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?
Let $C$ be a Cantor set. The union of countably many pairwise disjoint Cantor sets is homeomorphic to $C\times\Bbb N$, where $\Bbb N$ has the discrete topology, and is therefore not compact and not a Cantor set. It is also homeomorphic to $C\setminus\{x\}$ for any $x\in C$.
$[0,1]\setminus\Bbb Q$ is not $\sigma$-compact, so in particular it is not the union of countably many Cantor sets.