Compact space on ℤ

Question

I want to make $ℤ$ (or equivalently, any countably infinite set) a compact space, as long as it satisfies a separation axiom. What is the maximum possible value of $x$ such that the space satisfies $T_x$ axiom? Could the space be metric, or even completely metric?

(I'm going to make this a challenge to post on Code Golf SE, tbh.)

Answer 1

One way to make $\Bbb Z$ a compact space is as follows. Define $d(z_1,z_2) = |f(z_1) - f(z_2)|$ where $$ f(z) = \begin{cases} 0 & z=0\\ 1/z & z \neq 0.\\ \end{cases} $$ As far as separation axioms go, the resulting space is at least normal Hausdorff, so $x \geq 4$.