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 $$z_0$$.