Английская Википедия:Divisor topology
In mathematics, more specifically general topology, the divisor topology is a specific topology on the set <math>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>X</math>.
Construction
The sets <math>S_n = \{x \in X : x\mathop|n \} </math> for <math>n = 2,3,...</math> form a basis for the divisor topology[1] on <math>X</math>, where the notation <math>x\mathop|n</math> means <math>x</math> is a divisor of <math>n</math>.
The open sets in this topology are the lower sets for the partial order defined by <math>x\leq y</math> if <math>x\mathop|y</math>. 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 <math>x\in X</math> is the set of all multiples of <math>x</math>.
- Given a point <math>x\in X</math>, there is a smallest neighborhood of <math>x</math>, namely the basic open set <math>S_x</math> of divisors of <math>x</math>. So the divisor topology is an Alexandrov topology.
- <math>X</math> is a T0 space. Indeed, given two points <math>x</math> and <math>y</math> with <math>x<y</math>, the open neighborhood <math>S_x</math> of <math>x</math> does not contain <math>y</math>.
- <math>X</math> is a not a T1 space, as no point is closed. Consequently, <math>X</math> is not Hausdorff.
- The isolated points of <math>X</math> are the prime numbers.
- The set of prime numbers is dense in <math>X</math>. In fact, every dense open set must include every prime, and therefore <math>X</math> is a Baire space.
- <math>X</math> is second-countable.
- <math>X</math> is ultraconnected, since the closures of the singletons <math>\{x\}</math> and <math>\{y\}</math> contain the product <math>xy</math> as a common element.
- Hence <math>X</math> is a normal space. But <math>X</math> is not completely normal. For example, the singletons <math>\{6\}</math> and <math>\{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>S_6\cap S_4=S_2</math>.
- <math>X</math> is not a regular space, as a basic neighborhood <math>S_x</math> is finite, but the closure of a point is infinite.
- <math>X</math> is connected, locally connected, path connected and locally path connected.
- <math>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>X</math> are the finite subsets, since any set <math>A\subseteq X</math> is covered by the collection of all basic open sets <math>S_n</math>, which are each finite, and if <math>A</math> is covered by only finitely many of them, it must itself be finite. In particular, <math>X</math> is not compact.
- <math>X</math> is locally compact in the sense that each point has a compact neighborhood (<math>S_x</math> is finite). But points don't have closed compact neighborhoods (<math>X</math> is not locally relatively compact.)
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