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 {{nowrap|1=X = R² ∐ {0*} }}, where ∐ denotes the disjoint union.

Construction

Given a point x belonging to X, such that {{nowrap|1=x ≠ 0}} and {{nowrap|1=x ≠ 0*}}, the neighbourhoods of x are those given by the standard metric topology on {{nowrap|1=R²−{0}.}}{{Citation|first1=L. A.|last1=Steen|first2=J. A.|last2=Seebach|title=Counterexamples in Topology|pages=92 − 93|publisher=Dover|year=1995|isbn=0-486-68735-X}} 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:

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

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

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

Properties

The space {{nowrap|1=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 {{nowrap|1=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.{{Citation|first1=L. A.|last1=Steen|first2=J. A.|last2=Seebach|title=Counterexamples in Topology|pages=198–199|publisher=Dover|year=1995|isbn=0-486-68735-X}}

References

Category:General topology