A Topological Proof of the Infinitude of Primes

The following is a topological proof of the infinitude of primes due to Furstenberg.

Consider a topology \mathop {\mathcal{T}} on {\mathbb{Z}} generated by family of sets U(a,b)=\{an+b\vert n\in\mathbb{Z}\}=a\mathbb{Z}+b. It is easy to verify that this collection forms a basis for the topology. Now every non empty open set in \mathcal{T} is infinite so that no finite set is open. Also the sets U(a,b) are  both open and closed as they are the basis sets and U(a,b)=\mathbb{Z}\backslash \bigcup_{i=1}^{a-1} U(a,b+j) so that they are closed.

Since -1and 1 are the only integers that are not multiples of primes we have \mathbb{Z}\backslash\{-1,1\}=\bigcup_{p\,prime}U(p,0). Note that since \{-1,1\} is not open ( it is finite ), its complement is not closed. And if there were only finitely many primes, the right side of the above equality will be a finite union of closed sets and hence would be closed which would be a contradiction. This proves the result.

The fact that certain topological property of arithmetic sequences gives you this result is interesting. The topology considered is called Evenly spaced integer topology. There are several other interesting topologies on \mathbb{Z} like prime integer topology and the relatively prime integer topology, divisor topology, etc which give several important results in arithmetic.

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