Some new results on $\star$-metric spaces

Abstract

The concept of $\star$-metric, based on the relaxation of triangle inequality of metric axioms by using a t-definer, was introduced by Khatami and Mirzavaziri. This paper extends and generalizes some well-known results of classical metric space. Considering the definition of $\star$-metric space, it studies the notion of a closed ball. The paper proves some results related to closed sets, convergent sequences, Cauchy sequences, and the diameter of a set. This paper contains the study on the metrizability of $\star$-metric space and provides an alternative approach to the proof of metrizability for $\star$-metric space using the famous `Niemytski and Wilson's metrization theorem'.