In mathematics, more specifically general topology, the divisor topology is an example of a topology given to the set X of positive integers that are greater than or equal to two, i.e., X = {2, 3, 4, 5, …}. The divisor topology is the poset topology for the partial order relation of divisibility on the positive integers.
Contents
To give the set X a topology means to say which subsets of X are "open", and to do so in a way that the following axioms are met:
- The union of open sets is an open set.
- The finite intersection of open sets is an open set.
- The set X and the empty set ∅ are open sets.
Construction
The set X and the empty set ∅ are required to be open sets, and so we define X and ∅ to be open sets in this topology. Denote by Z+ the set of positive integers, i.e., the set of positive whole number greater than or equal to one. Read the notation x|n as "x divides n", and consider the sets
Then the set Sn is the set of divisors of n. For different values of n, the sets Sn are used as a basis for the divisor topology.
The open sets in this topology are the lower sets for the partial order defined by x ≤ y if x | y.