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]
- [math]\displaystyle{ \ N(0,n) = \{ (x,y) \in {\mathbf R}^2 : x^2 + y^2 \lt 1/n^2, \ y \gt 0\} \cup \{0\} . }[/math]
In a similar way, the basis of neighbourhoods of 0* is defined to be:[1]
- [math]\displaystyle{ N(0^*,n) = \{ (x,y) \in {\mathbf R}^2 : x^2 + y^2 \lt 1/n^2, \ y \lt 0\} \cup \{0^*\} . }[/math]
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
- ↑ 1.0 1.1 1.2 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
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