# Definition:Closed Set/Real Analysis

< Definition:Closed Set(Redirected from Definition:Closed Set (Real Analysis))

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*This page is about Closed Set in the context of Real Analysis. For other uses, see Closed.*

## Definition

### Real Numbers

Let $S \subseteq \R$ be a subset of the set of real numbers.

Then $S$ is **closed (in $\R$)** if and only if its complement $\R \setminus S$ is an open set.

### Real Euclidean Space

Let $n \ge 1$ be a natural number.

Let $S \subseteq \R^n$ be a subset.

Then $S$ is **closed (in $\R^n$)** if and only if its complement $\R^n \setminus S$ is an open set.

## Also see

- Results about
**closed sets**can be found**here**.