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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.

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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
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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
    
Thanks for the comment, I edited the problem. Hope now is clear. –  Romeo Aug 7 '12 at 13:08
    
I think you're asking about sets of Hausdorff dimension smaller than 1. –  tomasz Aug 8 '12 at 1:39

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