In mathematics, more specifically general topology, the divisor topology is a specific topology on the set 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 .

Construction

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The sets   for   form a basis for the divisor topology[1] on  , where the notation   means   is a divisor of  .

The open sets in this topology are the lower sets for the partial order defined by   if  . The closed sets are the upper sets for this partial order.

Properties

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

  • The closure of a point   is the set of all multiples of  .
  • Given a point  , there is a smallest neighborhood of  , namely the basic open set   of divisors of  . So the divisor topology is an Alexandrov topology.
  •   is a T0 space. Indeed, given two points   and   with  , the open neighborhood   of   does not contain  .
  •   is a not a T1 space, as no point is closed. Consequently,   is not Hausdorff.
  • The isolated points of   are the prime numbers.
  • The set of prime numbers is dense in  . In fact, every dense open set must include every prime, and therefore   is a Baire space.
  •   is second-countable.
  •   is ultraconnected, since the closures of the singletons   and   contain the product   as a common element.
  • Hence   is a normal space. But   is not completely normal. For example, the singletons   and   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  .
  •   is not a regular space, as a basic neighborhood   is finite, but the closure of a point is infinite.
  •   is connected, locally connected, path connected and locally path connected.
  •   is a scattered space, as each nonempty subset has a first element, which is an isolated element of the set.
  • The compact subsets of   are the finite subsets, since any set   is covered by the collection of all basic open sets  , which are each finite, and if   is covered by only finitely many of them, it must itself be finite. In particular,   is not compact.
  •   is locally compact in the sense that each point has a compact neighborhood (  is finite). But points don't have closed compact neighborhoods (  is not locally relatively compact.)

References

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  1. ^ a b 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