Let $A$ 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_{...