Closed Upper Half-Space in Euclidean Space

Let $n\in\mathbb{N}$. In the Euclidean space $\mathbb{R}^n$, the closed upper half-space is the subset

$$H^n=\{x=(x_1,\dots,x_n)\in\mathbb{R}^n : x_n\ge 0\}.$$

A subset $\Omega\subseteq H^n$ is said to be open in $H^n$ if there exists an open subset $U\subseteq\mathbb{R}^n$ such that

$$\Omega = H^n\cap U.$$

The boundary hyperplane of $H^n$ is the subset

$$\partial H^n = \{x=(x_1,\dots,x_n)\in H^n : x_n=0\}.$$

The interior of $H^n$ is the subset

$$\operatorname{int}(H^n)=\{x=(x_1,\dots,x_n)\in H^n : x_n>0\}.$$