The irrationals, the Cantor Set, and $F_\sigma$ sets I know that $\mathbb{R}\setminus\mathbb{Q}$ is not an $F_\sigma$ set.  However, what about $\mathbb{R}\setminus\mathbb{Q}\cap$ Cantor set? Is that an $F_\sigma$?
HINT: Let $C$ be the Cantor set. $(\mathbb{R}\setminus\mathbb{Q})\cap C=C\setminus\mathbb{Q}$, which in some ways is a lot like $\mathbb{R}\setminus\mathbb{Q}$. How do you prove that $\mathbb{R}\setminus\mathbb{Q}$ is not an $F_\sigma$? Can you use the same basic idea on $C\setminus\mathbb{Q}$?
Do you know the properties of the Cantor set, i.e., is it closed, open, etc? Find out, and that will answer your question.