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Q: A set with more than $n$ components has $n+1$ pairwise separated subsets.

HeManLet $A$ be a subset of $X$ which is a compact connected metric space, and A with more than $n$ components. Do exist $C_1, \ldots, C_{n+1}$ pairwise separated subsets such that $A = \bigcup_{i=1}^{n+1}C_i$? I have tried to take the first $n$ components of $A$ as $C_1, \ldots, C_n$ and define $C_{...

 

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