Prime integer topology

In mathematics, specifically general topology, the prime integer topology and the relatively prime integer topology are topologies on the set of positive integers, i.e. the set Z⁺ = {1, 2, 3, 4, ...}.

Construction

Given two positive integers a and b, define the following congruence class:

$U_a(b)=\{b+na\in {\mathbf{𝐙}}^{+}\mid n\in \mathbf{𝐙}\}.$

Then the relatively prime integer topology is the topology generated from the basis

$\mathfrak{𝔅}=\{U_a(b)\mid a,b\in {\mathbf{𝐙}}^{+},\gcd (a,b)=1\},$

where $\gcd$ is the greatest common divisor function, and the prime integer topology is the topology generated from the subbasis

$\mathfrak{𝔓}=\{U_p(b)\mid p,b\in {\mathbf{𝐙}}^{+},p\text{ is prime},\gcd (p,b)=1\}.$

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.

See also

• Fürstenberg's proof of the infinitude of primes

References

it:Topologia degli interi equispaziati