Posts Tagged ‘prime numbers’

Idle remarks on the Furstenburg topology

July 10, 2010

There are numerous proofs of the infinitude of primes in the literature, but for my money the “cutest” is the “topological proof” due to Hillel Furstenburg.

Make $\mathbb{Z}$ into a topological space by taking as a neighborhood basis the set of arithmetic progressions $A(a,d):=\{a+nd:n\in\mathbb{Z}\}$ (where d>0, of course).  It’s not hard to check that this really is a topology, and that it has the following interesting properties.

  1. The union of finitely many arithmetic progressions is both closed and open.  (Look modulo the gcd of the various differences in the progressions; this set and its complement are each unions of residue classes.)
  2. Any open set which is not empty is infinite.  (Obvious, since the neighborhoods are all infinite.)
  3. The set S=\bigcup_{p\mbox{ prime}} (p\mathbb{Z}) is open; by 1, it is also closed if there are only finitely many primes.
  4. But \mathbb{Z}\setminus S=\{\pm 1\} is finite, so \mathbb{Z}\setminus S is not open, and S is not closed.

That’s the whole proof.  Once you introduce the topology, the theorem practically proves itself!

(If someone has written about this topology and given it an official name, then I don’t know about it; “Furstenburg topology” seems as good a name as any for now.)


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