Divisor topology

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

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 ${\displaystyle x\mathop {|} y}$. The closed sets are the upper sets for this partial order.

Properties

All the properties below are proved in ^[1] or follow directly from the definitions.

• The closure of a point ${\displaystyle x\in X}$ is the set of all multiples of ${\displaystyle x}$.
• Given a point ${\displaystyle x\in X}$, there is a smallest neighborhood of ${\displaystyle x}$, namely the basic open set ${\displaystyle S_{x}}$ of divisors of ${\displaystyle x}$. So the divisor topology is an Alexandrov topology.
• ${\displaystyle X}$ is a T₀ space. Indeed, given two points ${\displaystyle x}$ and ${\displaystyle y}$ with ${\displaystyle x<y}$, the open neighborhood ${\displaystyle S_{x}}$ of ${\displaystyle x}$ does not contain ${\displaystyle y}$.
• ${\displaystyle X}$ is a not a T₁ space, as no point is closed. Consequently, ${\displaystyle X}$ is not Hausdorff.
• The isolated points of ${\displaystyle X}$ are the prime numbers.
• 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.
• ${\displaystyle X}$ is second-countable.
• ${\displaystyle X}$ is ultraconnected, since the closures of the singletons ${\displaystyle \{x\}}$ and ${\displaystyle \{y\}}$ contain the product ${\displaystyle xy}$ as a common element.
• Hence ${\displaystyle X}$ is a normal space. But ${\displaystyle X}$ is not completely normal. For example, the singletons ${\displaystyle \{6\}}$ and ${\displaystyle \{4\}}$ are separated sets (6 is not a multiple of 4 and 4 is not a multiple of 6), but have no disjoint open neighborhoods, as their smallest respective open neighborhoods meet non-trivially in ${\displaystyle S_{6}\cap S_{4}=S_{2}}$.
• ${\displaystyle X}$ is not a regular space, as a basic neighborhood ${\displaystyle S_{x}}$ is finite, but the closure of a point is infinite.
• ${\displaystyle X}$ is connected, locally connected, path connected and locally path connected.
• ${\displaystyle X}$ is a scattered space, as each nonempty subset has a first element, which is an isolated element of the set.
• The compact subsets of ${\displaystyle X}$ are the finite subsets, since any set ${\displaystyle A\subseteq X}$ is covered by the collection of all basic open sets ${\displaystyle S_{n}}$, which are each finite, and if ${\displaystyle A}$ is covered by only finitely many of them, it must itself be finite. In particular, ${\displaystyle X}$ is not compact.
• ${\displaystyle X}$ is locally compact in the sense that each point has a compact neighborhood (${\displaystyle S_{x}}$ is finite). But points don't have closed compact neighborhoods (${\displaystyle X}$ is not locally relatively compact.)

References

1. Steen & Seebach, example 57, p. 79-80

• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover Publications reprint of 1978 ed.), Berlin, New York: Springer-Verlag, ISBN   978-0-486-68735-3, MR   0507446