TITLE: Some topologies are more equal than others
QUESTION [8 upvotes]: On MathOverflow 5 years ago, I answered a question about Awfully sophisticated proofs for simple facts. I answered Fürstenberg's topological proof of the infinitude of primes. While the answer ultimately got several votes, there was criticism that the topology that Fürstenberg used was somehow not "real". I'd like to explore this.
The topology in question happens to be the profinite topology on $\mathbb{Z}$, yet still there were complaints. This topology was even dismissed as just "words". But I thought the point of the abstract definition of topology was to deal with the most essential properties of "open" sets (arbitrary unions and finite intersections). If sets that don't appear to be "naturally" open (e.g. open sets of real numbers) can be defined as open and satisfy the topology axioms, why should I be bothered? If anything, I would have thought that the topological connection would have to be interesting, which was the content of my answer.
So my question is: Is there a good reason why some topologies are more important and "natural" than others? Given the abstract definition, I'm not seeing it.

REPLY [6 votes]: Another view of the objection to the "topological" proof can be based on the following extract from the introduction to Peter Freyd's book "Abelian Categories":
"If topology were publicly defined as the study of families of sets closed under finite intersection and infinite unions a serious disservice would be perpetrated on embryonic students of topology.  The mathematical correctness of such a definition reveals nothing about topology except that its basic axioms can be made quite simple. $\dots$ A better (albeit not perfect) description of topology is that it is the study of continuous maps; $\dots$"
In other words, the aspects of "topology" that are used in the proof of the infinitude of the primes are not "really" topology but rather a convenient framework in which to study the real subject matter of topology, which begins with continuous maps.  I think this attitude accounts for the feeling, on the part of many people, that (1) we are not dealing here with a topological proof at all but rather an encoding of a number-theoretic proof and (2) the possibility of such an encoding tells us nothing interesting about topology.