Cantor Set

August 21, 2024

What is the Cantor Set?

The Cantor set is the limit of a process that involves repeatedly dividing a given line segment into three equal parts and removing the middle segment.

Here, we assume that all segments are closed intervals.

Cardinality of the Cantor Set

Although it might seem like the segments will eventually become empty as their lengths decrease, this is not the case.

For example, if we start with the closed interval $C_{0} = [0, 1)$ , the process yields the following sets:

$C_{1} = [0, \frac{1}{3}] \oplus [\frac{2}{3}, 1]$

$C_{2} = [0, \frac{1}{9}] \oplus [\frac{2}{9}, \frac{1}{3}] \oplus [\frac{2}{3}, \frac{5}{9}] \oplus [\frac{8}{9}, 1]$

$C_{3} = [0, \frac{1}{27}] \oplus [\frac{2}{27}, \frac{1}{9}] \oplus [\frac{2}{9}, \frac{7}{27}] \oplus [\frac{8}{27}, \frac{1}{3}] \oplus [\frac{2}{3}, \frac{19}{27}] \oplus [\frac{20}{27}, \frac{7}{9}] \oplus [\frac{8}{9}, \frac{25}{27}] \oplus [\frac{26}{27}, 1]$

To make it clearer, using ternary notation, the sets are:

$C_{1} = [0, 0.1] \oplus [0.2, 1]$

$C_{2} = [0, 0.01] \oplus [0.02, 0.1] \oplus [0.2, 0.21] \oplus [0.22, 1]$

$C_{3} = [0, 0.001] \oplus [0.002, 0.01] \oplus [0.02, 0.021] \oplus [0.022, 0.1] \oplus [0.2, 0.201] \oplus [0.202, 0.21] \oplus [0.22, 0.221] \oplus [0.222, 1]$

For instance, $0.2$ is included in $C_{0}$ , $C_{1}$ , $C_{2}$ , and $C_{3}$ , and it will never be removed in future iterations.

As can be seen from the initial diagram, any point that becomes the endpoint of some segment will always remain in the set.

Thus, $0.2 \in C_{\infty}$ , and the Cantor set is not empty.

In fact, it contains a very large number of elements.

Specifically, the Cantor set is composed of numbers that can be expressed using only the digits 0 and 2 in their ternary representation.

Note that this also includes infinite decimals such as

$0.02020202 \dots (3) = \frac{2}{22} (3) = \frac{1}{4} (10)$

$0.022222222 \dots = 0.1$

$0.202002000200002000002 \dots$

Since these numbers can be put into a one-to-one correspondence with all real numbers in binary representation, the Cantor set has the cardinality of the continuum $2^{ℵ_{0}}$ .

This is far greater than the countable infinity $ℵ_{0}$ , which is the cardinality of the integers.

Length of the Cantor Set

If we define the length of $C_{n}$ as the sum of the lengths of the segments, then:

$| C_{0} | = 1$

$| C_{1} | = 1 - \frac{1}{3} = \frac{2}{3}$

$| C_{2} | = (\frac{2}{3})^{2} = \frac{4}{9}$

$| C_{n} | = (\frac{2}{3})^{n}$

Therefore, as $n \to \infty$ , $| C_{n} | \to 0$ .

Thus, the length of the Cantor set is $0$ .

In terms of measure theory, the Cantor set is a null set.

Interest in the Cantor Set

The Cantor set is fascinating because, despite having the cardinality of the continuum (i.e., being composed of an incredibly large number of elements), its measure is 0.