Interior of closure of an open set

Question

The question is is the interior of closure of an open set equal the interior of the set?

That is, is this true:

$(\overline{E})^\circ=E^\circ$

($E$ open)

Thanks.

Answer 1

HINT: Try $E=(0,1)\cup(1,2)$ in $\Bbb R$.

Answer 2

Let $\varepsilon>0$, I claim there is an open set of measure (or total length, if you like) less than $\varepsilon$ whose closure is all of $\mathbb R$.

To see this, simply enumerate the rationals $\{r_n\}$ and then for each $n\in\mathbb N$ choose an open interval about $r_n$ of length $\varepsilon/2^n$. The union of those intervals has the desired property.