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 iff it is either:
 * the empty set $$\varnothing$$ or
 * a union of sequences $$S(a, b)$$, where:
 * $$S(a, b) = \{ a n + b : n \in \Z \} = a \Z + b$$.

In other words, $$U$$ is open iff 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:


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

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


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

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.