Hereditary coreflective subcategories of the categories of Tychonoff and zero-dimensional spaces
type: Item
instance of: scholarly article
title: Hereditary coreflective subcategories of the categories of Tychonoff and zero-dimensional spaces
zbMATH Open document ID: 1287.54008
author: Juraj Činčura
published in: Applied Categorical Structures
publication date: 2014-01-30T00:00:00Z
review text: The author investigates hereditary coreflective subcategories in the categories \textbf{Tych} of Tychonoff and \textbf{ZD} of zero-dimensional spaces. It is shown that there are numerous hereditary additive and divisible subcategories in \textbf{Tych} and \textbf{ZD} that are not coreflective. If a hereditary coreflective subcategory \({\mathcal B}\) of \textbf{Tych} (\textbf{ZD}) contains all countable Tychonoff (zero-dimensional) spaces, then \({\mathcal B}\) contains all Tychonoff (zero-dimensional) spaces of nonmeasurable cardinality (in the sense of Ulam). It is also proved, in the case of \textbf{Tych} under Martin's axiom or under the continuum hypothesis, that if \({\mathcal B}\) is a hereditary coreflective subcategory of \textbf{Tych} (\textbf{ZD}), then either the topologies of all spaces belonging to \({\mathcal B}\) are closed under countable intersections or \({\mathcal B}\) contains all Tychonoff (zero-dimensional) spaces of nonmeasurable cardinality. Finally, it is established that if \({\mathcal A}\) is an epireflective subcategory of the category \textbf{Top} of topological spaces which is not bireflective and \({\mathcal B}\) is an additive and divisible subcategory of \({\mathcal A}\) which is not coreflective, then the coreflective hull of \({\mathcal B}\) in \({\mathcal A}\) is not hereditary.
Mathematics Subject Classification ID: 54B30
Mathematics Subject Classification ID: 18D15
Mathematics Subject Classification ID: 18A40
zbMATH DE Number: 6252057
zbMATH Keywords: coreflective subcategory
zbMATH Keywords: epireflective subcategory
zbMATH Keywords: hereditary coreflective subcategory
zbMATH Keywords: hereditary additive and divisible subcategory
reviewed by: Hans Peter Künzi
MaRDI profile type: Publication
full work available at URL: s10485-012-9285-4
OpenAlex ID: W2069203699
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DOI: 10.1007/S10485-012-9285-4
description: scientific article; zbMATH DE number 6252057