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 ${\displaystyle y\geq 0}$.^[1] 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.^[1] 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).^[1]

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