Relators generating the same generalized topology
type: Item
instance of: scholarly article
title: Relators generating the same generalized topology
zbMATH Open document ID: 0769.54026
DOI: 10.1007/BF00051647
author: S. Singh
published in: Acta Mathematica Hungarica
publication date: 1993-04-01T00:00:00Z
review text: Relators were introduced quite recently, as a general approach to uniformity: a relator is a non-void collection of reflexive relations on a set --- see [\textit{Á. Száz}, ibid. 50, 177-201 (1987; Zbl 0643.54033)]. With each relator \(\mathcal R\) on a set \(X\), one associates its interior operator \(\text{int}_{\mathcal R}\) on \(X\) and its topological refinement \(\widehat{\mathcal R}\). According to Száz, (i) \(x\in\text{int}_{\mathcal R}(A)\) iff \({\mathcal R}(x)\subset A\) for some \(R\in{\mathcal R}\), (ii) \(A\) is an \({\mathcal R}\)-open set if \(A\subset\text{int}_{\mathcal R}(A)\), (iii) \(S\in\widehat{\mathcal R}\) iff \(x\in\text{int}_{\mathcal R}(S(x))\), for all \(x \in X\), and (iv) \(\widehat{\mathcal R}\) is the largest relator on \(X\) with the same interior operator as \(\mathcal R\). Here, the author deals with the collection of \({\mathcal R}\)-open sets, obtaining analogous results with the help of the transitive closure of \(\mathcal R\) and/or of \(\widehat{\mathcal R}\) --- in particular, the latter is the largest relator with the same open sets as \(\mathcal R\).
Mathematics Subject Classification ID: 54E15
Mathematics Subject Classification ID: 54A05
Mathematics Subject Classification ID: 54A20
zbMATH DE Number: 149640
zbMATH Keywords: uniformity
zbMATH Keywords: relator
zbMATH Keywords: interior operator
zbMATH Keywords: transitive closure
MaRDI profile type: Publication
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cites work: Q4180760
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full work available at URL: https://doi.org/10.1007/bf00051647
OpenAlex ID: W1977384069
description: scientific article; zbMATH DE number 149640