Open/Closed set in Lebesgue measuer

Question

Let $E\subset \mathbb{R}$ such that $\lambda(E)=0$ (the Lebesgue measure on $\mathbb{R}$). Can $E$ be open? Must it be closed?

Answer 1

Hint 1: if $E \subset \mathbb{R}$ is a nonempty open set, it contains an interval of positive length.

Hint 2: Consider the rationals.

Answer 2

Hint: open sets contain an interval of the form $(a,b)$. Another hint: find countable everywhere dense subset in $\Bbb R$.