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 The sets for form a basis for the divisor topology 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 All the properties below are proved in 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