relatively prime integer topology

Let X be the set of strictly positive integers. The relatively prime integer topology on X is the topologyMathworldPlanetmathPlanetmath determined by a basis consisting of the sets


for any a and b are relatively prime integers. That this does indeed form a basis is found in this entry. (

Equipped with this topology, X is T0 (, T1 (,and T2 (, but satisfies none of the higher separation axiomsMathworldPlanetmathPlanetmath (and hence meet very few compactness criteria).

We can define a coarserPlanetmathPlanetmath topology on X by considering the subbasis of the above basis consisting of all U(a,b) with a being a prime. This is called the prime integer topology on +.


  • 1 L.A. Steen, J.A.Seebach, Jr., Counterexamples in topology, Holt, Rinehart and Winston, Inc., 1970.
Title relatively prime integer topology
Canonical name RelativelyPrimeIntegerTopology
Date of creation 2013-03-22 14:42:07
Last modified on 2013-03-22 14:42:07
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 7
Author mathcam (2727)
Entry type Definition
Classification msc 54E30
Defines prime integer topology