On the topology of \(D\)-metric spaces and generation of \(D\)-metric spaces from metric spaces
If \(X\) is a non-empty set, then a \(D\)-metricon \(X\) is a function \(\rho :X\times X\times X\rightarrow [ 0,\infty )\) such that the following are true: 1) \(\rho (x,y,z)=0\) if and only if \(x=y=z\). 2) \(\rho (x,y,z)=\rho (p(x,y,z))\) for any permutation \(p(x,y,z)\) of \( (x,y,z)\). 3) \(\rho (x,y,z)\leq \rho (a,y,z)+\rho (x,a,z)+\rho (x,y,a)\) for all \( x,y,z,a\). The third condition is called the \textit{tetrahedral inequality}. If \(\rho (x,y,z)\) is the area of the triangle with vertices \(x,y,z\), then the reason for the name becomes obvious. Unfortunately, the idea that a \(D\)-\textit{metric} represents area is a not really accurate, since the first condition fails in that case. The author who introduced \(D\)-\textit{metrics} in his original paper claimed that with the natural notion of convergence, a \(D\)-\textit{metric} generates Hausdorff topology with respect to which the \(D\)-\textit{metric} is continuous in all three variables. In this paper the authors construct examples which show the following: \(D\)-\textit{metric} convergence does not always define a topology. If a \(D\)-\textit{metric} does define a topology, it need not be Hausdorff. If a \(D\)-\textit{metric} generates a metrizable topology, it need not be continuous in the individual variables.
scientific article; zbMATH DE number 2171628