Countable subsets of an uncountable set 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?
 A: 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.
A: The set of non-negative numbers is uncountable, and its complement in $\mathbb R$, the set of negative numbers, is also uncountable.
A: 
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.

8 years later... I suppose my answer will help someone else down the road, as I aim to prove your belief that if $X$ is uncountable and $A \subset X$ is countable, then $A^C$ is uncountable.
I am assuming you've already proven that the union of countable sets is countable. Others have already given you excellent counterexamples to your question.
Suppose that $A^C$ is countable, as is $A$ by definition. Then $A^C \cup A = X$ is countable. But $X$ is uncountable which we defined. So $A^C$ must be uncountable.
Hope this solidifies your belief.
