Cantor Set

Cantor Ternary Set

A Cantor ternary set $C$ of base-3 can be constructed from $[0,1]$ through the iteratively process of removing the open middle third from each closed interval. Specifically, starting from a closed interval $C_{0}=[0,1]$, one can first remove the open middle third, $(1/3,2/3)$, to get the remaining union of closed intervals $C_{1}=C_{0}\setminus (1/3,2/3)=[0,1/3]\cup [2/3,1]$. Next, one repeat the process of removing open middle thirds from each closed interval, ie $C_{2}=C_{1}\setminus ((1/9,2/9)\cup (7/9,8/9))=[0,1/9]\cup [2/9,1/3]\cup [2/3,7/9]\cup [8/9,1]$. Each $C_{n}$ is then constructed iteratively by removing the middle one third of the closed intervals of $C_{n-1}$. The Cantor set $C$ is then obtained when one repeats the process infinitely many times, or equivalently:

$C=\cup _{n=1}^{+\infty }C_{n}.$

Properties of Cantor Sets

A Cantor set $C$ constructed with the iterating process above has the following properties.

• $C$ is compact, nowhere dense, and totally disconnected. Moreover, $C$ has no isolated points.
• Denote $\lambda$ as the Lebesgue measure and ${\mathcal {B}}_{\mathbb {R} }$ as the Borel set defined on $\mathbb {R}$. Then $C$ is measurable, and $\lambda (C)=0$.
• Cantor set is in bijection with $[0,1]$, giving us a counterexample of a noncountable set having zero measure.

Cantor Function

The Cantor set can be used to define Cantor Function, an increasing function which is continuous but has zero derivative almost everywhere.