Open Set/Complex Analysis/Examples/Open Unit Circle

Example of Open Set in the context of Complex Analysis
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.

Proof
By definition, $S$ is closed $S$ consists only of interior points.

Let $z_1 \in S$.

Then $\cmod {z_1} < 1$.

Let $\epsilon \in \R: \epsilon < 1 - \cmod {z_1}$.

Then:
 * $\map {\N_\epsilon} {z_1} \cap S \subseteq S$

and so $z_1$ is an interior points of $S$.

As $z_1$ is arbitrary, the result follows.