Closure of interior of closed set

Closure of interior of closed set

If $D$ is a closed set, what is the relation in general between the set
$D$ and the closure of $\operatorname{Int}D$?
We know that $\operatorname{Int}D\subseteq D$, so
$\overline{\operatorname{Int}D}\subseteq \overline{D}$, but since $D$ is
closed, we have $\overline{D}=D$, so that
$\overline{\operatorname{Int}D}\subseteq D$.
Now, is it true as well that $D\subseteq \overline{\operatorname{Int}D}$?
I can't seem to prove it, or give an example of $D$ such that this doesn't
hold.