Double origin topology

In mathematics, more specifically general topology, the double origin topology is an example of a topology 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 X = R2 ∐ {0*}, where ∐ denotes the disjoint union.

Construction
Given a point x belonging to X, such that x ≠ 0 and x ≠ 0*, the neighbourhoods of x are those given by the standard metric topology on R2−{0}. We define a countably infinite basis of neighbourhoods about the point 0 and about the additional point 0*. For the point 0, the basis, indexed by n, is defined to be:
 * $$ \ N(0,n) = \{ (x,y) \in {\mathbf R}^2 : x^2 + y^2 < 1/n^2, \ y > 0\} \cup \{0\} . $$

In a similar way, the basis of neighbourhoods of 0* is defined to be:
 * $$N(0^*,n) = \{ (x,y) \in {\mathbf R}^2 : x^2 + y^2 < 1/n^2, \ y < 0\} \cup \{0^*\} . $$

Properties
The space R2 ∐ {0*}, along with the double origin topology is an example of a Hausdorff space, although it is not completely Hausdorff. In terms of compactness, the space R2 ∐ {0*}, along with the double origin topology fails to be either compact, paracompact or locally compact, however, X is second countable. Finally, it is an example of an arc connected space.