Some topologies are more equal than others

Question

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.

Answer 1

You misunderstand the criticism. Brian Conrad is not saying that the evenly spaced integer topology, meaning the actual family of open sets, is just words: he’s saying that the proof is just Euclid’s proof in a topological disguise, that the topological trappings are entirely superficial, and that the argument does not use topology in any essential way. In particular, he’s not saying that this topology is unimportant or unnatural, and your title does not address his actual objection.

That said, there’s much truth in his objection. It’s quite true that the argument does not use any actual topological theorems and is from a topological point of view trivial. It’s also true that the demonstration that the sets $a\Bbb Z+b$ actually are a base, and indeed a clopen base, for a topology on $\Bbb Z$ is basically number- or group-theoretic.

On the other hand, the demonstration uses such very elementary number-theoretic facts that they could just as well be viewed as not really belonging to any particular area. In the unlikely event that one encountered Furstenberg’s proof first, one might well then see Euclid’s as a restatement of it in more obviously number-theoretic terms. More important, the language in which a problem or argument is cast does affect the way we naturally think about it, and the correspondence between Euclid’s and Furstenberg’s argument is probably not instantly obvious to most people. Thus, while Furstenberg’s argument is arguably not essentially topological or essentially different from Euclid’s, it does in fact use the machinery of topology (albeit in a fairly trivial way) and does thereby display the argument in significantly a different light. It also begins to suggest a way of looking at certain kinds of combinatorial number-theoretic problems that has proved quite productive.

In short, his pronouncement from on high is a bit overdrawn, but the basic objection is not without justification.

Answer 2

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.

Answer 3

I'm going to post this as an answer rather than a long comment to @Brian M. Scott's answer, which I appreciate and may well accept.

I did not misunderstand the criticism, and I do understand that the use of topology is trivial in this example. But my point was that the existence of a result like Furstenberg is one of those instances where we can see that math "got the abstraction right," in this case the abstract definition of a topology. The fact that the number-theoretic approach makes more sense as the right setting for the problem does not seem to weaken the case that the topological properties exist by definition, and the fact that they can get to the same result is at least interesting, even if trivial. This seems to be Baby Dragon's point.

Now against this, the abstract definition of topology is very general, so general that it is an important part of topology to know and understand the many counterexamples, which exist precisely because an arbitrary topology can have very little structure. And perhaps many of those counterexamples are interesting only as counterexamples and nothing more.

To sum up, my initial thought was that there was no reason to think that such a connection between a problem of number theory and topology should exist at all, so that such a connection exists ought to be more interesting. The fact that the proof uses nothing more than the basic definitions of topology and no other deeper result does make the problem topologically trivial, but that it has any topological cast at all still seems relevant. I think this means I basically agree with Brian M. Scott's answer with a bit more of a tilt towards finding the connection more interesting than many did on MathOverflow. So this was helpful--thanks.

EDIT: I wanted to add that Andreas Blass's answer is excellent. He notes that the Furstenberg proof is not categorical, which is a major insight, at least to me! Admittedly my category theory is weak, but I know that it is important, so I appreciate this insight.