Let us take complex plane to be the space.
In complex plane, the boundary of a set is defined as $\partial D = \overline{D} \cap (\overline{\mathbb{C}\setminus D})$.
I want to prove such assertion holds:
if $D\neq \emptyset$ and $D\neq \mathbb{C}$, then $\partial D \neq \emptyset$.
Intuitively it is straightforward, however I failed to find a formal proof.
I suspect it should be proved by contradiction and using the property of complex plane.
Anyone help me to formulate the proof?
Cheers!