Exercise 1886Topology › Topology of ℝL1Added 3 Nov 2026

Compact sets in $\mathbb{R}$

#compact sets#Heine-Borel#closed bounded

Statement

State the Heine–Borel theorem and give two examples of compact and two non-compact sets.

Hint

A subset of $\mathbb{R}$ is compact iff it is closed and bounded.

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Theorem (Heine–Borel). $K\subseteq\mathbb{R}$ is compact iff $K$ is closed and bounded.

Compact. $[0,1]$ , $\{0\}\cup\{1/n:n\ge1\}$ .

Non-compact. $]0,1[$ (not closed), $[0,+\infty[$ (not bounded).

Conclusion. Compactness = closed + bounded in $\mathbb{R}$ .