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