Double origin topology

Template:Short description Template:Distinguish 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 Template:Nowrap, where ∐ denotes the disjoint union.

Construction

Given a point x belonging to X, such that Template:Nowrap and Template:Nowrap, the neighbourhoods of x are those given by the standard metric topology on Template:Nowrap 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:

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

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

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

Properties

The space Template:Nowrap}, 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 Template:Nowrap}, 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.