Definition:Open Set/Complex Analysis

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 $\C$) if and only if every point of $S$ is an interior point.

Examples

Open Unit Circle

Let $S$ be the subset of the complex plane defined as:

$\cmod z < 1$

where $\cmod z$ denotes the complex modulus of $z$.

Then $S$ is open.