\begin{definition}[Definition:Open Set/Complex Analysis/Definition 1]
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 $z_0$.
\end{definition}