Defining the Cantor Set

Question

I'm having trouble understanding the construction of the Cantor Set as defined by wikipedia.

In particular, we have that

$$ C_0 = [0,1] $$

and then

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

My problem is that wikipedia then states that we can view $C_n$ as the following

$$ C_{n} = \frac{C_{n-1}}{3} \cup \left(\frac{2}{3}+\frac{C_{n-1}}{3}\right) $$

so that $C = \bigcap_{n = 0}^{\infty} C_n$. But above, we have that the interval $C_{n-1}$ is being multiplied by $\frac{1}{3}$ and then in the second instance is being added onto $\frac{2}{3}$. I'm not aware of any standard method to add and multiply numbers by intervals. What is here being designated?

Answer 1

Generally speaking, if $f:\mathbb{R}\to\mathbb{R}$ is a function and $S\subset\mathbb{R}$ is any subset, then $f(S)$ is just the image of $S$ under the function $f$. In the context of your question, the functions of multiplying or adding by a fixed real number are not given names, for the sake of simplicity, and are being denoted with binary operations the way they would if everything involved were a number.

Thus, given a real number $r$ and a subset $S\subset\mathbb{R}$, we define the expression "$r\cdot S$" by $$r\cdot S\overset{\text{def}}{=}m_r(S)=\{rs:s\in S\}$$ where $m_r:\mathbb{R}\to\mathbb{R}$ is the "multiplication function" defined by $m_r(x)=rx$, and similarly, $$S+r\overset{\text{def}}{=}a_r(S)=\{s+r:s\in S\}$$ where $a_r:\mathbb{R}\to\mathbb{R}$ is the "addition function" defined by $a_r(x)=x+r$.

P.S. As Andres points out in a comment above, $C_{n-1}$ is not an interval for any $n\geq 2$.

Answer 2

It means that the given operation is applied to every number in the interval.

For example, $\frac{1}{3}\cdot[0,1] = \left[0,\frac{1}{3}\right]$ and $1+[0,1] = [1,2]$.

An example where this is commonly used is $n\Bbb Z$.