I was wondering if every compact subset of $\Bbb R^n$ could be written as a disjoint union of compact subsets, where each of them are path-connected, i.e. :
If $X \subset \Bbb R^n$, $n \ge 1$, $X$ is compact, then there exists a finite (or infinite) collection of compact sets $U_i,$ $i= 1, 2, ...$, such that each $U_i$ is path-connected (i.e. $\forall x, y \in U_i$, there exists $\gamma : [0, 1] \rightarrow U_i$ continuous such that $\gamma(0) = x$ and $\gamma(1) = y$), $X = \bigcup_i U_i$, and $U_i \cap U_j = \varnothing$ if $i \ne j$.
I don't really know if this is true and how one could prove it. If it is not, do you have any counterexample(s) ?