# Equal boundaries

I am given that $A \subset B \subset \mathbb{R}$, $A$ is open, $B$ is closed, and that $\partial A = \partial B$.

Can I prove from this that $B$ is either equal to the closure of $A$ or it is unbounded or some stronger result relating $A$ and $B$?

Okay so I know now that $\bar{A} \neq B$ in general. Is there any statement I can say about them though?

Let $U=\bigcup_{n\in\Bbb N}U_n$ be a union of bounded open intervals $U_n$ with disjoint closures, which are chosen such that $(-1)^n+(-\frac12)^n\in U_n$ for all$~n$. Then $\overline U$ of course also contains the boundary points of all those intervals, but also the limit points $-1$ and $1$ which are not of that form. Now $C=\overline U\cup[-1,1]$ is another closed set with the same boundary as $U$ (and as $\overline U$), in particular $\partial C$ contains $\{-1,1\}$ since these are not interior points of $C$.