Prime integer topology

In mathematics, and especially general topology, the prime integer topology and the relatively prime integer topology are examples of topologies on the set of positive whole numbers, i.e. the set Z⁺ = {1, 2, 3, 4, …}. To give the set Z⁺ a topology means to say which subsets of Z⁺ are "open", and to do so in a way that the following axioms are met:

1. The union of open sets is an open set.
2. The finite intersection of open sets is an open set.
3. Z⁺ and the empty set ∅ are open sets.

Construction

Given two positive integers a, b ∈ Z⁺, define the following congruence class:

$U_{a}(b)=\{b+na\in \mathbf {Z} ^{+}\,|\,n\in \mathbf {Z} \}$

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

${\mathfrak {B}}=\{U_{a}(b)\,|\,a,b\in \mathbf {Z} ^{+},(a,b)=1\}$

and the prime integer topology is the sub-topology generated from the sub-basis

${\mathfrak {P}}=\{U_{p}(b)\,|\,p,b\in \mathbf {Z} ^{+},p{\text{ is prime}}\}$

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