Let $X$ be a topological space and let $A \subseteq X$. Let $D(A)$ denote the boundary of $A$, i.e. the set of points in the closure of $A$ which are not in the interior of $A$. A closed set is nowhere dense if its interior is the empty set. Pick out the true statements:
(a) If $A$ is open, then $D(A)$ is nowhere dense.
(b) If $A$ is closed, then $D(A)$ is nowhere dense.
(c) If $A$ is any subset, then $D(A)$ is always nowhere dense.
if I consider open and closed intervals in real line both (a) and (b) are true, but what is the general case?