Divisor topology

In mathematics, more specifically general topology, the divisor topology is a specific topology on the set formula_1 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 formula_2.
Construction.
The sets formula_3 for formula_4 form a basis for the divisor topology on formula_2, where the notation formula_6 means formula_7 is a divisor of formula_8.
The open sets in this topology are the lower sets for the partial order defined by formula_9 if formula_10. 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.