Cantor Set

Cantor Ternary Set

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

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

Properties of Cantor Sets

A Cantor set ${\displaystyle C}$ has the following properties.[1]

• ${\displaystyle C}$ is closed, compact, nowhere dense, and totally disconnected. Moreover, ${\displaystyle C}$ has no isolated points.
• ${\displaystyle C}$ is Lebesgue measurable, with Lebesgue measure ${\displaystyle \lambda (C)=0}$.
• Cantor set is in bijection with ${\displaystyle [0,1]}$, 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.

References

1. Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, second edition, §1.5
2. Craig, Katy. MATH 201A HW 5. UC Santa Barbara, Fall 2020.