On the topological equivalence of S-metric and cone S-metric spaces

TL;DR

This paper demonstrates the topological equivalence between S-metric and cone S-metric spaces, introduces a new TVS-cone S-metric space concept, and shows how existing results transfer between these frameworks.

Contribution

It establishes the topological equivalence of S-metric and cone S-metric spaces and introduces the TVS-cone S-metric space concept, unifying different approaches.

Findings

Topological equivalence between S-metric and cone S-metric spaces

Introduction of TVS-cone S-metric space concept

Known results on cone S-metric spaces derived from S-metric space studies

Abstract

The aim of this paper is to establish the equivalence between the concepts of an $S$-metric space and a cone $S$-metric space using\ some topological approaches. We introduce a new notion of $TVS$-cone $S$-metric space using some facts about topological vector spaces. We see that the known results on cone $S$-metric spaces (or $N$-cone metric spaces) can be directly obtained from the studies on $S$-metric spaces.