Some remarks on the metrizability of some well known generalized metric-like structures

TL;DR

This paper refines the understanding of when generalized metric-like spaces are metrizable, removing previous assumptions and providing simpler proofs for several classes of such spaces.

Contribution

It improves existing metrizability results for b-metric, -metric, and ta-metric spaces by removing assumptions and offering shorter, more straightforward proofs.

Findings

Removed the continuity assumption for b-metric spaces.

Provided shorter proofs for -metric space metrizability.

Presented an alternative proof for ta-metric space metrizability.

Abstract

In \cite[\, An, V.T., Tuyen, Q.L. and Dung, V.N., Stone-type theorem on $b$-metric spaces and applications, Topology Appl. 185-186 (2015), 50-64.]{an}, An et al. had provided a sufficient condition for $b$-metric spaces to be metrizable. However, their proof of metrizability relied on an assumption that the distance function is continuous in one variable. In this short note, we improve upon this result in a more simplified way without considering any assumption on the distance function. Moreover, we provide two shorter proofs of the metrizability of $\mathcal{F}$-metric spaces recently introduced by Jleli and Samet in \cite[\, Jleli, M. and Samet, B., On a new generalization of metric spaces, J. Fixed Point Theory Appl. (2018) 20:128]{JS1}. Lastly, in this short note, we give an alternative proof of the metrizability of $\theta$-metric spaces introduced by Khojasteh et al. in \cite[\, Khojasteh, F., Karapinar, E. and Radenovic, S., $\theta$-metric space: A Generalization, Math. Probl. Eng. Volume 2013, Article 504609, 7 pages]{ks}.