There's fairly well-known proof of infinitude of prime numbers by using topology by Furstenberg. (Here is K.Conrad's note about the proof). As we can see in the last section, Furstenberg's topological proof is actually not topological, but only uses the terminology. In fact, it is almost same as Euclid's famous argument.
So I have some questions about this proof, especially some possible generalizations.
1) How far can we generalize this to an arbitrary (infinite) ring? I think we can prove that there are infinitely many irreducible elements in $\mathcal{O}_{K}$, a ring of integer of some number field $K$, by using the same argument with suitable modification. Also, it seems possible to prove that any $\mathcal{O}_{K}$ has infinitely many prime ideals, at least if it has finitely many units (like totally imaginary fields).
2) Can we really prove something related to topology by using a similar argument? For example, by defining a suitable topology on some set $X$, can we prove that there are infinitely many irreducible subsets, or elements in $X$, where irreducible may depend on context? For example, can we prove that there are infinitely many prime geodesics on a hyperbolic surface by defining some suitable topology on it? (I think this is not a good example)
Thanks in advance.