On Frink's metrization technique and applications

In this paper, we give a simple counterexample to show again the limits of Frink's~construction and then use Frink's metrization technique to answer two conjectures posed by Berinde and Choban, and to calculate corresponding metrics induced by some $b$-metrics known in the literature. We also use that technique to prove a metrization theorem for 2-generalized metric spaces, and to deduce the Banach contraction principle in $b$-metric spaces and 2-generalized metric spaces from that in metric spaces.