Definition:Open Set/Complex Analysis

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Definition

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 $\C$).


Definition 2

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

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


Also see