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.

$\blacksquare$