Stone-type theorem on \(b\)-metric spaces and applications
type: Item
instance of: scholarly article
title: Stone-type theorem on \(b\)-metric spaces and applications
zbMATH Open document ID: 1322.54014
author: Tran Van An
author: Luong Quoc Tuyen
author: Nguyen Van Dung
published in: Topology and its Applications
publication date: 2015-03-23T00:00:00Z
review text: A triple \((X,d,k)\) is called a \(b\)-metric space if \(X\) is a non-empty set, \(k\geq 1\) is a constant, and \(d: X\times X\to[0,+\infty)\) is a function satisfying (1) \(d(x,y)= 0\) if and only if \(x=y\), (2) \(d(x,y)= d(y, x)\), and (3) \(d(x, z)\leq k[d(x,y)+ d(y, z)]\) for all \(x,y,z\in X\). For \(k=2\), \(b\)-metric spaces were introduced by \textit{S. Czerwik} [Acta Math. Inform. Univ. Ostrav. 1, 5--11 (1993; Zbl 0849.54036)]. They were later rediscovered (for arbitrary \(k\)) by \textit{M. A. Khamsi} and \textit{N. Hussain} [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 73, No. 9, 3123--3129 (2010; Zbl 1321.54085)]. Obviously, every metric space is a \(b\)-metric space. As for metric spaces, the underlying topology \(\tau_d\) of a \(b\)-metric space \((X,d,k)\) consists of all open subsets of \(X\), where a subset \(U\) of \(X\) is open if and only if, for every \(x\in U\), there exists an \(\varepsilon>0\) such that \(\{y\in X\mid d(x,y) <\varepsilon\}\subset U\). The authors show that if \((X,d,k)\) is a \(b\)-metric-space, then \((X,\tau_d)\) is semi-metrizable. They point out that several results in the above-mentioned paper of Khamsi and Hussain [loc. cit.] are an immediate consequence of this observation. The main result of the paper is an analogue to Stone's theorem for metric spaces, namely it is shown that if \((X,d,k)\) is a \(b\)-metric space such that \(d\) is continuous in one variable, then \((X,\tau_d)\) is regular and paracompact, hence metrizable. Following \textit{S. Sedghi} et al. [Mat. Vesn. 4, No. 3, 258--266 (2012; Zbl 1289.54158)] a pair \((X,S)\) is called an \(S\)-metric space if \(X\) is a non-empty set and \(S: X\times X\times X\to[0,+\infty)\) is a function satisfying (S.1) \(S(x,y,z)= 0\) if and only if \(x=y=z\), and (S.2) \(S(x,y,z)\leq S(x,x,a)+ S(y,y,a)+ S(z,z, a)\) for all \(x,y,z,a\in X\). The authors of the reviewed paper show that if \((X,S)\) is an \(S\)-metric space and \(d_S: X\times X\to[0,+\infty)…
Mathematics Subject Classification ID: 54E35
Mathematics Subject Classification ID: 54E25
zbMATH DE Number: 6418569
zbMATH Keywords: \(b\)-metric space
zbMATH Keywords: \(S\)-metric space
zbMATH Keywords: 2-metric space
zbMATH Keywords: semi-metrizable space
zbMATH Keywords: paracompact space
zbMATH Keywords: metrization theorem
reviewed by: Harald Brandenburg
MaRDI profile type: Publication
full work available at URL: j.topol.2015.02.005
OpenAlex ID: W2004781734
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DOI: 10.1016/J.TOPOL.2015.02.005
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description: scientific article; zbMATH DE number 6418569