Some properties of bornological convergences
type: Item
instance of: scholarly article
title: Some properties of bornological convergences
zbMATH Open document ID: 1213.54035
published in: Topology and its Applications
publication date: 2011-01-12T00:00:00Z
full work available at URL: http://hdl.handle.net/10251/37064
review text: Let \({\mathcal S}\) be an ideal in a quasi-uniform space \((X,{\mathcal U})\), i.e. a family of non-empty subsets which is closed under the formation of non-empty subsets and finite unions. A net \((A_\lambda)_{\lambda\in \Lambda}\) of non-empty subsets of \(X\) is said to \({\mathcal S}^+_{{\mathcal U}}\)-converge to \(A\) if \(A_\lambda\cap S\subset U(A)\) residually for each \(S\in{\mathcal S}\) and \(U\in{\mathcal U}\). It is said to \({\mathcal S}^-_{{\mathcal U}}\)-converge to \(A\) if \(A\cap S\subset U^{-1}(A_\lambda)\) residually for each \(S\in{\mathcal S}\) and \(U\in{\mathcal U}\). If it both \({\mathcal S}^+_{{\mathcal U}}\)-converges and \({\mathcal S}_{{\mathcal U}}\)-converges to \(A\), it is said to \({\mathcal S}^-_{{\mathcal U}}\)-converge to \(A\). In the first part of this paper, the authors consider the question under which circumstances these convergences are topological. They introduce pretopologies \({\mathcal N}^+_{{\mathcal S},{\mathcal U}}\), \({\mathcal N}^-_{{\mathcal S},{\mathcal U}}\), and \({\mathcal N}_{{\mathcal S},{\mathcal U}}\) on the the set \({\mathcal P}_0(X)\) of all non-empty subsets of \(X\) and prove the following: Theorem 1. Let \({\mathcal S}\) be an ideal in a quasi-uniform space \((X,{\mathcal U})\) and \({\mathcal M}\) a collection of non-empty subsets of \(X\) containing all non-empty finite subsets. Then \({\mathcal N}^+_{{\mathcal S},{\mathcal U}}\) is a topology on \({\mathcal M}\) if and only if the \({\mathcal S}^+_{{\mathcal U}}\)-convergence is topological on \({\mathcal M}\). Theorem 2. If additionally \({\mathcal S}\subset{\mathcal M}\), then \({\mathcal N}^-_{{\mathcal S},{\mathcal U}}\) is a topology on \({\mathcal M}\) if and only if the \({\mathcal S}^-_{{\mathcal U}}\)-convergence is topological on \({\mathcal M}\). Theorem 3. \({\mathcal N}_{{\mathcal S},{\mathcal U}}\) is a topology on \({\mathcal M}\) if and only if the \({\mathcal S}_{{\mathcal U}}\)-convergence is topological on \({\mathcal M}\).…
reviewed by: Harald Brandenburg
Mathematics Subject Classification ID: 54E15
Mathematics Subject Classification ID: 54A05
zbMATH DE Number: 5835579
zbMATH Keywords: quasi-uniform space
zbMATH Keywords: bornological convergence
zbMATH Keywords: hyperspace
zbMATH Keywords: pretopological space
zbMATH Keywords: precompact space
zbMATH Keywords: totally bounded space
author: Jesús Rodríguez-López
author: Miguel Angel Sanchez Granero
MaRDI profile type: Publication
OpenAlex ID: W2068398385
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DOI: 10.1016/J.TOPOL.2010.10.009
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