Furstenberg and Margulis won the Abel Prize today. In honor of this, I spent the better part of the evening trying to prove Furstenberg’s topological proof of the infinitude of primes. I was going down the wrong road at first, but then, after ample hints from Wikipedia and elsewhere, I was able to come up with Furstenberg’s original argument.
Furstenberg’s argument: Consider , and a topology on it in which the open sets are generated by , where . It is easy to see that such sets are also closed. Open sets, being the union of infinite generators, have to be infinite. However, if there are a finite number of primes , then the open set is finite, which is a contradiction.
My original flawed proof: Let be connected sets in this topology. Then, as one can see clearly, ; in other words, it is the union of two open disjoint sets. Therefore, it is not connected. If the number of primes is finite, then , which is itself an open connected set. Hence, as all have a non-empty intersection which is open and connected, the union of all such open sets must lie in a single component. This contradicts the fact that .
This seemed too good to be true. Upon thinking further, we realize the fact that our original assumption was wrong. can never be a connected set, as it is itself made up of an infinite number of open sets. In fact, it can be written as a union of disjoint open sets in an infinite number of ways. This topology on is bizarrely disconnected.