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Hi everyone: Suppose $V$ is a bounded open set and $F$ a closed set with non-empty interior, both in $\mathbb{R}^{k}$. If the boundary of $V$ is included in the boundary of $F$, can we conclude that $V$ is included either in $F$ or in the complement of $F$?

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No. For k = 1, let V = (0,4), F = [0,1] union [3,4] which can be extended for k > 1. If V and F were simply connected, would the conjecture hold?

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