Is there a countable space that is $T_1$ and not metacompact? (A space $(X,\tau)$ is not metacompact iff there is on open cover $\cal{U}_0$ such that for every open refinement $\cal V$ there is $x\in X$ such that $x$ is contained in infinitely many members of $\cal V$.)

Note that $(\omega,\tau)$ with $\tau=\{\emptyset,\omega\}\cup\big\{\{0,n\}:n\in\omega\big\}$ is $T_0$ and not metacompact.