# Connectedness of the complement of a compact “small” subset of $\mathbb R^n$

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?

What do you mean by "and totally bounded"? If $C$ is compact, then it is necessarily totally bounded according to the usual definition, so maybe you mean something different from this? – Sam Aug 7 '12 at 13:00
If $C$ is just compact, there is a simple counter-example to a): The unit sphere $S^{n-1}$ is obviously compact, as it is a closed set in $\mathbb R^n$ and bounded, but it clearly divides $\mathbb R^n$ into an inside and outside part. – Johannes Kloos Aug 7 '12 at 13:03