My problem is that intuitively I would think the whole $\mathbb{R}$ is the only open supset of $\mathbb{Q}$. However this is not true since I can take out for example $\pi$ an I still have an open subset. Now I have two questions, first how to construct a minimal open supset of $\mathbb{Q}$. And the other one is this set countable?
The reason I come up with this is, that I shall prove that the smallest open supset of any given subset of $[0,1]$ has the same Lebesgue measure as the original set.