📌 The Cantor set consists exactly of the real numbers in the unit interval whose ternary expansions use only the digits 0 and 2.
Digits 0 and 2 remain; digit 1 marks a removed middle third.
★ Must-know
If F_n consists of 2^n intervals of the form [k/3^n,(k+1)/3^n], then F_{n+1} is obtained by removing the open middle third of each of those intervals.
At each stage, the open middle third of every surviving closed interval is removed.
Further detail
After the first removal, F_1 is the union of the two closed intervals [0,1/3] and [2/3,1].
At stage n, the surviving set F_n is the union of 2^n closed intervals.
Remove, split, repeat: each surviving interval produces two smaller intervals.
1. How is the Cantor set defined in the middle-third construction?
2. Which ternary-expansion property characterizes membership in the Cantor set?
3. What is removed from each surviving interval at every stage of the Cantor construction?
How is the Cantor set constructed from the unit interval?
By intersecting sets formed by successively removing open middle thirds.
Which digits appear in ternary expansions of Cantor set numbers?
Only the digits 0 and 2 appear.
What happens at each stage in the successive middle-third removal process?
The open middle third of every surviving closed interval is removed.
What is the form of the surviving set F_n at stage n?
It is the union of 2^n closed intervals.
Which points belong to every F_n and the Cantor set F?
0, 1/3, 2/3, and 1 belong to every F_n and the Cantor set F.
What does the Cantor set contain regarding intervals?
It contains no nonempty interval.
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