Metrizability of $b$-metric space and $\theta$-metric space via Chittenden's metrization theorem

TL;DR

This paper provides a concise proof that both $b$-metric and $ heta$-metric spaces are metrizable using Chittenden's theorem, removing previous assumptions about the continuity of the distance function.

Contribution

It offers a direct, assumption-free proof of metrizability for $b$-metric and $ heta$-metric spaces, simplifying and generalizing prior results.

Findings

Established metrizability of $b$-metric spaces without continuity assumptions.

Proved $ heta$-metric spaces are metrizable, extending their theoretical foundation.

Simplified proof approach enhances understanding of metric space generalizations.

Abstract

In [An, V.T., Tuyen, Q.L., Dung, V.N., Stone-type theorem on $b$-metric spaces and applications, Topology Appl. 185-186 (2015) 50-64], Tran Van An et al. provide a sufficient condition for $b$-metric space to be metrizable. They proved the metrizability by assuming that the distance function is continuous in one variable. The main purpose of this manuscript is to provide a direct short proof of the metrizability of $b$-metric space introduced by Khamsi and Hussain in \cite[\, Khamsi, M.A and Hussain, N., KKM mappings in metric type spaces, Nonlinear Anal. 73 (9) (2010) 3123-3129]{kh} via Chittenden's metrization theorem without any assumption on the distance function. Further in this short note, we prove the metrizability of $\theta$-metric space introduced by Khojasteh et al. in [Khojasteh, F., Karapinar, E., Radenovic, S., $\theta$-metric space: A Generalization, Mathematical problems in Engineering, Volume 2013, Article 504609, 7 pages].