  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}.
