# Why the infinity of prime numbers can be proved topologically?

I was reading Proofs from THE BOOK by Martin Aigner, Günter M. Ziegler and was very impressed by the following proof of infinity of prime numbers with topology: Edit: The proof can also be found here.

Though I understand every step of this proof logically, I still find using topology to prove this classical result in Number Theory amazing. I wonder why there is a topological proof to the infinity of prime numbers. What is the intuition that motives this proof? Are there any topological proofs to other famous results in other fields of mathematics?

• To me, the topology almost feels like a wasted and unnecessary layer of abstraction here because the proof is so similar to Euclid's in philosophy. I think of this proof as a direct analogue of Euclid to a topological argument – Cameron Williams Dec 31 '17 at 3:47
• @CameronWilliams: Covering the integers with arithmetic progressions is something number theorists are actually interested in; it's not an artifice invented solely for this proof. – Hurkyl Dec 31 '17 at 4:18
• @Zuriel: Abstract point-set topology is actually fairly useful for more general things than studying things like the continuum; this pattern of "collection of things closed under finite intersections and arbitrary unions" actually crops up in a lot of places (as do some other patterns that also correspond to doing topology). Better to use the existing theory as much as applicable rather than to reinvent the wheel! Also I do not believe this reduces to Euclid's proof either; the combinatorial flavor feels quite different – Hurkyl Dec 31 '17 at 4:20
• Related discussions on MO: mathoverflow.net/questions/42512/…, mathoverflow.net/questions/42589/… – Hans Lundmark Dec 31 '17 at 9:57
• This is the profinite topology on the integers. – Musa Al-hassy Dec 31 '17 at 13:18

## 2 Answers

One answer to the last question (topological proofs to other famous results in other fields of mathematics) can be found in the following example: V.I. Arnold's topological proof of the Abel–Ruffini theorem. A topological proof of this theorem can be found here. Arnold's approach was the starting point for topological Galois theory.

You can sometimes understand a proof and its origin if you summarise it fast, like so. There's a topology on $\mathbb{Z}$ whose open sets are unions of arithmetic progressions; then open sets are nonempty or finite so $S:=\{n\in\mathbb{Z} \mid \not\exists p\in\mathbb{P}:p|n\}=\{-1,\,1\}$ isn't one of them, and each AP is also closed; with finitely many primes, $\{-1,\,1\}$ would be open. OK, so what triggers such thoughts?

It's funny that, while the contents of $S$ as defined above are trivial to work out, the set is defined as a complement with respect to the union of APs whose contents are unknowable without working out the primes. But as soon as you notice this role of unions of APs, you can't help but wonder whether they have any interesting properties that unions of sets do, such as those topology discuss. And since "unions of open sets are open" isn't open$\mapsto$closed-true unless you add "finitely many" into it, it's natural to explore whether the infinitude of the primes comes into this, especially when you notice that the topology you're now thinking of is not only valid, but one in which your open sets are also closed.

As to why it worked out in the end, sadly sometimes the answer is just "when I tried it, it did".