Is all of $\mathbb{R}$ the only open set containing $\mathbb{Q}$?
False, take any irrational $p$ in $\mathbb{R}$. Then $\mathbb{Q} \subset \mathbb{R} \setminus \{p\}$ and $\mathbb{R} \setminus \{p\}$ is open.
Of course we can take any subset $A$ of $\mathbb{R} \setminus \mathbb{Q}$ that is closed in $\mathbb{R}$ and take $\mathbb{R} \setminus A$.
Is what I got right? or did I forget something?
What fact or theorem can I use?