Is there a dense set that is a countable intersection of open sets with measure $0$ on $\mathbb{R}$

Question

Is there a dense set $A$ that is a countable intersection of open sets, so that $m(A)=0$ on $\mathbb{R}$?

I've been thinking that $\mathbb{Q}$ is a perfect candidate, but it's not a countable intersection of open sets.

Answer 1

I wonder if this is a valid answer:

If we look at $\cup_{q\in \mathbb{Q}} (q-\frac{1}{n},q+\frac{1}{n})$ this is an open set and $\cap_{n\in N} \cup_{q\in \mathbb{Q}} (q-\frac{1}{n},q+\frac{1}{n}) = \mathbb{Q}$ which has a measure $0$? is this correct?