What results can be found from point set topology?

[Kontilera]

Hello!
I'm currently teaching an advanced course in mathematics at high school.
The first half treats discrete mathematics, e.g. combinatorics, set theory for finite sets, and some parts of number theory.

Next year I would like to change some of the subjects in the course. My question is: Are there any interesting results that are not to difficult to reach from point set topology? If that is the case, maybe we could introduce some topology as well.

Thanks!

 

[Number Nine]

There's a very nice (and relatively simple) proof of the infinitude of prime numbers using point-set topology (by Furstenberg):

Define a topology on $\mathbb{Z}$ by taking the open sets to be unions of arithmetic sequences
$$S(a,b) = \{an+b \ | \ n \in \mathbb{Z}, a \neq 0\}$$
Note that every open set contains infinitely many integers, and so no finite set is open, and no complement of a finite set is closed. Conversely, every $S(a,b)$ is closed, because we can construct its complement by taking the union of $S(a,b+i)$ for $i \in (1,2, \dots, a-1)$.

We know that every integer (except 1 and -1) can be written as a product of prime numbers, so
$$\bigcup_\text{p prime} S(p,0) = \mathbb{Z}\setminus \{1,-1\}$$
If there were only finitely many prime numbers, then $\mathbb{Z}\setminus \{1,-1\}$ is a union of finitely many closed sets, and so is closed, but it can't be closed, since it's the complement of a finite set. By contradiction, the set of primes must be infinite.

 

[disregardthat]

That is a cool proof. Should clarify though that every open set is either empty or contains infinitely many integers, and that no complement of a non-empty finite set is closed. Since $\{1,-1\}$ is non-empty, the proof is done!