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. Define a topology on the integers , called the evenly spaced integer topology, by declaring a subset U ⊆ to be an open set if and only if it is a union of arithmetic sequences S(a, b) for a ≠ 0, or is empty (which can be seen as a nullary union (empty union) of arithmetic sequences), where Equivalently, U is open if and only if for every x in U there is some non-zero integer a such that S(a, x) ⊆ U. The axioms for a topology are easily verified: ∅ is open by definition, and is just the sequence S(1, 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, x) ⊆ Ui also shows that S(ai, 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, x) ⊆ S(ai, x) ⊆ Ui. This topology has two notable properties: 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. The basis sets S(a, b) are both open and closed: they are open by definition, and we can write S(a, b) as the complement of an open set as follows: The only integers that are not integer multiples of prime numbers are −1 and +1, i.e. Now, by the first topological property, the set on the left-hand side cannot be closed.