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 Template:Math with an extra point, say Template:Math, added. In this case, the double origin topology gives a topology on the set Template:Math, where Template:Math denotes the disjoint union.

Construction

Given a point Template:Mvar belonging to Template:Mvar, such that Template:Math and Template:Math, the neighbourhoods of Template:Mvar are those given by the standard metric topology on Template:Math We define a countably infinite basis of neighbourhoods about the point 0 and about the additional point Template:Math. For the point Template:Math, the basis, indexed by Template:Mvar, 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 Template:Mvar is defined to be:

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

Properties

The space Template:Math, 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 Template:Math, along with the double origin topology fails to be either compact, paracompact or locally compact, however, Template:Mvar is second countable. Finally, it is an example of an arc connected space.