Another example can be found in a disconnected topological space.
A topological space X is disconnected iff X has a proper subset A which is both open and closed.
For instance, the subspace of R is a disconnected space.
Both (0,1) and (2,3) are open in X.
Definition: A set C is closed in X if X\C is open.
Now, a complement of (0,1) in X is (2,3) which is open in X. Thus, (0,1) is a closed set in X. So, we can conclude (0,1) is a clopen set in X.
You can check that a boundary of (0,1) in X is empty as well (same with (2,3)).