If $X$ is a countable set, does there exist a metric that makes it compact?

Question

If $X$ is a countable set, does there exist a metric that makes it compact? I know there is always a metric that does not make it compact and that there are examples of compact countable spaces but I am unable to figure out if every countable set has a compact metric or not.

Answer 1

Take a countable set $X$. There is a bijection $f:X\to A$ where $$A=\{1/n~\vert~n\geq 1\}\cup \{0\}$$ Give $A$ the metric induced by the metric on $\Bbb R$ (it's just the absolute value). Then define the metric on $X$ by $$d(x,y)=\vert f(x)-f(y)\vert$$ You can check that this is a metric on $X$ and that with this metric, $f$ is an isometry between $X$ and $A$. Therefore $X$ and $A$ are homeomorphic spaces. Because $A$ is compact, so is $X$.

Answer 2

By similar argument X can be homeomorphic to {1/n : n is natural no. } having usual metric. Then A and this set {1/n : n is natural no. } are homeomorphic. But the 1st one is compact where 2nd one is not compact.