TITLE: Uncountable family of uncountable compact subsets of $\mathbb{R}$
QUESTION [9 upvotes]: Does there exists an uncountable family $\mathcal{A}$ consisting of uncountable compact subsets of $\mathbb{R}$ and pairwise disjoint?

REPLY [14 votes]: Let $C$ be any Cantor set in the line. $C \times C$ is again a Cantor set, so let $h:C \times C \cong C$ be a homeomorphism. For each $x \in C$ let $C_x = \{x\} \times C$; clearly each $C_x$ is yet another Cantor set, and $\{h[C_x]:x \in C\}$ is a partition of $C$ into $2^\omega$ Cantor sets.