Let $ X $ be a non empty set and $ \tau= \{d\mid d$ is a metric on $X\}$ Define the relation $\sim $ on $\tau$ by $ d \sim d' $ iff $ d $ and $ d'$ are equivalent metrics on $X$. Show that $\sim $ is an equivalence relation on $\tau $. Identify equivalence classes.
I could prove that it is an equivalence relation but I couldn't understand how to identify equivalence classes? Any help would be very much appreciated.
If $X$ is a metric space,you can determine equivalent classes of this equivalent relation with metrizable topologies on $X$ Because,equivalent metrics generate same topology on $X$.