Hi everyone: Suppose $A\subset \mathbb{R}^{m}$ ($m>1$) is a closed set with empty interior. Suppose moreover that $A$ is unbounded and has no bounded component. Which are the necessary and sufficient conditions that $A$ have a neighborhood $V$ such that every bounded component of $ \mathbb{R}^{m}\setminus A$ has a point of $\mathbb{R}^{m}\setminus V$?