Double origin topology
In mathematics, more specifically general topology, the double origin topology is an example of a topology given to the plane R^{2} with an extra point, say 0*, added. In this case, the double origin topology gives a topology on the set X = R^{2} ∐ {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 R^{2}−{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 R^{2} ∐ {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 R^{2} ∐ {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
- ^ ^{a} ^{b} ^{c} Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, pp. 92 − 93, ISBN 0-486-68735-X
- ^ Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, pp. 198 – 199, ISBN 0-486-68735-X