Definition:Open Set/Complex Analysis/Definition 1

Definition
Let $S \subseteq \C$ be a subset of the set of complex numbers.

Let:
 * $\forall z_0 \in S: \exists \epsilon \in \R_{>0}: N_{\epsilon} \left({z_0}\right) \subseteq S$

where $N_{\epsilon} \left({z_0}\right)$ is the $\epsilon$-neighborhood of $z_0$ for $\epsilon$.

Then $S$ is an open set (of $\C$), or open (in $\C$).

Note that $\epsilon$ may depend on $z_0$.

Also see

 * Equivalence of Definitions of Open Set (Complex Analysis)