Infinitude of Primes: A Topological Proof

Most of us have long known that there are infinitely many prime numbers. Euclid’s classical argument — dating back over two millennia — elegantly shows this by contradiction. Countless other proofs exist, each unveiling a different aspect of number theory’s depth and beauty.

But what if we could prove the infinitude of primes topologically?

Yes, topology — the study of shapes, continuity, and structure — can step into the arena of arithmetic and reveal the same timeless truth from an entirely new vantage point. The proof, originally due to Hillel Furstenberg, is as breathtakingly simple as it is conceptually profound.

Let’s dive in.

Defining a Topology on the Integers

We begin by equipping the set of integers $\mathbb{Z}$ , with an unusual but elegant topology.

For any integers $a\in\mathbb{Z}$ and positive integer $b\in\mathbb{Z}_{+}$ , define the following subset of $\mathbb Z$ :
$B_{a,b}=\{\,a+kb:k\in\mathbb{Z}\,\}$ .

Each $B_{a,b}$ is nothing but an arithmetic progression with initial term $a$ and common difference $b$ . Alternatively, one can think of $B_{a,b}$ as the residue class of $a$ modulo $b$ .
Now we declare the collection $\mathcal{B}=\{\,B_{a,b}:a,b\in\mathbb{Z},\,b>0\,\}$ as a basis for our topology on $\mathbb{Z}$ .

Checking the Basis Axioms

To ensure this truly defines a topology, let’s verify the basis properties.

Covering:
Every integer lies in some basic open set.
Indeed, for any $x\in\mathbb{Z}$ , we have $x=x+0\cdot b\in B_{x,b},\forall b\in\mathbb Z$ .
Hence, $\bigcup\limits_{a,b\in\mathbb Z,b>0}B_{a,b}=\mathbb{Z}$ .
Intersection Property:
If $x\in B_{a,b}\cap B_{c,d}$ , then one can show that $B_{a,b}\cap B_{c,d}=B_{x,b}\cap&space;B_{x,d}=B_{x,\mathrm{lcm}(b,d)}$ . Thus, the intersection of two basis elements containing $x$ is again a basis element containing $x$ .

Hence, $\mathcal B$ indeed forms a basis for a topology on $\mathbb Z$ .

A Few Curious Observations

This topology behaves quite differently from the familiar Euclidean one. Let’s note a few key features:

Neighborhoods:
The neighborhoods of an integer $x$ are precisely all arithmetic progressions passing through $x$ .
Clopen Sets:
Each $B_{a,b}$ is both open and closed.
Why? Because its complement can be expressed as a union of other basic open sets:
if $x\notin B_{a,b}$ , then $x\in B_{x,b}\subseteq\mathbb{Z}\setminus B_{a,b}$ .
Infinite Open Sets:
Every non-empty open set is infinite, since every arithmetic progression contains infinitely many integers.
Not Discrete:
The topology is not discrete because singleton sets like $\{x\}$ are not open.

Interestingly, this topology turns out to be Hausdorff and regular, despite being so far removed from our intuitive notions of “closeness.”

Towards the Proof — Connecting Primes and Topology

Now comes the magic.

For every integer $x\in\mathbb{Z}\setminus\{\pm 1\}$ , there exists a prime $p$ dividing $x$ .
That means $x=kp$ for some integer $k$ , so $x\in B_{0,p}$ .

Hence, $\mathbb{Z}\setminus\{\pm 1\}=\bigcup_{p\in\mathbb{P}}B_{0,p}$

where $\mathbb P$ denotes the set of all prime numbers.

The Contradiction Argument

Suppose, for the sake of argument, that there are only finitely many primes. Then the union $\bigcup_{p\in\mathbb{P}}B_{0,p}$ would be a finite union of closed sets, and therefore closed itself. That would make $\mathbb{Z}\setminus\{\pm 1\}$ closed, implying that its complement $\{\pm 1\}$ is open.

But here’s the catch — we already know that no non-empty open set can be finite in this topology.

Thus, $\{\pm 1\}$ cannot be open, and our assumption of finitely many primes collapses.

Therefore, the set of primes $\mathbb P$ must be infinite.

Closing Thoughts

Furstenberg’s proof does not rely on divisibility tricks or clever algebraic manipulations — instead, it unveils the structure of the integers through the lens of topology. Arithmetic progressions, residue classes, and primes all interlace naturally within this topological framework.

Hillel Furstenberg (American-Israeli Mathematician)

It’s a striking reminder that mathematics is not a collection of separate silos — number theory and topology, algebra and geometry — but a unified landscape where deep truths can be rediscovered through many different languages.

“Mathematics, rightly viewed, possesses not only truth, but supreme beauty.”
— Bertrand Russell

Manifolds Unfolded