Can we adapt Furstenberg's proof of the infinitude of primes to other interesting subsets of the integers?
In a textbook, I was introduced to Furstenberg's topological proof of the infinitude of primes, which goes basically as follows:
We define a topology on $\mathbb{Z}$ such that the sets $$S(a,b)=\set{an+b|n\in\mathbb{N}}$$ for $a$ and $b$ integers, with $a\neq0$, as well as the empty set, are open. You can show without too much work that these sets are also closed. Since the only numbers that are not multiples of some prime are $1$ and $-1$, we can write $$\mathbb{Z}\setminus\set{-1,1}=\bigcup_{p\text{ prime}}S(p,0)\tag{1}$$ $\set{-1,1}$ is not open, and $\mathbb{Z}\setminus\set{-1,1}$ is not closed. Therefore, the right-hand side must be the union of an infinite number of closed sets, which implies an infinite number of primes.
Now, this isn't truly a "topological" proof - it uses no theorems from topology. In fact, we can show that Furstenberg's proof is quite similar to Euclid's classic proof of the infinitude of primes, just somewhat couched in topological language. All the same, I found it helpful to think of the argument in Furstenberg's terms, which led me to a question:
Can we define a different topology on $\mathbb{Z}$ from which we can directly prove the infinitude of another interesting, non-trivial subset of the integers - perhaps a particular type of prime? I would guess maybe not, as the left-hand side of our new version of $(1)$ might end up being of the form $\mathbb{Z}\setminus\set{\text{some open set}}$, and so the proof would break down - but I don't really have anything to support this, and I'm not well-enough versed in topology to have confidence in it.
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