Evenly spaced integer topology

In general topology, a branch of mathematics, the evenly spaced integer topology is the topology on the set of integers $\mathbb {Z}$ = {…, −2, −1, 0, 1, 2, …} 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 Furstenberg (1955) where it was used to prove the infinitude of primes.

Construction

The arithmetic progression associated to two (possibly non-distinct) numbers a and k, where $a,k\in \mathbb {Z} :k\neq 0$, is the set of integers

$a+k\mathbb {Z} :=\{a+k\lambda :\lambda \in \mathbb {Z} \}.$

To give the set $\mathbb {Z}$ a topology means to say which subsets of $\mathbb {Z}$ are open in a manner that satisfies the following axioms:

1. The union of open sets is an open set.
2. The finite intersection of open sets is an open set.
3. $\mathbb {Z}$ and the empty set ∅ are open sets.

The family of all arithmetic progressions does not satisfy these axioms: the union of arithmetic progressions need not be an arithmetic progression itself, e.g., {1, 5, 9, …} ∪ {2, 6, 10, …} = {1, 2, 5, 6, 9, 10, …} is not an arithmetic progression. So 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 as open subsets the family of all arithmetic progressions: 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 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.

References

• Furstenberg, Harry (1955), "On the infinitude of primes", American Mathematical Monthly, 62 (5), Mathematical Association of America: 353, doi:10.2307/2307043, JSTOR 2307043, MR 0068566.
• Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, pp. 80–81, ISBN 0-486-68735-X.