Divisor topology

In mathematics, more specifically general topology, the divisor topology is a specific topology on the set [math]\displaystyle{ X = \{2, 3, 4,...\} }[/math] 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 [math]\displaystyle{ X }[/math].

Construction

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

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

References

• 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