Stone-type theorem on \(b\)-metric spaces and applications
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 \(d_S: X\times X\to[0,+\infty)\) is defined by \(d_S(x,y)= S(x,x,y)\), then \((X,d_S,2)\) is a \(b\)-metric space whose underlying topology \(\tau_{d_S}\) is homeomorphic to the underlying topology of \((X,S)\). Since \(d_S\) is continuous, it follows that \((X,S)\) is metrizable. Similarly, they show that if \((X,\sigma)\) is a bounded 2-metric space in the sense of \textit{S. Gähler} [Math. Nachr. 26, 115--148 (1963; Zbl 0117.16003)], then there is a natural function \(d_\sigma: X\times X\to[0,+\infty)\) on \(X\) such that \((X,d_\sigma,2)\) is a \(b\)-metric space whose induced topology \(\tau_{d_\sigma}\) is stronger than the topology induced by \(\sigma\). Since \(d_\sigma\) is continuous in one variable, it follows that \((X,\tau_{d_\sigma})\) is metrizable.
scientific article; zbMATH DE number 6418569
scientific article