Exercise 1885Topology › Topology of ℝL1Added 3 Nov 2026

The Cantor set is compact

#Cantor set#compactness#closed

Statement

Show that the Cantor set $K$ (intersection of nested triadic sets) is compact.

Hint

$K$ is the intersection of closed bounded sets.

Show solutionHide solution

Construction. $K=\bigcap_n K_n$ with $K_0=[0,1]$ , $K_{n+1}$ removing the middle thirds of each interval of $K_n$ .

Closed. Each $K_n$ is a union of closed intervals, hence closed; the intersection is closed.

Bounded. $K\subseteq[0,1]$ .

Conclusion. $K$ is closed and bounded, hence compact (Heine–Borel).

Remark. $K$ is also uncountable, perfect, and has empty interior.