# Definition:Closed Set/Complex Analysis

< Definition:Closed Set(Redirected from 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

- 1981: Murray R. Spiegel:
*Theory and Problems of Complex Variables*(SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Point Sets: $3.$