Half-disk topology

In mathematics, and particularly general topology, the half-disk topology is an example of a topology given to the set ${\displaystyle X}$, given by all points ${\displaystyle (x,y)}$in the plane such that ${\displaystyle y\geq 0}$. The set ${\displaystyle X}$can be termed the closed upper half plane.

We consider ${\displaystyle X}$to consist of the open upper half plane ${\displaystyle P}$, given by all points ${\displaystyle (x,y)}$in the plane such that ${\displaystyle y>0}$; and the x-axis ${\displaystyle L}$, given by all points ${\displaystyle (x,y)}$in the plane such that ${\displaystyle y=0}$. Clearly ${\displaystyle X}$is given by the union ${\displaystyle P\cup L}$. The open upper half plane ${\displaystyle P}$has a topology given by the Euclidean metric topology. We extend the topology on ${\displaystyle P}$to a topology on ${\displaystyle X=P\cup L}$by adding some additional open sets. These extra sets are of the form ${\displaystyle {(x,0)}\cup (P\cap U)}$, where ${\displaystyle (x,0)}$is a point on the line ${\displaystyle L}$and ${\displaystyle U}$is a neighbourhood of ${\displaystyle (x,0)}$in the plane, open with respect to the Euclidean metric (defining the disk radius).

This topology results in a space satisfying the following properties.

• ${\displaystyle X}$is Hausdorff (and thus also ${\displaystyle T_{0}}$and ${\displaystyle T_{1}}$).

• ${\displaystyle X}$is also regular and thus ${\displaystyle T_{3}}$. (Taking the convention that ${\displaystyle T_{3}={\text{regular}}+T_{0}}$.)

• By the Urysohn metrization theorem, ${\displaystyle X}$is in fact metrizable. Alternatively, one can see this by noting that ${\displaystyle X}$is simply the subspace of ${\displaystyle \mathbb {R} ^{2}}$obtained by removing the open lower half plane.

• ${\displaystyle L}$with the topology inherited from ${\displaystyle X}$is a subspace homeomorphic to the real line ${\displaystyle \mathbb {R} }$.

• List of topologies