Week 1 Tutorial

Cantor sets

The middle-thirds Cantor set is defined as follows. Put $$C_1=[\frac{0}{3},\frac{1}{3}]\cup [\frac{2}{3},\frac{3}{3}]$$ and define $C_{k+1}$ from $C_k$ by removing the open middle-third of every interval in $C_k$. The intersection $$\mathcal{C}=\bigcap _{n=1}^{\mathrm{\infty }}{C}_n$$ is the middle-thirds Cantor set.

1. Write down $C_2$ and $C_3$.
2. Why is $\mathcal{C}$ a Borel subset of $\mathbb{R}$?
3. Every $x\in [0,1]$ can be written in the form $$x=\sum _{n=1}^{\mathrm{\infty }}{\displaystyle \frac{d(n)}{3^n}}$$ where each $d(n)\in \{0,1,2\}$. Describe in terms of the $d(n)$ what it means for $x$ to belong to $C_1$, to $C_2$, and to $\mathcal{C}$.
4. Come up with a bijection between $\{0,1{\}}^{\mathbb{N}}$ and $\mathcal{C}$.
5. Prove that $\mathrm{\Lambda }(\mathcal{C})=0$.

Put $$S_1=[\frac{0}{8},\frac{3}{8}]\cup [\frac{5}{8},\frac{8}{8}]$$ and define $S_{k+1}$ from $S_k$ by removing from every interval in $S_k$ an interval of length $\frac{1}{4^{k+1}}$. The intersection $$\mathcal{S}=\bigcap _{n=1}^{\mathrm{\infty }}{S}_n$$ is a Smith-Volterra-Cantor set.

5. Write down $S_2$ and $S_3$.
6. Is there a bijection between $\mathcal{S}$ and $\{0,1{\}}^{\mathbb{N}}$?
7. What is $\mathrm{\Lambda }(\mathcal{S})$?