Can the boundary of a subset be open? Let $X$ be a topological space and $A$ a subset of $X$. Can $\partial A$ be an open subset of $X$?
Of course the answer is no if $A$ were closed; that is $\partial A\subset A$, because in this case we would have $int(\partial A)\subset int(A)$ hence by definition of $\partial A$ no point of $\partial A$ can be in $int(\partial A)$ because in that case we would have a point in $\partial A$ which is also in $int(A)$, something that is not allowed. 
Now what if $A$ is not closed, that is there  are boundary points in $X$ to $A$ that are not contained in $A$?  

If $X$ is a discrete space, then every subset of $X$ is open. In particular, every boundary of every subspace is open.
Also, viewing $\Bbb Q$ as a subspace of $\Bbb R$ equipped with the classical topology, we have
$$
\partial\Bbb Q=\bar{\Bbb Q}\setminus\Bbb Q^\circ=\Bbb R\setminus\varnothing=\Bbb R
$$
