Is the complement of a set 1st category set $X$ of 2nd category?

Question

If I have some metric space $M$ and find that $X \subset M$ is of 1st category (resp. 2nd category), is the complement of $X$,

$X^c$ of second cateogry (resp. 1st category)? Thanks!

Answer 1

Consider $\Bbb Q$ with the Euclidean metric. $\{0\}$ is of first category and so is its complement.

Consider $[0, 1] \cup [2, 3]$ with the Euclidean metric. Both $[0, 1]$ and $[2, 3]$ are of second category.

Answer 2

Any countable set is of first category. This settles first question.

For the second question, take $M = \mathbb{R}$, $X$ to be irrationals in $[0,1]$.

If you need elaboration, please ask.