Definition:Closed Set/Complex Analysis

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This page is about closed sets in the context of complex analysis. For other uses, see Definition: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


Sources