Cantor Set Homeomorphic to Itself

Question

This question is partly taken from Hatcher’s notes on point-set topology.

Given standard $1/3$ Cantor set on $[0,1]$, we need to show that a function $f_S:C \to C$ is a homeomorphism, where $f_S$ is defined as follows: $S\subset \mathbb{N}$ and $f_S$ changes the $i^{th}$ coordinate of a ternary number $x=0.a_1 a_2 ... \in C, \forall j: a_j\in\{0,2\}$ from 0 to 2 or vice verse if $i \in S$ and leaves it unchanged otherwise. Note that $f_S=f_S^{-1}$.

To help prove that $f_S$ is indeed a homeomorphism, Hatcher gives two hints:

1. Basis for $C$ is given by intersecting $C$ with interval components of elements $C_n$, whose intersection gives us $C$, i.e., $C_0=[0,1],C_1=[0,\frac{1}{3}]\cup[\frac{2}{3},1]...$

2. We can describe basis elements of C in terms of ternary numbers $0.a_1 a_2 ..., \forall j: a_j\in\{0,2\}$.

I have done proving the hints, but I have no idea how to approach last part about proving that $f_S$ is a homeomorphism.

Any help or guidance is appreciated.

Answer 1

Before answering the question, let me alter its notation a bit, because it has too many $C$'s, creating a clash of variables.

I'll use ${\cal I}_i$ to stand for what the question calls $C_i$. The key properties we need are:

1. ${\cal I}_i$ is a disjoint union of $2^i$ intervals.
2. For each $i$, the component intervals of ${\cal I}_i$ are indexed by length $i$ finite ternary expansions $0.a_1a_2...a_i$ such that $a_j \in \{0,2\}$ for each $j=1,...,i$.
3. Each component interval of ${\cal I}_i$ has of length $3^{-i}$.
4. For any two distinct component intervals of ${\cal I}_i$, the distance between their closest points is $\ge 3^{-i}$.

Intersecting each element of ${\cal I}_i$ with $C$ we obtain the following partition of $C$: $${\cal C}_i = \{C \cap I \mid \text{$I$ is a component interval of ${\cal I}_i$}\} $$ I'll refer to ${\cal C}_i$ as the "level $i$ partition of $C$", and I'll note that it is indexed exactly as in item 2 above.

The key observation is that $f_S$ permutes the elements of ${\cal C}_i$: it takes the level $i$ partition element indexed by $0.a_1a_2...a_i$ to the level $i$ partition element indexed by the sequence which is obtained by changing the $j$th coordinate $a_j$ if and only if $j \in S$ ($1 \le j \le i$).

From this observation, the proof of continuity of $f_S$ is completed as follows.

For each $\epsilon > 0$, choose $i$ large enough that $3^{-i} < \epsilon$. Let $\delta = 3^{-i}$. If $x,y \in C$ and if $|x-y|<\delta=3^{-i}$ then $x,y$ must lie in the same element of ${\cal C}_i$. Therefore $f_S(x)$ and $f_S(y)$ lie in the same element of ${\cal C}_i$. Therefore $d(f_S(x),f_S(y)) \le 3^{-i} < \epsilon$.

If I wanted to summarize this in a one line proof, I would say: $f_S$ is a $1$-Lipschitz function, therefore it is continuous.