Arithmetic progression topologies

The arithmetic progression associated to two (possibly non-distinct) integers a and k, where k ≠ 0, is the set of integers

${\displaystyle a+k\mathbb {Z}$ :=\{a+km:m\in \mathbb {Z} \}.}

To give the set ${\displaystyle \mathbb {Z} }$ a topology means to specify which subsets of ${\displaystyle \mathbb {Z} }$ are "open" in a manner that satisfies the following axioms:[5]

1. Arbitrary unions of open sets are open.
2. Finite intersections of open sets are open.
3. ${\displaystyle \mathbb {Z} }$ and the empty set ∅ are open.

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,

${\displaystyle \{1,5,9,\dots \}\cup \{2,6,10,\dots \}=\{1,2,5,6,9,10,\dots \}}$

is not an arithmetic progression.

Therefore, the evenly spaced integer topology is defined to be the topology generated by the family of arithmetic progressions: it is the coarsest topology that includes the family of all arithmetic progressions as open subsets.[6] Because the intersection of any finite collection of arithmetic progressions is again an arithmetic progression, arithmetic progressions form a basis for this topology: every open set is a union of arithmetic progressions.

Similar constructions give other arithmetic progression topologies:

• The evenly spaced integer topology on ℤ has basis

${\displaystyle \{a\mathbb {Z} +b:a,b\in \mathbb {Z} \}}$ (as discussed above).

• The prime integer topology on ℤ⁺ has basis

${\displaystyle \{a\mathbb {N} +b:a{\text{ prime}},b\in \mathbb {Z} ^{+}\}}$.[1]

• The relatively prime integer topology on ℤ⁺,[1] or Golomb space[7] has basis

${\displaystyle \{a\mathbb {N} +b:a,b\in \mathbb {Z} ^{+},\gcd {\!(a,b)}=1\}}$.

• Kirch space is a topology on ℤ⁺ with basis

${\displaystyle \{a\mathbb {N} +b:a,b\in \mathbb {Z} ^{+},\gcd {\!(a,b)}=1,b<a,a{\text{ squarefree}}\}}$.[7][8]

Broughan showed that the evenly spaced integer topology is closely related to the p-adic completion of the rational numbers. Indeed, the Furstenberg integers are homeomorphic to the rationals with the subspace topology inherited from the real line. Thus they are separable and metrizable, but incomplete,[9] and (by Urysohn's metrization theorem) regular and Hausdorff.[10]

All four topologies defined above are Hausdorff.[7]

On the other hand, the Furstenberg integers are totally disconnected.[7] The other topologies above are more connected, but admit fewer metrization axioms. For example, the set of positive integers with the relatively prime integer topology or with the prime integer topology are examples of topological spaces that are Hausdorff but not regular.[1] Both the latter two are connected (but not locally so).[7] Kirch space is both connected and locally connected.[7][8]

The Furstenberg integers are a homogeneous space, because they are a topological ring — in some sense, the only topology on ${\displaystyle \mathbb {Z} }$ for which it is a ring.[11] Conversely, the Golomb integers are rigid — the only self-homeomorphism is the trivial one.[4]

Both the Furstenberg integers and Golomb space furnish a proof that there are infinitely many prime numbers.[3][2] A sketch of the proof runs as follows:

1. Fix a prime p and note that the (positive, in the Golomb space case) integers are a union of finitely residue classes modulo p. Each residue class is an arithmetic progression, and thus clopen.
2. Consider the multiples of each prime. These multiples are a residue class (so closed), and the union of these sets is all (Golomb: positive) integers except the units ±1.
3. If there are finitely many primes, that union is a closed set, and so its complement ({±1}) is open.
4. Every nonempty open set is infinite, so {±1} is not open.

The Furstenberg integers are a special case of the profinite topology on a group.

The notion of an arithmetic progression makes sense in arbitrary ${\displaystyle \mathbb {Z} }$-modules, but the construction of a topology on them relies on closure under intersection. Instead, the correct generalization builds a topology out of ideals of a Dedekind domain.[12] This procedure produces a large number of countably infinite, Hausdorff, connected sets, but whether different Dedekind domains can produce homeomorphic topological spaces is a topic of current research.[12][13][14]

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