Divisor topology

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

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

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

Construction

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

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

Properties

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

See also

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