# 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

 $\displaystyle U(a,b)=\{ax+b\mid x\in X\}$

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 RelativelyPrimeIntegerTopology 2013-03-22 14:42:07 2013-03-22 14:42:07 mathcam (2727) mathcam (2727) 7 mathcam (2727) Definition msc 54E30 prime integer topology