Open Set/Complex Analysis/Examples/Open Unit Circle

From ProofWiki
Jump to navigation Jump to search

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 if and only if $S$ 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.

$\blacksquare$


Sources