Half-plane

All points on one side of a line, e.g. $\operatorname{Re}(z)>0$ .

A half-plane is the set of all points lying strictly on one side of a fixed line in the complex plane, such as $\operatorname{Re}(z)>0$ (everything right of the imaginary axis) or $\operatorname{Im}(z)>1$ (everything above the horizontal line $y=1$ ). Including the boundary line itself gives a closed half-plane; excluding it gives an open one.

Examples

$\operatorname{Re}(z)<0$ is the open left half-plane.
$\operatorname{Im}(z)\ge 0$ is the closed upper half-plane, the boundary line included.