Half-disk topology

In mathematics, and particularly general topology, the half-disk topology is an example of a topology given to the set

X

, given by all points

in the plane such that

y\ge 0

. The set

X

Construction

We consider

X

to consist of the open upper half plane

P

, given by all points

in the plane such that

y>0

; and the x-axis

L

, given by all points

in the plane such that

y=0

. Clearly

X

is given by the union

P\cup L

. The open upper half plane

P

has a topology given by the Euclidean metric topology. We extend the topology on

P

to a topology on

X=P\cup L

by adding some additional open sets. These extra sets are of the form

\cup

, where

is a point on the line

L

and

U

This topology results in a space satisfying the following properties.

is Hausdorff.
is also regular and thus.
By the Urysohn metrization theorem, is in fact metrizable. Alternatively, one can see this by noting that is simply the subspace of obtained by removing the open lower half plane.
with the topology inherited from is a subspace homeomorphic to the real line.