Some metrization problem on \(\nu \)-generalized metric spaces (Q2314663)

Recently, \textit{N. Van Dung} and \textit{V. T. Le Hang} [Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 112, No. 4, 1295--1303 (2018; Zbl 1403.54020)] proved the following theorem: Let \((X, d)\) be a \(\nu\)-generalized metric space. Assume that every convergent sequence is Cauchy. Define a function \(\rho\) from \(X\times X\) into \([0, +\infty)\) by \(\rho(x,y)=\text{inf}\{D(u_0,\ldots,u_n): (u_0,\ldots,u_n)\in X^{n+1},u_0=x, u_n=y\}\). Then \((X,\rho)\) is a metric space. Moreover, the topology on \((X,\rho)\) is compatible with \(d\). In this paper, the author mainly gives sufficient and necessary conditions on the conclusion of the theorem, and also gives sufficient and necessary conditions on the assumption of the theorem.