State the Baire Category Theorem. A set $X \subseteq \mathbb{R}$ is said to be a $G_{\delta}$-set if it is the intersection of countably many open sets. Show that the set $\mathbb{Q}$ of rationals is not a $G_{\delta}$-set.

[You may assume that the rationals are countable and that $\mathbb{R}$ is complete.]