Divisor topology

In mathematics, more specifically general topology, the divisor topology is an example of a topology given to the set ${\displaystyle X}$ of positive integers that are greater than or equal to two, i.e., ${\displaystyle X=\{2,3,4,...\}}$. The divisor topology is the poset topology for the partial order relation of divisibility on the positive integers.

To give the set ${\displaystyle X}$ a topology means to say which subsets of ${\displaystyle X}$ are "open", and to do so in a way that the following axioms are met:[1]

1. The union of open sets is an open set.
2. The finite intersection of open sets is an open set.
3. The set ${\displaystyle X}$ and the empty set ${\displaystyle \varnothing }$ are open sets.

Construction

The sets ${\displaystyle S_{n}=\{x\in X:x\mathop {|} n\}}$ for ${\displaystyle n=2,3,...}$ form a basis for the divisor topology[1] on ${\displaystyle X}$, where the notation ${\displaystyle x\mathop {|} n}$ means ${\displaystyle x}$ is a divisor of ${\displaystyle n}$.

The open sets in this topology are the lower sets for the partial order defined by ${\displaystyle x\leq y}$ if ${\displaystyle x\mathop {|} y}$.

Properties

• The set of prime numbers is dense in ${\displaystyle X}$. In fact, every dense open set must include every prime, and therefore ${\displaystyle X}$ is a Baire space.[1]
• ${\displaystyle X}$ is a Kolmogorov space that is not T1. In particular, it is non-Hausdorff.
• ${\displaystyle X}$ is second-countable.
• ${\displaystyle X}$ is connected and locally connected.
• ${\displaystyle X}$ is not compact, since the basic open sets ${\displaystyle S_{n}}$ comprise an infinite covering with no finite subcovering. But ${\displaystyle X}$ is trivially locally compact.
• The closure of a point ${\displaystyle x}$ in ${\displaystyle X}$ is the set of all multiples of ${\displaystyle x}$.

See also

• Zariski topology: A topology on the integers whose open sets are the complements of prime ideals.

References

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