Let $C$ be a compact subset of $\mathbb R^n$ and suppose that for every $\varepsilon >0$ there exists a finite family of open disks $B_i$ s.t. $C \subset \bigcup_{i} B_i$ and $\sum_i r_i \le \varepsilon$ (where $r_i$ denotes the radius of $B_i$).
(a) Suppose $n\ge 2$. Is $\mathbb R^n \setminus C$ connected?
If the answer is yes, then
(b) For which $n \ge 2$ is $\mathbb R^n \setminus C$ path-connected? And simply-connected?
Similar questions can be found here (and also on MathOverflow, with a beautiful but difficult answer). Anyway, I do not know the answer to the questions. What do you think? Do you have any references? Do you know any elementary proofs?
Thanks in advance.