relatively prime integer topology
Let $X$ be the set of strictly positive integers. The relatively prime integer topology on $X$ is the topology^{} determined by a basis consisting of the sets
$U(a,b)=\{ax+b\mid x\in X\}$ |
for any $a$ and $b$ are relatively prime integers. That this does indeed form a basis is found in this entry. (http://planetmath.org/HausdorffSpaceNotCompletelyHausdorff)
Equipped with this topology, $X$ is ${T}_{0}$ (http://planetmath.org/T0Space), ${T}_{1}$ (http://planetmath.org/T1Space),and ${T}_{2}$ (http://planetmath.org/T2Space), but satisfies none of the higher separation axioms^{} (and hence meet very few compactness criteria).
We can define a coarser^{} 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 ${\mathbb{Z}}^{+}$.
References
- 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 |