Exercise 282Topology › Topology of ℝⁿL1Added 3 Nov 2026

Closed sets in $\mathbb{R}^n$

#closed sets#complement#examples

Statement

Define a closed set in $\mathbb{R}^n$ and give two examples.

Hint

Closed = complement open.

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Definition. $F\subseteq\mathbb{R}^n$ is closed iff $\mathbb{R}^n\setminus F$ is open.

Examples. Closed ball $\overline B(a,r)=\{x:\|x-a\|\le r\}$ ; spheres $\{x:\|x-a\|=r\}$ ; hyperplanes.

Conclusion. Closed sets are complements of open ones.