On Frink's metrization technique and applications

TL;DR

This paper explores Frink's metrization technique, providing counterexamples, resolving conjectures, and extending its application to 2-generalized metric spaces and $b$-metrics, including the Banach contraction principle.

Contribution

It offers new insights into Frink's construction, proves a metrization theorem for 2-generalized metric spaces, and connects $b$-metrics with classical metric space results.

Findings

Counterexample illustrating limits of Frink's construction

Resolution of two conjectures by Berinde and Choban

Extension of Banach contraction principle to $b$-metric and 2-generalized metric spaces

Abstract

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.