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 $S$-metric
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.