# Equivalence of Definitions of Open Set (Complex Analysis)

## Theorem

The following definitions of the concept of **Open Subset of Complex Plane** are equivalent:

### 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.

## Proof

Let $S \subseteq \C$.

### Definition 1 implies Definition 2

Let $S$ be an open set in $\C$ by definition 1.

That is:

- $\forall z_0 \in S: \exists \epsilon \in \R_{>0}: N_{\epsilon} \left({z_0}\right) \subseteq S$

Thus every point in $S$ has an $\epsilon$-neighborhood $N_{\epsilon} \left({z_0}\right)$ such that $N_{\epsilon} \left({z_0}\right) \subseteq S$.

That is, by definition, every point of $S$ is an interior point.

Thus $S$ is an open set in $\C$ by definition 2.

$\Box$

### Definition 2 implies Definition 1

Let $S$ be an open set in $\C$ by definition 2.

That is, every point of $S$ is an interior point.

This means:

- $\forall z_0 \in S: \exists \epsilon \in \R_{>0}: N_{\epsilon} \left({z_0}\right) \subseteq S$

Thus $S$ is an open set in $\C$ by definition 1.

$\blacksquare$