Furstenberg’s topological proof of the infinitude of primes

by ayushkhaitan3437

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 \Bbb{N}, and a topology on it in which the open sets are generated by \{a+b\Bbb{N}\}, where a,b\in\Bbb{N}. 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 p_1,\dots,p_n, then the open set \Bbb{N}\setminus (\cup_i \{p_i\Bbb{N}\})=\{1\} is finite, which is a contradiction.

My original flawed proof: Let \{a+b\Bbb{N}\} be connected sets in this topology. Then, as one can see clearly, \Bbb{N}=\{2\Bbb{N}\}\cup\{1+2\Bbb{N}\}; 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 \cap \{p_i\Bbb{N}\}=\{p_1p_2\dots p_n\Bbb{N}\}, which is itself an open connected set. Hence, as all \{p_i\Bbb{N}\} have a non-empty intersection which is open and connected, the union of all such open sets \cup \{p_i\Bbb{N}\} must lie in a single component. This contradicts the fact that \cup\{p_i\Bbb{N}\}=\Bbb{N}.

This seemed too good to be true. Upon thinking further, we realize the fact that our original assumption was wrong. \{a+b\Bbb{N}\} 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 \Bbb{N} is bizarrely disconnected.