Does topology apply to the integers? What is the natural topology (or topologies) on the integers. Can we define a metric on the integers? 

We could add (since I don't see it or any equivalent when skimming the comments) the symmetric set topology on $\mathbb{Z}$: $U \subseteq \mathbb{Z}$ is open if $$n \in U \iff -n \in U$$ holds. This topology disconnects the integers and makes them non-compact, but does admit a countable basis (correct me if I'm messing something up here).
