Boundary points of the complex plane

Question

There is no boundary points of the complex plane. Since complex plane $C$ is closed, it contains all of its boundary points. And, since $C$ is open, it cannot contain any of its boundary points. Then, set of all boundary points of $C$ must be empty to satisfy both conditions. Similarly, set of all boundary points of empty set is empty set.

Am I correct?

Answer 1

In general since $\partial A = \bar A - A^o,$
the boundary of any clopen subset, including
the empty set and the whole space, is empty.