# Definition:Closed Set/Real Analysis

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*This page is about closed sets in the context of real analysis. For other uses, see Definition: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 $n\geq1$ 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.