Bounded Open Domain in Euclidean Space

In the setting of Second-Order Equations on Euclidean Open Sets, this setting fixes a bounded open domain in Euclidean space together with the notation for its closure and its boundary.

Let $n$ be a natural number with $n\ge1$.

1. (The domain)(i) $\Omega\subseteq\mathbb{R}^{n}$ is nonempty, open and bounded.

2. (Closure and compactness)(ii) We write $\overline{\Omega}=\operatorname{cl}_{\mathbb{R}^{n}}(\Omega)$. Then $\Omega\subseteq\overline{\Omega}$ by The Closure is the Smallest Closed Superset, so that $\overline{\Omega}$ is nonempty; moreover $\overline{\Omega}$ is bounded by The Closure of a Bounded Subset of a Metric Space is Bounded and compact by The Closure of a Bounded Subset of $\mathbb{R}^n$ is Compact.

3. (Boundary)(iii) We write $\partial\Omega=\partial_{\mathbb{R}^{n}}\Omega$. Since $\Omega$ is open, claim 4 of The Interior is the Largest Open Subset gives $\operatorname{int}_{\mathbb{R}^{n}}(\Omega)=\Omega$, so that by the definition of the boundary

$$\partial\Omega=\overline{\Omega}\setminus\Omega .$$

In particular $\partial\Omega\subseteq\overline{\Omega}$, the sets $\Omega$ and $\partial\Omega$ are disjoint, and $\overline{\Omega}=\Omega\cup\partial\Omega$.