Hyperplanes and Half-Spaces in $\mathbb{R}^n$

In the setting of Separation in $\mathbb{R}^n$: Standing Notation, let $a\in\mathbb{R}^{n}$ with $a\neq0$, and let $\alpha\in\mathbb{R}$.

(i) Hyperplane. The hyperplane with normal $a$ and level $\alpha$ is

$$H(a,\alpha)=\{x\in\mathbb{R}^{n}:a\cdot x=\alpha\}.$$

(ii) Closed half-spaces. The closed half-spaces bounded by $H(a,\alpha)$ are

$$H^{\le}(a,\alpha)=\{x\in\mathbb{R}^{n}:a\cdot x\le\alpha\}\qquad\text{and}\qquad H^{\ge}(a,\alpha)=\{x\in\mathbb{R}^{n}:a\cdot x\ge\alpha\}.$$

(iii) Open half-spaces. The open half-spaces bounded by $H(a,\alpha)$ are

$$H^{<}(a,\alpha)=\{x\in\mathbb{R}^{n}:a\cdot x<\alpha\}\qquad\text{and}\qquad H^{>}(a,\alpha)=\{x\in\mathbb{R}^{n}:a\cdot x>\alpha\}.$$