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, ...}.[1] 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]

1. Arbitrary unions of open sets are open.
2. Finite intersections of open sets are open.
3. Z⁺ and the empty set ∅ are open.

Construction

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

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

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

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

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

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

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.[1]

See also

• Fürstenberg's proof of the infinitude of primes

References

1. Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, ISBN 0-486-68735-X