On $\star$-metric spaces

TL;DR

This paper investigates $ ext{ extsterling}$-metric spaces, proving their metrizability and exploring properties like total boundedness and completeness, thereby extending the understanding of generalized metric spaces.

Contribution

It demonstrates that every $ ext{ extsterling}$-metric space is metrizable and analyzes key properties such as total boundedness and completeness.

Findings

$ ext{ extsterling}$-metric spaces are metrizable

Conditions for total boundedness in $ ext{ extsterling}$-metric spaces

Conditions for completeness in $ ext{ extsterling}$-metric spaces

Abstract

Metric spaces are generalized by many scholars. Recently, Khatami and Mirzavaziri use a mapping called $t$-definer to popularize the triangle inequality and give a generalization of the notion of a metric, which is called a $\star$-metric. In this paper, we prove that every $\star$-metric space is metrizable. Also, we study the total boundedness and completeness of $\star$-metric spaces.