In this case, the double origin topology gives a topology on the set X = R2 ∐ {0*}, where ∐ denotes the disjoint union.
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}.
[1] 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:[1] In a similar way, the basis of neighbourhoods of 0* is defined to be:[1] The space R2 ∐ {0*}, along with the double origin topology is an example of a Hausdorff space, although it is not completely Hausdorff.
Finally, it is an example of an arc connected space.