Furstenberg's proof of the infinitude of primes

In mathematics, particularly in number theory, Hillel Furstenberg's proof of the infinitude of primes is a topological proof that the integers contain infinitely many prime numbers. When examined closely, the proof is less a statement about topology than a statement about certain properties of arithmetic sequences. Unlike Euclid's classical proof, Furstenberg's proof is a proof by contradiction. The proof was published in 1955 in the American Mathematical Monthly while Furstenberg was still an undergraduate student at Yeshiva University.

Furstenberg's proof
Define a topology on the integers $$\mathbb{Z}$$, called the evenly spaced integer topology, by declaring a subset U ⊆ $$\mathbb{Z}$$ to be an open set if and only if it is a union of arithmetic sequences S(a,&thinsp;b) for a ≠ 0, or is empty (which can be seen as a nullary union (empty union) of arithmetic sequences), where


 * $$S(a, b) = \{ a n + b \mid n \in \mathbb{Z} \} = a \mathbb{Z} + b. $$

Equivalently, U is open if and only if for every x in U there is some non-zero integer a such that S(a,&thinsp;x) ⊆ U. The axioms for a topology are easily verified:


 * ∅ is open by definition, and $$\mathbb{Z}$$ is just the sequence S(1,&thinsp;0), and so is open as well.
 * Any union of open sets is open: for any collection of open sets Ui and x in their union U, any of the numbers ai for which S(ai,&thinsp;x) ⊆ Ui also shows that S(ai,&thinsp;x) ⊆ U.
 * The intersection of two (and hence finitely many) open sets is open: let U1 and U2 be open sets and let x ∈ U1 ∩ U2 (with numbers a1 and a2 establishing membership). Set a to be the least common multiple of a1 and a2. Then S(a,&thinsp;x) ⊆ S(ai,&thinsp;x) ⊆ Ui.

This topology has two notable properties:


 * 1) Since any non-empty open set contains an infinite sequence, a finite non-empty set cannot be open; put another way, the complement of a finite non-empty set cannot be a closed set.
 * 2) The basis sets S(a,&thinsp;b) are both open and closed: they are open by definition, and we can write S(a,&thinsp;b) as the complement of an open set as follows:


 * $$S(a, b) = \mathbb{Z} \setminus \bigcup_{j = 1}^{a - 1} S(a, b + j).$$

The only integers that are not integer multiples of prime numbers are &minus;1 and +1, i.e.


 * $$\mathbb{Z} \setminus \{ -1, + 1 \} = \bigcup_{p \mathrm{\, prime}} S(p, 0).$$

Now, by the first topological property, the set on the left-hand side cannot be closed. On the other hand, by the second topological property, the sets S(p,&thinsp;0) are closed. So, if there were only finitely many prime numbers, then the set on the right-hand side would be a finite union of closed sets, and hence closed. This would be a contradiction, so there must be infinitely many prime numbers.

Topological properties
The evenly spaced integer topology on $$\Z$$ is the topology induced by the inclusion $$\Z\subset \hat\Z$$, where $$\hat\Z$$ is the profinite integer ring with its profinite topology.

It is homeomorphic to the rational numbers $$\mathbb{Q}$$ with the subspace topology inherited from the real line, which makes it clear that any finite subset of it, such as $$\{-1, +1\}$$, cannot be open.