Study sheet: 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

• The Cantor set contains countably many points, including the points of the form $1/3^n$ for $n=1{,}2{,}3,\ldots$.

💡 Memory Hook

N-C: non-empty and closed.

📖 3. Length Zero

📝 Essential Points

📐 Formula — At stage $n$, the set $I_n$ consists of $2^n$ closed intervals, each of length $1/3^n$.

📐 Formula — The length of the stage-$n$ set is $\ell(I_n)=2^n\cdot(1/3^n)=(2/3)^n$.

📐 Formula — The Cantor set has length zero because $\ell(C)=\lim_{n\to\infty}(2/3)^n=0$.

💡 Memory Hook

Each stage multiplies length by 2/3, so the limit is zero.

📖 4. Uncountability Proof

📝 Essential Points

• Every point of the Cantor set determines a unique infinite sequence of left and right choices through the nested intervals.

📌 Replacing left by 0 and right by 2 converts the left/right sequence into a ternary expansion containing only 0s and 2s.

• To prove uncountability, list the ternary expansions of the supposed elements as $a_1,a_2,a_3,\ldots$ and construct $b=b_1b_2b_3\ldots$ by choosing $b_i=0$ when $a_{ii}=2$ and $b_i=2$ when $a_{ii}=0$.

• The diagonal construction produces an element of the Cantor set that differs from the nth listed element at its nth ternary digit, so the Cantor set is uncountable.

💡 Memory Hook

Encode by L/R, list ternary digits, then change the diagonal.

📖 5. Topological Properties

🔑 Key Concepts & Definitions

• Nowhere dense : a nowhere dense subset of [0, 1]
• Totally disconnected : totally disconnected, meaning that it has no connected subset containing more than one point

★ Must-know

• The Cantor set contains no interval.

Further detail

• The Cantor set is compact in the usual topology on [0, 1].

💡 Memory Hook

The Cantor set is closed and compact, but contains no interval and is nowhere dense.

📖 6. Ternary Characterization

🔑 Key Concepts & Definitions

• Ternary characterization : a number $x\in[0{,}1]$ belongs to the Cantor set if and only if it has a ternary expansion containing only the digits 0 and 2

★ Must-know

📌 A real number may have more than one ternary expansion, as shown by $(0.1)_3=(0.022\ldots)_3=1/3$.

• To prove the converse direction, induction shows that a number with only 0 and 2 in its ternary expansion belongs to every stage $I_k$ and therefore belongs to $C$.

Further detail

📐 Formula — If the ternary digits satisfy $a_k\in\{0{,}2\}$, then the corresponding number is $x=\sum_{k=1}^{\infty}a_k3^{-k}$.

💡 Memory Hook

Ternary digits 0 and 2 remain; digit 1 is removed.

📊 Synthesis Tables

Cantor Set Properties

Property	Statement	Basis
Non-empty	Contains every endpoint of every construction interval	Endpoints remain at every stage
Closed	Is an intersection of finite unions of closed intervals	Each stage is closed
Length zero	Has $\ell(C)=0$	$\ell(I_n)=(2/3)^n$
Uncountable	Cannot be listed completely	Cantor diagonal argument