Definition:Closed Set/Real Analysis

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

• Definition:Open Set (Real Analysis)

• Results about closed sets can be found here.