A Couple of Other Connections Between Number Theory and Topology

an_amazing_connection_between_the_riemann_hypothesis_and_topology-1 Download

Following the video I made (link and notes above) about how RH can be restated via topology, I decided to follow it up with a little addendum- two other connections (not as deep) between number theory and topology.

Infinitude of primes

Firstly, one can show the infinitude of primes using topology. More precisely, we will put a topology on $\mathbb{Z}$ and show that there can’t be a finite number of primes. Firstly, let $N(a,b)=\{a+nb: n\in\mathbb{Z}\}$ . Then we define open sets as follows:

Definition:

A set $U$ is open if for every $a\in U$ , there exists $b>0$ such that $N(a,b)\subset U$ .

Note the following important facts:

Any non-empty, open set is infinite.
All of the $N(a,b)$ s are clopen.

Now here’s the punchline: since every number is a product of primes, we have that:

$\mathbb{Z}\setminus\{\pm 1\}=\bigcup N(0,p)$

If there were a finite number of primes, then the union on the right would be closed, which would directly imply that $\{\pm 1\}$ is open in this topology, but this would contradict that all open sets in this topology are open. Therefore, there are infinitely many primes.