In mathematics, and especially general topology, the prime integer topology and the relatively prime integer topology are examples of topologies on the set of positive whole numbers, i.e. the set Z+ = {1, 2, 3, 4, …}. To give the set Z+ a topology means to say which subsets of Z+ 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.
- Z+ and the empty set ∅ are open sets.
Construction
Given two positive integers a, b ∈ Z+, define the following congruence class:
Then the relatively prime integer topology is the topology generated from the basis
and the prime integer topology is the sub-topology generated from the sub-basis
The set of positive integers with the relatively prime integer topology or with the prime integer topology are examples of topological spaces that are Hausdorff but not regular.