What about two sets when the boundary of one is included in the boundary of the other

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 $F$?