Some remarks on the metrizability of some metric-like structures

Description

The main purpose of this article is to provide alternative proofs of the metrizability of metric-like spaces like $b$ -metric spaces, $\mathcal{F}$ -metric spaces, and $\theta$ -metric spaces. We improve upon the metrizability result of An et al. [Topology Appl. 185–186 (2015)] for $b$ -metric spaces. Moreover, we provide two shorter proofs of the metrizability of $\mathcal{F}$ -metric spaces, recently introduced by Jleli and Samet. Furthermore, we give a partial answer to an open problem regarding the openness of $\mathcal{F}$ -open balls in $\mathcal{F}$ -metric spaces. Finally, we give an alternative proof of the metrizability of $\theta$ -metric spaces.