Open Halfspace

Last updated Nov 1, 2022

Let $V$ be an Inner Product Space over $\mathbb{R}$. An Open Halfspace is a Set of the form $$H = {x : \langle x, a \rangle > b}$$ for any $a \in V$ and $b \in \mathbb{R}$.

1. An Open Halfspace is Open. TODO - prove this.
2. If $H$ is an Open Halfspace, then $\text{cl} H$ is a Closed Halfspace - TODO prove this.
3. The Boundary of a Closed Halfspace is a Hyperplane - TODO prove this.

• Real Numbers
• Closure