Cantor Set

Key idea

A perfect, uncountable set of measure zero.

In depth

Start with [0, 1], repeatedly remove the open middle third of each interval. What remains is the Cantor set: uncountable, nowhere dense, of Lebesgue measure 0, and self-similar with dimension ln 2/ln 3 ≈ 0.63.

Worked example

Points with ternary expansions using only 0 and 2, like 1/4 = 0.020202..., are in the set.

Real-world uses

• A model of gaps in signals and fractal dust in physics
• A standard example in analysis

Common mistake

Thinking measure zero means countable or empty.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related expert topics in Analysis & Topology

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• Density of Rationals – Between any two reals there is a rational.
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• Limit Theorems for Sequences – Algebra of limits and squeeze theorem.