Find an explicit homeomorphism between the Cantor set and a proper subset of the Cantor set

Question

Given the Cantor set C, find an explicit homeomorphism between C and a proper subset of C. The topology book I am using defines the Cantor set inductively, let $I_0=[0,1]$, let $I_1$ Be the complement of the open middle third of $I_0$, i.e. $I_1=[0,\frac{1}{3}]\cup[\frac{1}{3},1]$. Let $I_2=[0,\frac{1}{9}]\cup[\frac{2}{9},\frac{1}{3}]\cup[\frac{2}{3},\frac{7}{9}]\cup[\frac{8}{9},1]$. Inductively, define $I_k$ as the set obtained from $I_{k-1}$ by removing the open middle third of each closed interval in $I_{k-1}$. Define the Cantor set C to be $\bigcap_{k=0}^{\infty}I_{k}$. So far I proposed the proper subset of C to be the set {0,1}, namely, the endpoints. How do I map these two points to the Cantor set by a continuous bijective function with a continuous inverse. Perhaps there is another proper subset that is easier to use? Thanks in advance for the help.

Answer 1

Define $f(x)=\frac x3$. That's a homeomorphism between $C$ and $\left\{x\in C\,\middle|\,x\leqslant\frac13\right\}$.