Definition:Half-Plane/Upper
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Definition
Let $\PP$ be a Cartesian plane.
Let $\LL$ be the $x$-axis.
Open Upper Half-Plane
The open upper half-plane $\HH_{\text {OU} }$ is the area of $\PP$ above $\LL$.
That is, where $y > 0$:
- $\HH_{\text {OU} } := \set {\tuple {x, y}: y \in \R_{> 0} }$
Closed Upper Half-Plane
The closed upper half-plane $\HH_{\text {CU} }$ is the area of $\PP$ above and including $\LL$.
That is, where $y \ge 0$:
- $\HH_{\text {CU} } := \set {\tuple {x, y}: y \in \R_{\ge 0} }$