Property of Lebesgue measure in $\mathbb{R}^2$

Question

Let $I=[0,1]\times [0,1]$ and $E\subset \mathbb{R}^2,$ be a set of zero Lebesgue measure. Is it true that $$\overline{I\setminus E}=I?$$

I guess that the counterexample will be some form space filling curve.

Answer 1

Yes, it is true. Proving that $\overline{I\setminus E}\subseteq I$ is trivial.

For proving $I\subseteq\overline{I\setminus E}$ let $(x,y)\in I$ and assume that $(x,y)\notin\overline{I\setminus E}$.

Then some open set $U$ must exist with $(x,y)\in U$ and $U\cap(I\setminus E)=\varnothing$ or equivalently $U\cap I\subseteq E$.

But $U\cap I$ has positive Lebesgue measure.

So this contradicts that $E$ is a set with Lebesgue measure zero and we conclude that our assumption must be wrong.

That means that $(x,y)\in I$ implies that $(x,y)\in\overline{I\setminus E}$ and we are ready.

Answer 2

If the complement of $E$ is not dense in $I$, then $E$ contains some open rectangle, so it cannot be of measure zero.