Half-Spaces
Definition: Upper Half-Spaces
The \(n\)-dimensional closed upper-half space is the subset of \(\mathbb{R}^n\) defined as
\[ \left\{ \left. \begin{bmatrix} v_1 \\ \vdots \\ v_n \end{bmatrix} \in \mathbb{R}^n \right| v_n \ge 0 \right\} \]
and equipped with the Euclidean topology.
Notation
\[ \mathbb{H}^n \]