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

It is clear that if you look at the topology on $\Bbb Z$ induced by the natural (Euclidean) metric on the real line you get the discrete topology: every subset is open. This is not so exciting.
But that's not forced: you can define other topologies that turn out to be more interesting and useful.
An important metric that one can define in $\Bbb Z$ (actually on $\Bbb Q$)is the $p$-adic metric which requires fixing a prime number $p$ beforehand and whose basic idea is that numbers that are highly divisible by $p$ are "small", i.e. close to $0$. There's a vast literature on the $p$-adic numbers, which are what you get completing $\Bbb Q$ under the $p$-adic metric, pretty much in the same way one can construct $\Bbb R$ as the completion of $\Bbb Q$ under the Euclidean metric.

As another example you can consider the topology in $\Bbb Z$ whose closed set are those generated by the arithmetic progressions containing $0$. 
Then you can use this topology as follows: consider the union of all non-trivial closed sets. It is easy to check that this union is
$$
\bigcup_{p\ \text{prime}}\{...,-2p-p,0,p,2p,...\}=
\Bbb Z\setminus\{1,-1\}.
$$
Since it is the complementary set of a finite subset it cannot be closed (all basic open sets are infinite for this topology). Thus the union cannot be a finite union. Hence we proved that
$$
\text{there are infinitely many primes.}
$$
