Interior of closure of an open set 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.

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.
