Prove or disprove:
on every non empty set there exist at least two non topological equivalent metrics.
For finite set this is not true because every metric on finite set is topologicaly equivalent with discrete metric since every single point set is open.
For infinite set I think this is true. In special case in real number we have discrete metric , euclidean metric and ... .
Also for countable infinite sets for example $\mathbb{Q}$ we have discrete metric , euclidean metric , p-adic metric and .....
But in any arbitrary infinite set $X$ ( such that $X$ has not group structure and in my examples our metric is defined on group structure but for arbitrary sets we have not group structure) I can find discrete metric and I can't find another metric.
Please give me another example for arbitrary infinite set .
can we find all non topological equivalent metrics on an infinite set ? in This we can count the number of metrics . But can we classify all of them?
