Double origin topology

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Not to be confused with Line with two origins.

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[edit]

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]

 \ N(0,n) = \{ (x,y) \in {\bold 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]

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

Properties[edit]

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[edit]

  1. ^ a b c 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