This I found while studying sobolev spaces. In some results they have assumed that $\Omega\subseteq \mathbb{R}^n$ is with bounded boundary. Somebody told me the difference between the set with bounded boundary and bounded set but that thing is completely slipped from my mind.
2026-05-16 00:48:46.1778892526
What is the difference between bounded boundary and a bounded domain?
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The boundary of an $\Omega \subseteq \def\R{\mathbb R} \R^n$ is usually defined as $\partial\Omega := \bar\Omega - \Omega^°$. $\Omega$ has bounded boundary, iff $\partial \Omega \subseteq \R^n$ is a bounded set. Each bounded set $\Omega$ has bounded boundary (as $\Omega$ bounded by $R$ implies $\bar\Omega$ bounded by $R$), but the converse is false: As $\partial\Omega = \partial (\R^n - \Omega)$ holds for every $\Omega$, the complement of a bounded set has bounded boundary without being bounded itself.