbf{A})\).
reviewed by: Horst Herrlich
Mathematics Subject Classification ID: 54B30
Mathematics Subject Classification ID: 54A25
Mathematics Subject Classification ID: 18B30
Mathematics Subject Classification ID: 18A40
zbMATH DE Number: 2142288
zbMATH Keywords: Measurable cardinal
zbMATH Keywords: Epireflective subcategory of Hausdorff spaces
MaRDI profile type: Publication
full work available at URL: j.topol.2004.08.007
OpenAlex ID: W2028886045
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DOI: 10.1016/J.TOPOL.2004.08.007
description: scientific article; zbMATH DE number 2142288