The Cantor Set

📋 Course Outline

1. Cantor Set Construction
2. Basic Set Properties
3. Length Zero
4. Uncountability Proof
5. Topological Properties
6. Ternary Characterization

📖 1. Cantor Set Construction

🔑 Key Concepts & Definitions

• Cantor set: Georg Cantor, 1883 — the special subset of [0, 1] obtained by repeatedly removing the open middle third from every remaining closed interval

★ Must-know

• The construction starts with $I_0=[0{,}1]$, removes the open interval $(1/3{,}2/3)$, and then removes the open middle third from every closed interval at each subsequent stage.

📐 Formula — The Cantor set is defined by $C=\bigcap_{n=1}^{\infty} I_n$.

Further detail

• The first two stages are $I_1=[0{,}1/3]\cup[2/3{,}1]$ and $I_2=[0{,}1/9]\cup[2/9{,}3/9]\cup[6/9{,}7/9]\cup[8/9{,}9/9]$.

💡 Memory Hook

Start with [0,1], remove middle thirds, and repeat.

📖 2. Basic Set Properties

🔑 Key Concepts & Definitions

• Closed subset: a subset of [0, 1] because every stage $I_n$ is a finite union of closed intervals and $C$ is their intersection

★ Must-know

• The Cantor set is non-empty because every endpoint of every closed interval appearing at any stage belongs to the Cantor set.

Further detail