It's hard not to be amused and perhaps even amazed when first encountering Furstenberg's clever "topological" [proof][1] that there are infinitely many primes. Closer inspection, however, reveals the disappointing truth that there really isn't anything topological going on there, as pointed out by BCnrd in a [comment](https://mathoverflow.net/posts/comments/100562) to [this answer][2].

Nevertheless, the topology on $\mathbb{Z}$ introduced in the proof, where an open set is defined as any union of arithmetic sequences, does seem both natural and interesting.

My question is this: Can anything useful be done with this topology? Useful would include a new theorem, a simplification to a proof of a known result, or even fresh insight into standard material.


  [1]: https://en.wikipedia.org/wiki/Furstenberg's_proof_of_the_infinitude_of_primes
  [2]: https://mathoverflow.net/questions/42512/awfully-sophisticated-proof-for-simple-facts/42517#42517