Open Set/Complex Analysis/Examples/Open Unit Circle
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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 open 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:
- $\map {N_\epsilon} {z_1} \subseteq S$
and so $z_1$ is an interior points of $S$.
As $z_1$ is arbitrary, the result follows.
$\blacksquare$
Sources
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Point Sets: $6.$