Cantor Ternary Set
A Cantor ternary set of base-3 can be constructed from through the iterative process of removing the open middle third from each closed interval. Specifically, starting from a closed interval , one can first remove the open middle third, , to get the remaining union of closed intervals . Next, one repeat the process of removing open middle thirds from each closed interval, ie . Each is then constructed iteratively by removing the middle one third from each closed intervals of . The Cantor set is then obtained when one repeats the process infinitely many times, or equivalently:
Properties of Cantor Sets
A Cantor set has the following properties.
- is closed, compact, nowhere dense, and totally disconnected. Moreover, has no isolated points.
- is Lebesgue measurable, with Lebesgue measure .
- Cantor set is in bijection with , giving us a counterexample of a noncountable set having zero measure.
The Cantor set can be used to define Cantor Function, an increasing function which is continuous but has zero derivative almost everywhere.
- Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, second edition, §1.5
- Craig, Katy. MATH 201A HW 5. UC Santa Barbara, Fall 2020.