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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 $\mathbb{R}^{m}\setminus V$?

Hi everyone: Suppose $A\subset \mathbb{R}^{m}$ ($m>1$) is a closed set with empty interior. 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 $\mathbb{R}^{m}\setminus V$?

Hi everyone: Suppose $A\subset \mathbb{R}^{m}$ ($m>1$) is a closed set with empty interior. We know that $A$ does not necessarily have a neighborhood $V$ such that every bounded component of $ \mathbb{R}^{m}\setminus A$ has a point of $\mathbb{R}^{m}\setminus V$. 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 $\mathbb{R}^{m}\setminus V$?

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 $\mathbb{R}^{m}\setminus V$?

Hi everyone: Suppose $A\subset \mathbb{R}^{m}$ ($m>1$) is a closed set with empty interior. We know that $A$ has not necessarily a neighborhood $V$ such that every bounded component of $ \mathbb{R}^{m}\setminus A$ has a point of $\mathbb{R}^{m}\setminus V$. 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 $\mathbb{R}^{m}\setminus V$?

Hi everyone: Suppose $A\subset \mathbb{R}^{m}$ ($m>1$) is a closed set with empty interior. We know that $A$ does not necessarily have a neighborhood $V$ such that every bounded component of $ \mathbb{R}^{m}\setminus A$ has a point of $\mathbb{R}^{m}\setminus V$. 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 $\mathbb{R}^{m}\setminus V$?