On s-metric spaces with some topological aspects

Abstract

The notion of a metric space is an important tool in functional analysis, nonlinear analysis and especially in topology. New generalizations of metric spaces have been introduced in recent years. For instance, S-metric and b-metric spaces are among the recent generalizations of a metric space. Fixed point theory has been intensively studied and generalized using various approaches on these new spaces. In this paper we consider the relationships among a metric, an S-metric and a b-metric. In this context, we define the topological equivalence between a metric and an S-metric. Especially, we focus on the fact that every S-metric does not always generate a metric. This is the main motivation of the recent fixed point studies for self-mappings on an Smetric space. Also we revisit the notion of a metric generated by an S-metric. We support our theoretical findings by necessary illustrative examples. As a consequence, existing studies based on the metric generated by an S-metric can be updated using a general S-metric whether generate a metric or not.