The Cantor Set is Compact

Topology · Axiom Academy

EXAMPLE The Cantor Set is Compact A detailed proof using the Heine-Borel theorem Prove that the Cantor set is a compact subset of . We'll use the Heine-Borel theorem, which states that a subset of is compact if and only if it is closed and bounded. The Cantor set is constructed by repeatedly removing the middle third of intervals: Remove middle third from each remaining interval to get Cantor Set Construction Visualization Current step: C₀ | Intervals remaining: 1 Closed (intersection of closed sets) The Heine-Borel theorem gives us a practical way to prove compactness in : just show the set is closed and bounded. Intersections of closed sets are always closed, even if the intersection is infinite. This is crucial for the Cantor set proof. The Cantor set is uncountable despite having measure zero , showing that "small" sets can still be topologically rich. Compactness means every open cover has a finite subcover. For the Cantor set, this property follows from it being closed and bounded. The construction process shows that contains no intervals, yet it's still an uncountable perfect set.

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