Evenly spaced integer topology

In general topology, a branch of mathematics, the evenly spaced integer topology is the topology on the set of integers } generated by the family of all arithmetic progressions. It is a special case of the profinite topology on a group. This particular topological space was introduced by where it was used to prove the infinitude of primes.
Construction.
The arithmetic progression associated to two (possibly non-distinct) integers "a" and "k", where formula_1, is the set of integers
To give the set formula_3 a topology means to specify which subsets of formula_3 are "open" in a manner that satisfies the following axioms:
The family of all arithmetic progressions does not satisfy these axioms, e.g. since the union of arithmetic progressions need not be an arithmetic progression itself; for example, } is not an arithmetic progression. Therefore the evenly spaced integer topology is defined to be the topology "generated by" the family of arithmetic progressions. This is the coarsest topology that includes the family of all arithmetic progressions as open subsets: that is, arithmetic progressions are a subbase for the topology. Since the intersection of any finite collection of arithmetic progressions is again an arithmetic progression, the family of arithmetic progressions is in fact a base for the topology, meaning that every open set is a union of arithmetic progressions.
Properties.
The Furstenberg integers are separable and metrizable, but incomplete. By Urysohn's metrization theorem, they are regular and Hausdorff.