Let $\Omega$ be an open subset of $\mathbb R^n$, with $\Omega \neq \emptyset$ and $\Omega \neq \mathbb R^n$.
Can you give an example where $\partial\Omega \neq \partial\bar{\Omega}$, and how can one exclude this situation?
Let $\Omega$ be an open subset of $\mathbb R^n$, with $\Omega \neq \emptyset$ and $\Omega \neq \mathbb R^n$.
Can you give an example where $\partial\Omega \neq \partial\bar{\Omega}$, and how can one exclude this situation?
Take any open subset $G \subsetneq \mathbb{R}^n$ nonempty. Pick $x\in G$. Let $\Omega = G\setminus \{x\}$ which is clearly open. Clearly $x\in \partial\Omega$ but $x\not\in\partial\bar{\Omega}$.
A necessary and sufficient condition to rule this out is to require that $\partial\Omega = \partial\left( \bar\Omega^c\right)$. In other words, it suffices that every neighborhood of any point $y\in \partial\Omega$ contains a point $z$ in the interior of the complement of $\Omega$.
If you just want sufficient conditions:
Do you have an application in mind?