# Infinitely Many Primes (Using Topology)

DevFeed: [Infinitely Many Primes (Using Topology)](<https://devfeed.tech/articles/infinitely-many-primes-using-topology-40287.md>)

Original publisher: [Read original article](<https://www.jeremykun.com/2012/09/26/infinitely-many-primes-using-topology/>)

Published: 2012-09-26T11:51:28Z

Content type: tutorial

Language: en

Sources: [Jeremy Kun](<https://devfeed.tech/sources/jeremy-kun.md>)

Topics: [Sequences](<https://devfeed.tech/topics/sequences.md>), [structure](<https://devfeed.tech/topics/structure.md>)

Tags: [arithmetic](<https://devfeed.tech/tags/arithmetic.md>), [numbers](<https://devfeed.tech/tags/numbers.md>), [primes](<https://devfeed.tech/tags/primes.md>), [sequence](<https://devfeed.tech/tags/sequence.md>), [using](<https://devfeed.tech/tags/using.md>)

## AI overview

A proof that there are infinitely many prime numbers using a topology on the integers whose basis consists of two-sided arithmetic progressions. Assuming finitely many primes leads to a finite open complement, contradicting the fact that no finite set is open in this topology.

## Source excerpt

Problem: Prove there are infinitely many prime numbers. Solution: First recall that an arithmetic progression with difference $ d$ is a sequence of integers $ a_n \subset \mathbb{Z}$ so that for every pair $ a_k, a_{k+1}$ the difference $ a_{k+1} - a_k = d$. We proceed be defining a topology on the set of integers by defining a basis $ B$ of unbounded (in both directions) arithmetic progressions. That is, an open set in this topology is an arbitrary union of arithmetic progressions from $ -\infty$ to $ \infty$.