Suppose $K$ is a compact subset of $\mathbb{R}^m$ ($m>1$), and $0<r<R$ are fixed numbers. Let $A$ be the set of points having a distance $<R$ and $>r$ from $K$. My questions are

1. If $K$ and its complement are connected, is it possible that $A$ have an infinite number of components?

2. If yes, are there extra conditions that imposed to $K$ will exclude that $A$ can have an infinite number of components?

Suppose $K$ is a compact subset of $\mathbb{R}^m$ ($m>1$), and $0<r<R$ are fixed numbers. Let $A$ be the set of points having a distance $<R$ and $>r$ from $K$. My questions are

1. If $K$ and its complement are connected, is it possible that $A$ have an infinite number of components?

2. If yes, which are some extra conditions that imposed to $K$ will exclude that $A$ can have an infinite number of components?