Divisor topology

In mathematics, more specifically general topology, the divisor topology is a specific topology on the set $$X = \{2, 3, 4,...\}$$ 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 $$X$$.

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

Properties
All the properties below are proved in or follow directly from the definitions.


 * The closure of a point $$x\in X$$ is the set of all multiples of $$x$$.
 * Given a point $$x\in X$$, there is a smallest neighborhood of $$x$$, namely the basic open set $$S_x$$ of divisors of $$x$$. So the divisor topology is an Alexandrov topology.
 * $$X$$ is a T0 space. Indeed, given two points $$x$$ and $$y$$ with $$x<y$$, the open neighborhood $$S_x$$ of $$x$$ does not contain $$y$$.
 * $$X$$ is a not a T1 space, as no point is closed. Consequently, $$X$$ is not Hausdorff.
 * The isolated points of $$X$$ are the prime numbers.
 * 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 second-countable.
 * $$X$$ is ultraconnected, since the closures of the singletons $$\{x\}$$ and $$\{y\}$$ contain the product $$xy$$ as a common element.
 * Hence $$X$$ is a normal space. But $$X$$ is not completely normal.  For example, the singletons $$\{6\}$$ and $$\{4\}$$ 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 $$S_6\cap S_4=S_2$$.
 * $$X$$ is not a regular space, as a basic neighborhood $$S_x$$ is finite, but the closure of a point is infinite.
 * $$X$$ is connected, locally connected, path connected and locally path connected.
 * $$X$$ is a scattered space, as each nonempty subset has a first element, which is an isolated element of the set.
 * The compact subsets of $$X$$ are the finite subsets, since any set $$A\subseteq X$$ is covered by the collection of all basic open sets $$S_n$$, which are each finite, and if $$A$$ is covered by only finitely many of them, it must itself be finite. In particular, $$X$$ is not compact.
 * $$X$$ is locally compact in the sense that each point has a compact neighborhood ($$S_x$$ is finite). But points don't have closed compact neighborhoods ($$X$$ is not locally relatively compact.)