Definition:Closed Set/Complex Analysis
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This page is about Closed Set in the context of Complex Analysis. For other uses, see Closed.
Definition
Let $S \subseteq \C$ be a subset of the complex plane.
$S$ is closed (in $\C$) if and only if every limit point of $S$ is also a point of $S$.
That is: if and only if $S$ contains all its limit points.
Examples
Closed Unit Circle
Let $S$ be the subset of the complex plane defined as:
- $\cmod z \le 1$
where $\cmod z$ denotes the complex modulus of $z$.
Then $S$ is closed.
Also see
- Results about closed sets can be found here.
Sources
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Point Sets: $3.$