TITLE: An open set containing the boundary of another open set
QUESTION [0 upvotes]: Suppose $W$ is a bounded open set of $\mathbb{R}^m$ ($m>1$) such that  $\mathbb{R}^m\setminus W$ is connected. If $V$ is an open set having its boundary $\partial V$ in $W$, can we conclude that $V\subset W$?

REPLY [2 votes]: No. Consider $W=\{x\in\Bbb R^m\,:\, \lVert x\rVert<1\}$ and $V=\{x\in\Bbb R^m\,:\, \lVert x\rVert>1/2\}$.
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