Abstract: 
This article gives a formal version of Furstenberg's
topological proof of the infinitude of primes. He defines a topology
on the integers based on arithmetic progressions (or, equivalently,
residue classes). Using some fairly obvious properties of this
topology, the infinitude of primes is then easily obtained.
Apart from this, this topology is also fairly ‘nice’ in
general: it is second countable, metrizable, and perfect. All of these
(wellknown) facts are formally proven, including an explicit metric
for the topology given by Zulfeqarr. 
BibTeX: 
@article{Furstenberg_TopologyAFP,
author = {Manuel Eberl},
title = {Furstenberg's topology and his proof of the infinitude of primes},
journal = {Archive of Formal Proofs},
month = mar,
year = 2020,
note = {\url{http://isaafp.org/entries/Furstenberg_Topology.html},
Formal proof development},
ISSN = {2150914x},
}
