Exercise 1188Topology › Topology of ℝⁿL1Added 3 Nov 2026

Heine–Borel in $\mathbb{R}^n$

#compactness#Heine-Borel#closed bounded

Statement

State the Heine–Borel theorem in $\mathbb{R}^n$ and give examples.

Hint

In $\mathbb{R}^n$ , compact = closed + bounded.

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Solution

Theorem. $K\subseteq\mathbb{R}^n$ is compact iff $K$ is closed and bounded.

Examples.
- Compact: closed ball $\overline B(0,1)$ , the cube $[0,1]^n$ , spheres.
- Non-compact: open ball $B(0,1)$ (not closed), hyperplane (not bounded).

Conclusion. Compactness in $\mathbb{R}^n$ = closed and bounded.

Heine–Borel in $\mathbb{R}^n$

Statement

Solution

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