How to prove that $\mathbb{Q}$ is not the intersection of a countable collection of open sets.

One version of the Baire category theorem says that if $X$ is a complete metric space, and $\{G_n:n\in\Bbb N\}$ is a countable family of dense open sets in $X$, then $\bigcap_{n\in\Bbb N}G_n$ is dense in $X$. $\Bbb R$ with the usual metric is complete.

Suppose that $\Bbb Q=\bigcap_{n\in\Bbb N}G_n$, where each $G_n$ is open in $\Bbb R$. $\Bbb Q$ is dense in $\Bbb R$, so each $G_n$ is dense in $\Bbb R$. For each $q\in\Bbb Q$ let $U_q=\Bbb R\setminus\{q\}$; clearly $U_q$ is a dense open set in $\Bbb R$. Let $$\mathscr{U}=\{G_n:n\in\Bbb N\}\cup\{U_q:q\in\Bbb Q\}\;;$$ then $\mathscr{U}$ is a countable family of dense open subsets of $\Bbb R$, so its intersection is non-empty. However, the construction clearly ensures that $\bigcap\mathscr{U}=\varnothing$. This contradiction shows that $\Bbb Q$ cannot be a $G_\delta$ in $\Bbb R$.