Study sheet: The Cantor Set

📋 Course Outline

1. Definition and Ternary Characterization
2. Successive Middle-Third Removal
3. Cardinality of the Cantor Set
4. Two Senses of Thinness

📖 1. Definition and Ternary Characterization

🔑 Key Concepts & Definitions

• Cantor set : the intersection of the sets F_n obtained by successively removing open middle thirds from the unit interval

📝 Essential Points

📌 The Cantor set consists exactly of the real numbers in the unit interval whose ternary expansions use only the digits 0 and 2.

💡 Memory Hook

Digits 0 and 2 remain; digit 1 marks a removed middle third.

📖 2. Successive Middle-Third Removal

★ Must-know

• If F_n consists of 2^n intervals of the form [k/3^n,(k+1)/3^n], then F_{n+1} is obtained by removing the open middle third of each of those intervals.

• At each stage, the open middle third of every surviving closed interval is removed.

Further detail

• After the first removal, F_1 is the union of the two closed intervals [0,1/3] and [2/3,1].

• At stage n, the surviving set F_n is the union of 2^n closed intervals.

💡 Memory Hook

Remove, split, repeat: each surviving interval produces two smaller intervals.

📖 3. Cardinality of the Cantor Set

★ Must-know

• The points 0, 1/3, 2/3, and 1 belong to every F_n and therefore belong to the Cantor set F.

• The Cantor set contains a non-denumerable number of elements and its points can be put into one-to-one correspondence with the points of the unit interval.

Further detail

📌 The Cantor set is relatively thin geometrically but large in cardinality because it has as many points as the unit interval.

💡 Memory Hook

Thin in intervals and length, yet large in number of points.

📖 4. Two Senses of Thinness

📝 Essential Points

📌 Because the Cantor set is contained in every F_n and the lengths of F_n tend to zero, the Cantor set cannot have positive length.

• The Cantor set contains no nonempty interval.

• Every open interval containing a point of the Cantor set contains some middle thirds removed during the construction and therefore contains infinitely many points in the complement of the Cantor set.

📐 Formula — The length of the stage-n set is $\left(\frac{2}{3}\right)^n$.

💡 Memory Hook

Middle thirds disappear at every scale → no interval remains and the length tends to zero.

📊 Synthesis Tables

Geometric Thinness and Cardinality

Property	Cantor set F
Intervals	Contains no nonempty interval
Complement near each point	Every open interval around a point of F contains infinitely many points outside F
Cardinality	Non-denumerable, with a one-to-one correspondence with the unit interval
Length	Cannot have positive length