Properties of Lebesgue measure.

Question

Let $A,B\subset \mathbb{R}$ such that $A$ is a set of positive Lebesgue measure and $B$ is a set of zero Lebesgue measure (hence $B^c$ is dense in $\mathbb{R}$). Is it true that $$\overline{A\setminus B}=\overline{A}?$$ ($\overline{A} $ denotes the closure of $A$ in $\mathbb{R}$)

Answer 1

A counter-example is $B=$ Cantor set and $A = B\cup (1,2)$.

Answer 2

As a counterexample, let $B=\mathbb{Q}\cap (0,1]$ and let $A=[-1,0]\cup B$.

Then ${\overline{A}}=[-1,1]$, whereas ${\overline{A{\setminus}B}}=[-1,0]$.