Splitting closure operators
type: Item
instance of: scholarly article
title: Splitting closure operators
zbMATH Open document ID: 1082.18003
author: G. C. L. Brümmer
author: Eraldo Giuli
author: David Holgate
published in: Acta Mathematica Hungarica
publication date: 2006-01-26T00:00:00Z
review text: The authors extend Arkhangel'skiĭ's notion of splitting space to complete \((\mathcal{E, M})\)-categories: Suppose that \(\mathcal{X}\) is such a category and \(\mathcal{P}\) is a subclass of \(\mathcal{X}\). An object \(X\) belongs to the splitting class of \(\mathcal{P}\) if and only if for every subobject \(m\) of \(X\) there exist an object \(P\in\mathcal{P}\) and a morphism \(f:X\to P\) such that \(m=f^{-1}(f(m))\) (i.e., every subobject is a \(\mathcal{P}\)-piece). In section 2 of the paper basic properties of splitting classes are presented. In Section~3 an idempotent closure operator \(\text{spl}^{\mathcal{P}}\) is constructed. Then a splitting closure operator is defined as a closure operator \(\mathbf{c}\) for which there exists a class \(\mathcal{P}\) such that \(\mathbf{c}= \text{spl}^{\mathcal{P}}\). The authors illustrate every notion introduced in this paper with examples in classical categories as the category of left modules, the category of topological spaces and the category of Tychonoff spaces.
Mathematics Subject Classification ID: 18A32
Mathematics Subject Classification ID: 18A40
Mathematics Subject Classification ID: 54B30
Mathematics Subject Classification ID: 13C13
zbMATH DE Number: 5001793
zbMATH Keywords: splitting space
zbMATH Keywords: splitting class
zbMATH Keywords: splitting closure operator
zbMATH Keywords: regular closure operator
zbMATH Keywords: morphism factorization structure
reviewed by: Simion Sorin Breaz
MaRDI profile type: Publication
full work available at URL: s10474-005-0001-9
OpenAlex ID: W2034563530
DOI: 10.1007/S10474-005-0001-9
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description: scientific article; zbMATH DE number 5001793