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Let $C$ be a compact subset of $R^n$ such that for all $\epsilon>0$ there exists a finite number of open balls $\{B_i\}$ that covers $C$ and $\sum_i r_i\leq \epsilon$, where $r_i$ is the radius of $B_i$. Prove that $R^n \backslash C$ is connected for $n\geq 2$ and that it's simply connected if $n\geq 3$.

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What is the number $r_i$? –  Daniel Rust Aug 15 '13 at 9:38
    
$r_i$ is the radius of $B_i$. –  user62138 Aug 15 '13 at 9:39

1 Answer 1

hint: prove first that C is totally disconnected, then try to prove directly or make use of Alexander duality theorem.

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Alexander duality will not suffice to establish simple connectivity. Otherwise, it is a good hint. –  studiosus Aug 15 '13 at 23:44
    
Can I do it without Alexander duality theorem? –  user62138 Aug 16 '13 at 11:16
    
The connectivity case $n\geq2$ follows directly from the classical Kuratowski theorem, implying that if $M\subset\mathbb{R}^{n}$ and $\dim M\leq n-2$, then $\mathbb{R}^{n}\backslash M$ is connected. Now, as $C$ is totally disconnected, $\dim C=0$ and we are done. –  simeon Aug 16 '13 at 16:03
    
@simeon: Do you mean the Kuratowski theorem on planar graphs. I don't find the relation with that. –  user62138 Aug 19 '13 at 6:58

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