Definition:Open Set (Complex Analysis)

Definition

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

Suppose that $\forall z_0 \in S: \exists \epsilon > 0: N_{\epsilon} \left({z_0}\right) \subseteq S$

where $N_{\epsilon} \left({z_0}\right)$ is the $\epsilon$-neighborhood of $z_0$ for some real $\epsilon > 0$.

Then $S$ is described as open (in $\C$).

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