Let $U$ be an open subset of $\mathbb{R}^n$ and $K$ a compact subset of $U$ such that $U\setminus K$ is connected. Does there exist an open set $V$ such that $K\subseteq V \subseteq \overline{V} \subseteq U$ with $U\setminus\overline{V}$ connected?
Intuitively it seems like there is such a $V$. For a fix path between two points in $U\setminus K$ we may choose $V$ such that the path is contained in $U\setminus \overline{V}$. I cannot see how to solve the general case, which presumably involves a compactness argument, though. Possibly the statement is even false.