Discrete Topology definition question

Question

Definition: The topology in which every set is open (and therefore every set is closed).

So is the set of complex numbers a discrete topology?

Answer 1

This doesn't work. The definition of discrete topology is that every subset of $X$ be open, and consequently closed. While it's true that $\mathbb{C}$ is both open and closed, we can build subsets thereof that are only open or only closed (or neither!). An easy example is the unit ball $\{ z : |z| < 1\}$, open but certainly not closed!