Metrizable and Hausdorff space.

Question

Consider $(\mathbb{Z}, T)$, where $T$ is the topology generated by sets of the form $$A_{m, n}=\{m+nk\ | k\in \mathbb{Z}\}$$ for $m, n\in \mathbb{Z}$ and $n\neq 0$. Then, show that

1. $(\mathbb{Z}, T)$ is Hausdorff.

2. $(\mathbb{Z}, T)$ is meterizable.

For Hausdorff, let $z_1, z_2$ be any two elements of $\mathbb{Z}$. Then, we need to look for two disjoint open sets that act as neighborhoods of $z_1$ and $z_2$. We can construct such open sets. This means $(\mathbb{Z}, T)$ is Hausdorff. But I am not getting how to check (\mathbb{Z}, T)$ is meterizable. Please help.