I'd like to show that the set of irrational numbers in $[0,1]$ cannot be represented as a countable union of closed sets.
The hint says to use Baire's category theorem. I know two versions of such theorem:
- Baire's category theorem for complete metric spaces. If $(X,d)$ is a complete metric space, then $X$ is a Baire space.
- Baire's category theorem for locally compact spaces. If $(X, \mathcal{T})$ is a locally compact Hausdorff space, then $X$ is a Baire space.