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 $(x,y)$ in the plane such that $y\ge 0$. The set $X$ can be termed the closed upper half plane.

To give the set $X$ a topology means to say which subsets of $X$ are "open", and to do so in a way that the following axioms are met:

1. The union of open sets is an open set.
2. The finite intersection of open sets is an open set.
3. The set $X$ and the empty set $\mathrm{\varnothing }$ are open sets.

Construction

We consider $X$ to consist of the open upper half plane $P$, given by all points $(x,y)$ in the plane such that $y>0$; and the x-axis $L$, given by all points $(x,y)$ 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 $(x,0)\cup (P\cap U)$, where $(x,0)$ is a point on the line $L$ and $U$ is a neighbourhood of $(x,0)$ in the plane, open with respect to the Euclidean metric (defining the disk radius).

Properties of $X$

This topology results in a space satisfying the following properties.

• $X$ is Hausdorff (and thus also $T_0$ and $T_1$).

• $X$ is also regular and thus $T_3$. (Taking the convention that $T_3=\text{regular}+T_0$ .)

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

• $L$ with the topology inherited from $X$ is a subspace homeomorphic to the real line $ℝ$.

See also

• List of topologies