[Math] Countable subsets of an uncountable set

Question

I believe it's true that if I have an uncountably infinite set $X$ and a countable subset $A$, then it's complement, $A^c$ is uncountable.

Is it also true that if I have an uncountable subset of $X$, called $B$, the complement of this set, $B^c$, is countable?

Answer 1

Not in general, no. For a simple example, consider the uncountable set $[0,2)\subseteq\Bbb R$: it’s the union of the complementary subsets $[0,1)$ and $[1,2)$, which are clearly both uncountable.