In application of Runge type theorems on approximation of functions with some regularity on a neighborhood of a compact, it is interesting to know whether the complement of a compact has bounded components or even is connected. Suppose $F$ is a closed and unbounded set in $\mathbb{R}^m$ with $m>1$. For $r>0$ let $F_r$ be the set of all points in $\mathbb{R}^m$ having a distance $=r$ to $F$. We set $A=(F\cup F_r\cup F_R)\cap B$, where $0<r<R$ and $B$ is a closed ball. Suppose no set here is empty. Are there some known conditions on $F$ so that the complement of $A$ in $\mathbb{R}^m$ is connected? Any reference or suggestion is welcome.

In the application of Runge type theorems on the approximation of functions with some regularity on a neighborhood of a compact, it is interesting to know whether the complement of a compact has bounded components or even is connected. 

Suppose $F$ is a closed and unbounded set in $\mathbb{R}^m$ with $m>1$. For $r>0$ let $F_r$ be the set of all points in $\mathbb{R}^m$ having a distance $\:=r\:$ to $\,F$. We set $A=(F\cup F_r\cup F_R)\cap B$, where $0<r<R$ and $B$ is a closed ball. Suppose that no set here is empty.

My question: Are there some known conditions on $F$ so that the complement of $A$ in $\mathbb{R}^m$ is connected? Any reference or suggestion is welcome.

In application of Runge type theorems on approximation of functions with some regularity on a neighborhood of a compact, it is interesting to know whether the complement of a compact has bounded components or even is connected. Suppose $F$ is a closed and unbounded set in $\mathbb{R}^m$ with $m>1$. For $r>0$ let $F_r$ be the set of all points in $\mathbb{R}^m$ having a distance $=r$ to $F$. We set $A=(F\cup F_r\cup F_R)\cap B$, where $0<r<R$ and $B$ is a closed ball. Suppose no set here is empty. Are there some known conditions on $F$ so that the complement of $F$ in $\mathbb{R}^m$ is connected? Any reference or suggestion is welcome.

In application of Runge type theorems on approximation of functions with some regularity on a neighborhood of a compact, it is interesting to know whether the complement of a compact has bounded components or even is connected. Suppose $F$ is a closed and unbounded set in $\mathbb{R}^m$ with $m>1$. For $r>0$ let $F_r$ be the set of all points in $\mathbb{R}^m$ having a distance $=r$ to $F$. We set $A=(F\cup F_r\cup F_R)\cap B$, where $0<r<R$ and $B$ is a closed ball. Suppose no set here is empty. Are there some known conditions on $F$ so that the complement of $A$ in $\mathbb{R}^m$ is connected? Any reference or suggestion is welcome.