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Hey again. This is just an “extra” text, meant to be some kind of spiritual sucessor to my other text “On primes in some arithmetic progressions”, since we’ll be talking about primes and arithmetic progressions again. But today, I just want to talk about this topological proof of the infinitude of prime numbers made by Furstenberg in his article “On the Infinitude of Primes”. Since Furstenberg’s text is only 12 lines long and I have an exam incoming in a few days, I also want to be brief on this one. So here we go.

First of all, defining some new stuff:

• We’ll denote the domain of the integer numbers by \(\mathbb{Z}\).
• For this text, we’ll consider an arithmetic progression \(a\) to be a function \(a : \mathbb{Z} \to \mathbb{Z}\) such that \(a_n = An+B\), where \(A\) and \(B\) are integers. So we’ll always consider our arithmetic progressions to be infinite and going in both directions, instead of being strict sequences in \(\mathbb{N}\).
• I’ll also frequently denote both the arithmetic progression \(a\) and its image by \(A\mathbb{Z}+B\).

Alright, so the idea of this proof is to first define a topology over \(\mathbb{Z}\) such that a non-empty subset \(B\) of \(\mathbb{Z}\) will be considered open when for every element \(b\) of \(B\) there is an arithmetic progression \(a\mathbb{Z}+b\) for some \(a \geq 1\) such that \(a\mathbb{Z}+b \subset B\). Recalling the definition of a topology, we can verify that such structure is indeed a topology:

• Both \(\emptyset\) and \(1\mathbb{Z}+0 = \mathbb{Z}\) are trivially defined as an open.
• Consider \(\{B_i\}\) to be a collection of open subsets of . Then \(\bigcup B_i\) is open, because for each \(x \in \bigcup B_i\) we can find some \(i\) such that \(b \in B_i\), so \(a\mathbb{Z}+b \subset B_i\) for some \(a \geq 1.\) So, in fact, \(a\mathbb{Z}+b \subset \bigcup B_i\).
• Consider \(X_1, ..., X_n\) to be a finite number of open subsets of \(\mathbb{Z}\). Since we can assume that \(\bigcap_{i=1}^{n} X_i \neq \emptyset\), we have that for some \(b \in \bigcap_{i=1}^{n} X_i\), there are arithmetic progressions \(\alpha_i \mathbb{Z} + b \subset X_i\), where \(\alpha_i \geq 1\). Therefore, we have that \((\alpha_1 \cdots \alpha_n) \mathbb{Z} + b\) is an arithmetic progression contained within each \(\alpha_i \mathbb{Z} + b\), so \((\alpha_1 \cdots \alpha_n) \mathbb{Z} \subset \bigcap_{i=1}^n X_i\). Then, we can see that every element of \(\bigcap_{i=1}^n X_i\) is in an arithmetic progression that’s entirely contained in \(\bigcap_{i=1}^n X_i\), so \(\bigcap_{i=1}^n X_i\) is open.

On this topology we defined over \(\mathbb{Z}\), every non-empty open subset of \(\mathbb{Z}\) is infinite, because every non-empty open subset of \(\mathbb{Z}\) contains an arithmetic progression. Also, every arithmetic progression is both open and closed. To show that it is open, we can immediately see that for every \(an+b\) in \(a\mathbb{Z}+b\), we have that \(an + a\mathbb{Z} + b = a\mathbb{Z} + b\). To show that it is closed, we can show that its complement is open. Its complement is the finite union of arithmetic progressions \(a\mathbb{Z} + b'\), with \(0 \leq b' \leq a-1\) and \(a \not\equiv b'\mod a\). Since every \(a\mathbb{Z}+b'\) is open, the union of \(a-1\) such arithmetic progressions is also open.

Now that we fully defined our topology over \(\mathbb{Z}\), we can quickly prove the Euclid’s theorem. Consider the union \(\bigcup p\mathbb{Z}\) where p is a prime number. Since every integer except \(-1\) and \(1\) have a prime factor, we have that \(\bigcup p\mathbb{Z} = \mathbb{Z} \setminus \{-1,1\}\). Since this subset \(\{-1,1\}\) is not empty and is finite, it is not open, so its complement in \(\mathbb{Z}\) is not closed. But every \(p\mathbb{Z}\) is closed, and a finite union of closed subsets is closed, so \(\bigcup p\mathbb{Z}\) cannot be an union over a finite set, so the primes are infinite.

Finally, just out of curiosity, this topology we defined for \(\mathbb{Z}\) is in fact known as the profinite topology, and was called the evenly spaced integer topology by Steen in Counterexamples in Topology. It was introduced first in infinite Galois Theory and it can be defined via finitely indexed subgroups in any group. It is actually closely related to the \(\mathcal{I}\)-adic topology in a commutative ring, defined using cosets of powers of an ideal \(\mathcal{I}\) in such ring. You can read more about it here if you want.

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Last edited on April 25, 2025}