Number of Primes is Infinite/Proof 2

Theorem
There is an infinite number of primes.

Proof
Define a topology on the integers $\Z$ by declaring a subset $U \subseteq Z$ to be an open set it is either:
 * the empty set $\varnothing$

or:
 * a union of sequences $S \left({a, b}\right)$, where:
 * $S \left({a, b}\right) = \{ a n + b : n \in \Z \} = a \Z + b$.

In other words, $U$ is open every $x \in U$ admits some non-zero integer $a$ such that $S(a,x) \subseteq U$.

The axioms for a topology are easily verified:


 * By definition, $\varnothing$ is open: $\Z$ 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 $U_i$ and $x$ in their union $U$, any of the numbers $a_i$ for which $S(a_i,x)\subseteq U_i$ also shows that $S (a_i,x) \subseteq U$.


 * The intersection of two (and hence finitely many) open sets is open:

Let $U_1$ and $U_2$ be open sets and let $x \in U_1 \cap U_2$ (with numbers $a_1$ and $a_2$ establishing membership).

Set $a$ to be the lowest common multiple of $a_1$ and $a_2$.

Then $S(a,x) \subseteq S(a_i,x)\subseteq U_1 \cap U_2$.

The topology is quite different from the usual Euclidean one, and has two notable properties:


 * 1) Since any non-empty open set contains an infinite sequence, no finite set can be open; put another way, the complement of a finite set cannot be a closed set.
 * 2) 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:


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

The only integers that are not integer multiples of prime numbers are $-1$ and $+1$, i.e.


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

By the first property, the set on the left-hand side cannot be closed. On the other hand, by the second property, the sets $S(p,0)$ 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.

Therefore by Proof by Contradiction, there must be infinitely many prime numbers.

Also see

 * Euclid's Theorem and its Corollary