Different metric structure on $\mathbb{Z}$

Question

Is it possible to equip $\mathbb{Z}$ with a metric such that the closed sets are precisely the finite subsets and $\mathbb{Z}$?

Answer 1

No, $\mathbb{Z}$ is closed (and not finite) in every topology on $\mathbb{Z}$.

If you meant to say "the closed sets other than $\mathbb{Z}$ are finite", then the answer is still no. If the finite sets and $\mathbb{Z}$ are the only closed sets, then the only open sets are the cofinite ones (i.e. complement of a finite set) and $\varnothing$. This is known as the cofinite topology.

You can prove that the cofinite topology on any infinite set is not Hausdorff, and since every metrizable space is Hausdorff, your question is answered.

Answer 2

If one excludes the trivial close set $\Bbb Z$, then the topology one's looking for is the cofinite topology. This topology is not Hausdoff, hence not a metric topology. So the answer is still "NO".