I need to ask this before I forget about it, although it might be trivial to answer; I'm not sure. I'm asking it out of curiosity. Somehow I am fascinated by Cantor sets.
Definition: The Cantor Set is $\subset \mathbb{R}$ the only totally disconnected, perfect, compact metric space up to a homeomorphism (Willard 1970). I think this definition applies not just to the ternary Cantor set, but to all Cantor sets. Am I right about this?
Question a) Is the countable union of disjoint Cantor sets necessarily a Cantor set?
Question b) Can $[0,1] \cap (\mathbb{R} $ \ $ \mathbb{Q} $) be written as the countable union of Cantor sets?
Related: Can the Interval be Covered by Disjoint Cantor Sets?