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]

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

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]

gollark: Well, some maths at school etc. is like that, but it isn't real maths™.
gollark: Wrong.
gollark: Terrorism is generally meant to be emotionally salient regardless of actual impact. That's basically the point. This should probably not affect your views on the importance and effectiveness of aeroplane security.
gollark: I had vaguely assumed it was basically radar or something, and the X-raying let them know density (ish).
gollark: Arguably it was me. However, this is you, as a result.

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
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.