Certain aspects of ideal convergence in topological spaces
type: Item
instance of: scholarly article
title: Certain aspects of ideal convergence in topological spaces
zbMATH Open document ID: 1440.54003
author: Pratulananda Das
author: Sayan Sengupta
author: Marek Bienias
author: Szymon Glab
published in: Topology and its Applications
publication date: 2020-04-07T00:00:00Z
review text: A family \(\mathcal I\) of subsets of a set \(S\) is called an ideal if it is closed under the formation of finite unions and subsets. If \(\mathcal{I}\) is an ideal, a mapping \(f : S \to X\), where \(X\) is a topological space, is called \(\mathcal I\)-convergent to some \(x \in X\) if for all neighbourhoods \(U\) of \(x\) one has \(f^{-1}(X \setminus U) \in \mathcal I\). In other words, \(\mathcal I\) is an ideal if \({\mathcal{F(I)}}= \{S \setminus A : A \in \mathcal{I}\}\) is a filter and \(f\) is \(\mathcal I\)-convergent to \(x\) if \(f(\mathcal{F(I)})\) converges to \(x\). If \(\mathcal K\) is another ideal and \(\mathcal A\) a family of infinite subsets of \(S\), the authors give conditions which guarantee that for each mapping \(f\) which \(\mathcal J\)-converges to some \(x \in X\), where \(X\) is first countable, there is some \(M \in \mathcal A\) such that \(f|M\) is \(\mathcal K\)-convergent to \(x\). Under mild additional conditions for \(X\) the converse is also true. These results generalize known results if, e.g., \(\mathcal A = \mathcal F(\mathcal I)\). In the second part of the paper the authors prove some results if \(S\) is the set of all natural numbers (and so they study sequences), and the third part deals with \(\mathcal I\)-cluster points.
reviewed by: Heinz-Peter Butzmann
Mathematics Subject Classification ID: 54A20
Mathematics Subject Classification ID: 40A35
zbMATH DE Number: 7187652
zbMATH Keywords: function
zbMATH Keywords: \( \mathcal{F} \)-subfunction
zbMATH Keywords: ideal
zbMATH Keywords: \( P^{\mathcal{F}}(\mathcal{K})\) ideals
zbMATH Keywords: reaping family
zbMATH Keywords: tall ideal
zbMATH Keywords: \( \mathcal{I} \)-cluster point
MaRDI profile type: Publication
full work available at URL: j.topol.2019.107005
OpenAlex ID: W2994794149
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cites work: Statistical convergence in topology
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cites work: \(\mathcal I\)-convergence
cites work: $I$ and $I^*$-convergence in topological spaces
cites work: \(I\) and \(I^*\)-convergence of nets
cites work: \({\mathcal I}^{\mathcal K}\)-convergence
cites work: Ideal QN-spaces
Wikidata QID: Q126590141
DOI: 10.1016/J.TOPOL.2019.107005
description: scientific article; zbMATH DE number 7187652