Alternative definition of the Cantor Set

Question

The Cantor Set $C$ can be defined as taking the interval $C_0=[0,1]$

then $C_1=[0, \frac{1}{3}] \cup [\frac{2}{3},1]$

then $C_2=[0, \frac{1}{9}] \cup [\frac{2}{9},\frac{1}{3}] \cup [\frac{4}{9},\frac{5}{9}] \cup [\frac{2}{3},\frac{7}{9}] \cup [\frac{8}{9},1]$

and so on... Basically this definition involves splitting up the interval and then taking every other segment then $C$ is the intersection of all these so it still creates the same Cantor set in the end.

My question is what is the formula for $C_n$?

Answer 1

$C_n$ is the union over $[k/3^n,(k+1)/3^n]$ where $k = 0,2,...$ up to the largest even number less than $3^n$.

Answer 2

It's $$\bigcap_{n=0}^\infty\left(\bigcup_{i=0}^{\lfloor 3^n/2\rfloor}\left[\frac{2i}{3^n},\frac{2i+1}{3^n}\right]\right)$$