Topological Proof of the Infinitude of Primes
Theorem. There exists infinitely many prime numbers.
Proof. Construct a topology on Z as follows: For a, b " Z with b > 0, define
Na,b = {a + nb : n " Z}.
Call set O open if O = " or if for every x " O, there exists some b > 0 such that Na,b Ä…" O.
Then it s obvious that arbitrary unions are again open. To see that finite intersections
are open, let m be some positive integer and whenever the sets O1, O2, . . . , Om are open
with y " O1 )" O2 )" · · · )" Om and Na,b Ä…" Oi, then take b = lcm{b1, b2, . . . , bm} so that
i
y " Na,b Ä…" O1 )" O2 )" · · · )" Om. Hence, we conclude that we have defined a topology on
Z.
Note two properties of the topology:
(1) Any non-empty open set is infinite.
(2) Any set Na,b is closed.
The first property follows from our definition of open set. The second property from the
fact that
b-1

Na,b = Z \ Na+i,b
i=1
proving that Na,b is the complement of an open set and so is closed.
Since every n " Z with n = 1, 0, -1 has at least one prime divisor p, every n is contained

in the set

Z \ {-1, 1} = N0,p
p"P
where p is taken over all the primes.

Punchline: If the primes P were finite, then N0,p would be a finite union of closed
p"P
sets and would have to be closed by property 2. But the complement of a closed set is
an open set, so the set {-1, 1} would have to be open and infinite by property 1. So we
get a contradiction and the primes must be infinite in number.
References
[1] H. Furstenberg, On the Infinitude of Primes, Amer. Math. Monthly 62, 1955, 353.
[2] I. Niven, An Introduction to the Theory of Numbers, John Wiley & Sons, New York,
1991, 34.
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