\begin{definition}[Definition:Half-Plane/Open/Upper]
Let $\PP$ be a Cartesian plane.
Let $\LL$ be the $x$-axis.
The '''open upper half-plane''' $\HH_{\text {OU} }$ is the area of $\PP$ above $\LL$.
That is, where $y > 0$:
:$\HH_{\text {OU} } := \set {\tuple {x, y}: y \in \R_{> 0} }$
\end{definition}