In mathematics, more specifically general topology, the double origin topology is an example of a topology given to the plane with an extra point, say , added. In this case, the double origin topology gives a topology on the set , where denotes the disjoint union.
Construction
Given a point belonging to , such that and , the neighbourhoods of are those given by the standard metric topology on We define a countably infinite basis of neighbourhoods about the point 0 and about the additional point . For the point , the basis, indexed by , is defined to be:
In a similar way, the basis of neighbourhoods of is defined to be:
Properties
The space , along with the double origin topology is an example of a Hausdorff space, although it is not completely Hausdorff. Since it has a base of regular open sets, it is an example of a semiregular space that is not regular. In terms of compactness, the space , along with the double origin topology fails to be either compact, paracompact or locally compact, however, is second countable. Finally, it is an example of an arc connected space.
References