Let $U$ be a bounded open set in $\mathbb{R}^n$ and $A$ be an open subset of $U$. Fixed $\epsilon >0$. Does there exist an open set $B \subset U$ such that $B \cap \overline{U} \ne \emptyset$ and $|B \setminus A|< \epsilon$ ?
2026-03-26 03:01:46.1774494106
question on existence of open set
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Yes. $B = A$ does the job, unless $A$ is empty. In that case let $B$ be a small enough ball.