What are some of the nontrivial way to define open set on natural numbers?

Question

We can define topology/open sets on real numbers by defining a metrics first. I'm curious of what we can do with natural numbers that is interesting/nontrivial.

Answer 1

As Noah says in the comments, if you don't require any compatibility with the usual arithmetic operations on the natural numbers then you're just asking what the countable topological spaces spaces. Of course there are many of these, e.g. any countable subset of any other topological space you care about. It's unclear to me what can be usefully said at this level of generality.

There are some topologies on $\mathbb{N}$ that interact nicely with arithmetic. One of them is the Furstenburg topology, whose open sets are unions of arithmetic progressions. Furstenburg famously gave a proof using this topology that there are infinitely many primes. As Henno says in the comments, there are also the $p$-adic topologies, where we only consider arithmetic progressions with common difference a power of a particular prime $p$. These are important in number theory.