# Definition:Half-Plane/Open

## Definition

Let $\PP$ denote the plane.

Let $\LL$ denote an infinite straight line in $\PP$.

Let $\HH$ be a half-plane whose edge is $\LL$.

$\HH$ is an open half-plane if and only if $\HH$ does not include $\LL$.

## Instances

### Open Left Half-Plane

The open left half-plane $\HH_{\text {OL} }$ is the area of $\PP$ on the left of $\LL$.

That is, where $x < 0$:

$\HH_{\text {OL} } := \set {\tuple {x, y}: x \in \R_{<0} }$

### Open Right Half-Plane

The open right half-plane $\HH_{\text {OR} }$ is the area of $\PP$ on the right of $\LL$.

That is, where $x > 0$:

$\HH_{\text {OR} } := \set {\tuple {x, y}: x \in \R_{>0} }$

### Open Upper Half-Plane

The open upper half-plane $\HH_{\text {OU} }$ is the area of $\PP$ above $\LL$.

That is, where $y > 0$:

$\HH_{\text {OU} } := \set {\tuple {x, y}: y \in \R_{> 0} }$

### Open Lower Half-Plane

The open lower half-plane $\HH_{\text {OL} }$ is the area of $\PP$ below $\LL$.

That is, where $y < 0$:

$\HH_{\text {OL} } := \set {\tuple {x, y}: y \in \R_{< 0} }$