Definition:Open Set/Complex Analysis/Definition 1
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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 $z_0$.