I am trying to prove, that given a metric on a finite set it induces exactly one topology. I have an idea which might lead to a proof, but am not sure:
For a finite set X with a given metric d we can prove it is a discrete topology:
$\forall x \in X$ take $r$ s.t. $ r := \min_{y \in X}(d(x, y))$. We can do this as $X$ is finite. This way we can construct open balls for each $x$ such that they contain only x. So each x is in a open subset which is a singleton. And therefore we have a discrete topology.
I am not sure how to proceed showing that it is only this topology that we can get from the metric space.
Thank you.