# Topological proof of infinity of prime numbers and topological properties of the used space

Solving a general topology test from a few semesters ago I found a very interesting exercise, the objective is to conclude that there are infinitely many prime numbers (among other things ). For this we define:

For $$a,b\in\mathbb{Z}$$ we define the arithmetic sequence $$\text{PA}_{a,b}:=a\mathbb{Z}+b:=\{an+b:n\in\mathbb{Z}\}$$ and we define $$\text{PA}_{b}:=\bigcup_{a\in\mathbb{Z}} \text{PA}_{a,b}$$

On the set $$\mathbb{Z}$$ we place the topology $$\mathcal{T}_{\mathbb{Z}}$$ given by the vacuum and by the arbitrary union of progressions $$\text{PA}_{a,b}$$ for $$a\neq0$$.

Is it possible to see that $$(PA_{a,b})^{c} = \bigcup_{j=1}^{a-1}\text{PA}_{a,b+j}$$ and with this we can verify that the sets $$PA_{a,b}$$ are open and closed.

From the above we can prove that $$\mathbb{Z}\setminus\{-1,1\} = \bigcup_{p\text{ prime}}\text{PA}_{p,0}$$ and conclude that there are infinitely many prime numbers (note that $$-1,1$$ they are not prime numbers), because the right hand union cannot be closed or $$\{-1,1\}$$ would be a finite open set, which cannot be.

Some properties that I would be interested in testing in this topological space are:

1. $$(\mathbb{Z},\mathcal{T}_{\mathbb{Z}})$$ satisfies the second axiom of countability?
2. $$(\mathbb{Z},\mathcal{T}_{\mathbb{Z}})$$ is separable?
3. $$(\mathbb{Z},\mathcal{T}_{\mathbb{Z}})$$ is $$T_{2}$$?
4. $$(\mathbb{Z},\mathcal{T}_{\mathbb{Z}})$$ is $$T_{4}$$?
5. In the space $$(\mathbb{Z},\mathcal{T}_{\mathbb{Z}})$$, $$x_{n}\to0$$ when $$n\to\infty$$ if and only if for all $$m>0$$ exist $$n_{0}>0$$ such that, if $$n\geq n_{0}$$, $$x_{n}$$ is divisible by $$m$$

Also using (5) we can prove that $$\displaystyle\lim_{n\to\infty}n!=0$$ (I have my doubts about this, but in the test they ask for it).

I have this: To prove (1), considering the countable base (verifiable): $$\mathcal{B} = \{\text{PA}_{a,b}:a,b\in\mathbb{Z}\}$$ it is possible to conclude that the topological space satisfies the second axiom of countability, then it is also separable (2).

For (3): Let $$x,y\in\mathbb{Z}$$. Since the prime numbers are infinite, we know that there exists a prime number such that $$\max\{x,y\}. We also have to $$x\in\text{PA}_{p,x}$$ and $$y\in\text{PA}_{p,y}$$ for $$n=0$$. Since $$p$$ is prime and greater than $$x,y$$, then $$\text{PA}_{p,x}\cap\text{PA}_{p,y}=\emptyset$$ With the above we can conclude that the topological space is $$T_2$$.

I don't see how to approach (4) and (5). An idea I had was to see $$T_{4}$$ with the urysohn lemma, but this topology seems to me not to be metrizable, since it is different from the usual distance topology in $$\mathbb{Z}$$

Pd1: Hope to upload detailed solution after solving this.

Pd2: Do you know if it is possible to obtain other results of number theory from the general topology?

• topology.jdabbs.com/spaces/S000053 May 20, 2021 at 12:57
• No, there are no infinite prime numbers. May 20, 2021 at 13:46
• @KennyLau no, it's this one instead. May 20, 2021 at 16:05

See the Wikipedia page for some of its properties, or its $$\pi$$-base page for even more of them.
Obviously separable as $$\Bbb Z$$ is itself countable and dense; the given defiing base is also countable.
All basic sets are clopen as remarked so together with $$T_2$$ this implies $$T_3$$ and that with second countable implies $$T_4$$ and $$T_5, T_6$$ too here. The space is even metrisable by Urysohn.
The fact about $$\lim n!$$ is quite easy: a basic neighbourhood of $$0$$ is of the form $$U=\text{PA}_{m,0}$$ and so for all $$n \ge m$$, $$n! \in U$$ etc.
No point of this space is isolated, so a classical theorem says it's homeomorphic to $$\Bbb Q$$. It's a "gimmick space" to get a topological proof of the infinitude of primes, and not important beyond that I think.