In mathematics, more specifically 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 double origin topology is an example of a topology

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...

 given to the plane R² with an extra point, say 0*, added. In this case, the double origin topology gives a topology on the set , where ∐ denotes the disjoint union

Disjoint union

In mathematics, the term disjoint union may refer to one of two different concepts:* In set theory, a disjoint union is a modified union operation that indexes the elements according to which set they originated in; disjoint sets have no element in common.* In probability theory , a disjoint union...

.

To give the set X 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 X 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. The set X 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 a point x belonging to X, such that and , the neighbourhood

Neighbourhood (mathematics)

In topology and related areas of mathematics, a neighbourhood is one of the basic concepts in a topological space. Intuitively speaking, a neighbourhood of a point is a set containing the point where you can move that point some amount without leaving the set.This concept is closely related to the...

s of x are those given by the standard metric topology on We define a countably infinite basis

Base (topology)

In mathematics, a base B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B. We say that the base generates the topology T...

 of neighbourhoods about the point 0 and about the additional point 0*. For the point 0, the basis, index

Index (mathematics)

The word index is used in variety of senses in mathematics.- General :* In perhaps the most frequent sense, an index is a number or other symbol that indicates the location of a variable in a list or array of numbers or other mathematical objects. This type of index is usually written as a...

ed by n, is defined to be:
In a similar way, the basis of neighbourhoods of 0* is defined to be:

Properties

The space }, along with the double origin topology is an example of a Hausdorff space

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...

, although it is not completely Hausdorff. In terms of compactness, the space }, along with the double origin topology fails to be either compact, paracompact or locally compact. Finally, it is an example of an arc connected space.