In mathematics, and especially general topology

General topology

In mathematics, general topology or point-set topology is the branch of topology which studies properties of topological spaces and structures defined on them...

, the prime integer topology and the relatively prime integer topology are examples of topologies

Topology

Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

 on the set of positive whole number

Integer

The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

s, i.e. the set }. To give the set Z⁺ a topology means to say which subset

Subset

In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...

s of Z⁺ are "open", and to do so in a way that the following axiom

Axiom

In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...

s are met:

1. The union of open sets is an open set.
2. The finite intersection of open sets is an open set.
3. Z⁺ and the empty set
   Empty set
   In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
   
    ∅ are open sets.

Construction

Given two positive integers , 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

Hausdorff space

In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...

 but not regular

Regular space

In topology and related fields of mathematics, a topological space X is called a regular space if every non-empty closed subset C of X and a point p not contained in C admit non-overlapping open neighborhoods. Thus p and C...

.