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}.[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]

${\displaystyle \ N(0,n)=\{(x,y)\in {\mathbf {R} }^{2}:x^{2}+y^{2}<1/n^{2},\ y>0\}\cup \{0\}.}$

In a similar way, the basis of neighbourhoods of 0* is defined to be:[1]

${\displaystyle N(0^{*},n)=\{(x,y)\in {\mathbf {R} }^{2}:x^{2}+y^{2}<1/n^{2},\ y<0\}\cup \{0^{*}\}.}$

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.[2]

References

1. Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, pp. 92 − 93, ISBN 0-486-68735-X
2. Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, pp. 198–199, ISBN 0-486-68735-X