On \(\mathcal{I}\)-neighborhood spaces and \(\mathcal{I}\)-quotient spaces
type: Item
instance of: scholarly article
title: On \(\mathcal{I}\)-neighborhood spaces and \(\mathcal{I}\)-quotient spaces
zbMATH Open document ID: 1479.54008
published in: Bulletin of the Malaysian Mathematical Sciences Society. Second Series
publication date: 2021-08-12T00:00:00Z
review text: In this paper the author studies properties of ideal topological spaces, based on the new notion of \(\mathcal{I}_{sn}\)-open sets and related meanings such as the \(\mathcal{I}_{sn}\)-closure and \(\mathcal{I}_{sn}\)-interior operators and \(\mathcal{I}_{sn}\)-continuous mappings. Also, the author defines the so-called \(\mathcal{I}\)-neighborhood spaces, the \(\mathcal{I}_{sn}\)-open topology, denoted by \(\tau_{\mathcal{I}_{sn}}\), and \(\mathcal{I}_{sn}\)-open topological spaces, denoted by \(X_{\mathcal{I}_{sn}}\). Some of the main results of the paper are stated as follows: \begin{itemize} \item[(1)] A topological space \((X,\tau)\) is an \(\mathcal{I}\)-sequential space if and only if \(X_{\mathcal{I}_{sn}}\) is an \(\mathcal{I}\)-sequential space and \(\tau=\tau_{\mathcal{I}_{sn}}\). \item[(2)] A mapping \(f:X\rightarrow Y\) preserves \(\mathcal{I}\)-convergence if and only if provided \(U\) is an \(\mathcal{I}_{sn}\)-open subset of \(Y\), then \(f^{-1}(U)\) is an \(\mathcal{I}_{sn}\)-open subset of \(X\). \item[(3)] A topological space \(X\) is an \(\mathcal{I}\)-neighborhood space if and only if every \(\mathcal{I}\)-continuous mapping on the space \(X\) is an \(\mathcal{I}_{sn}\)-continuous mapping. \end{itemize} Moreover, in order to characterize the finest topology of the range that makes the mapping from a topological space onto a set preserve \(\mathcal{I}\)-convergence (be \(\mathcal{I}_{sn}\)-continuous), the author introduces \(\mathcal{I}_{sn}\)-quotient mappings and discusses some topological properties of \(\mathcal{I}_{sn}\)-quotient spaces. Continuing with the essential results of the paper, the following is stated: \begin{itemize} \item[(4)] Suppose that both \(X\), \(Y\) are topological spaces and \(f:(X,\tau)\rightarrow (Y,\mu)\) is a surjective mapping. Then the following are equivalent. \begin{itemize} \item[(i)] The topology \(\mu\) of the space \(Y\) is the finest topology that makes \(f\) preserve \(\mathcal{I}\)-convergence. \item[(ii…
Mathematics Subject Classification ID: 54A05
Mathematics Subject Classification ID: 54A20
Mathematics Subject Classification ID: 54B15
Mathematics Subject Classification ID: 54C08
Mathematics Subject Classification ID: 54D55
zbMATH DE Number: 7381185
zbMATH Keywords: ideal convergence
zbMATH Keywords: \(G\)-method
zbMATH Keywords: \(\mathcal{I}\)-open set
zbMATH Keywords: \(\mathcal{I}_{sn}\)-open set
zbMATH Keywords: \(\mathcal{I}\)-continuous mapping
zbMATH Keywords: preserving \(\mathcal{I}\)-convergence
zbMATH Keywords: \(\mathcal{I}_{sn}\)-continuous mapping
zbMATH Keywords: \(\mathcal{I}\)-sequential space
zbMATH Keywords: \(\mathcal{I}\)-neighborhood space
zbMATH Keywords: \(\mathcal{I}\)-quotient space
zbMATH Keywords: \(\mathcal{I}_{sn}\)-quotient space
zbMATH Keywords: \(\mathcal{I}\)-topological space
author: Shou Lin
reviewed by: Dimitris Georgiou
MaRDI profile type: Publication
full work available at URL: s40840-020-01043-1
OpenAlex ID: W3099103043
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DOI: 10.1007/S40840-020-01043-1
description: scientific article; zbMATH DE number 7381185