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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 $\max\{x,y\}<p$. 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?

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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.

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