Questions: The Cantor Set: An Uncountable Nowhere Dense Example

Question 1

The Cantor set is constructed by iteratively removing middle-third intervals from [0,1]. After infinitely many steps, which statement correctly describes what remains?

• A. A finite set — only the endpoints of the removed intervals survive
• B. A countably infinite set — only countably many intervals were removed, so only countably many points remain
• C. An uncountable set with measure zero — the same cardinality as [0,1] but zero total length
• D. The empty set — removing intervals of total length 1 leaves nothing behind

Answer: (C)

The Cantor set is uncountable — it bijects with {0,1}^ℕ via ternary representations — yet has measure zero because the removed intervals sum to total length 1. Options B and D confuse cardinality with measure. Removing countably many open intervals eliminates all the 'length' but leaves uncountably many points, because points are dimensionless. Option A is also wrong: the Cantor set contains far more than just the endpoints of removed intervals — it contains all points whose ternary expansions use only digits 0 and 2.

Question 2

Which is the correct characterization of which points in [0,1] belong to the Cantor set?

• A. Points whose decimal (base-10) expansions use only the digits 0 and 1
• B. Points whose ternary (base-3) expansions use only the digits 0 and 2, never the digit 1
• C. Points whose ternary expansions are eventually periodic
• D. Points whose ternary expansions are purely terminating (finitely many nonzero digits)

Answer: (B)

The middle-third removal corresponds exactly to removing all points that require the digit 1 in their base-3 representation. Points in (1/3, 2/3) have ternary expansions starting with 0.1..., so they are removed in the first step. Points with only digits 0 and 2 survive every stage. This characterization enables the proof of uncountability: replacing each 2 with 1 gives a bijection between the Cantor set and {0,1}^ℕ (infinite binary sequences), which is uncountable by Cantor's diagonal argument.

Question 3

The Cantor set contains no open interval, yet it contains uncountably many points.

• T. True
• F. False

Answer: True

Both parts are true and together make the Cantor set paradoxical. 'Nowhere dense' means every open interval (a, b) ⊂ [0,1] contains points not in the Cantor set — in fact, it contains an entire removed middle-third interval. Yet the Cantor set bijects with {0,1}^ℕ and has the same cardinality as [0,1] itself. This shows that 'size' in the sense of density (containing an interval) is completely decoupled from 'size' in the sense of cardinality. A set can be simultaneously nowhere dense and uncountable.

Question 4

Since the Cantor construction removes intervals whose lengths sum to 1 — equal to the full length of [0,1] — the Cantor set is expected to be empty.

• T. True
• F. False

Answer: False

Measure zero means zero total length, not zero points. Points are dimensionless and do not contribute to measure. The removed intervals are open (they exclude their endpoints), and the Cantor set consists of all points not in any removed interval. Endpoints of removed intervals are countably many, but the Cantor set also contains irrational points like 1/4 (whose ternary expansion 0.020202... uses only digits 0 and 2). Removing intervals that collectively cover the 'length' of a set does not remove all points.

Question 5

Explain why the Cantor set having measure zero does not contradict it being uncountable. What does this reveal about the relationship between measure and cardinality?

Model answer: Measure and cardinality are independent properties. Measure counts 'how much length' a set occupies; cardinality counts 'how many elements' it contains. The removed intervals account for all 1 unit of length in [0,1], so the Cantor set has measure zero. But uncountably many points can each have size zero and still sum to zero total measure — there is no contradiction. The bijection with {0,1}^ℕ (via ternary representations) proves uncountability independently of measure. A set can be large in cardinality and small in measure simultaneously.

This decoupling is foundational to real analysis and measure theory. The naive intuition — that a 'large' set must have positive measure — fails here. Equally, the Cantor set's complement (the removed intervals) has measure 1 and is dense in [0,1], yet is only countably many disjoint open intervals. The Cantor set thus demolishes two intuitions at once: uncountable sets need not have positive measure, and measure-1 sets need not contain all the points. Understanding this is why Lebesgue measure theory cannot be replaced by cardinality arguments — they answer fundamentally different questions about 'how big' a set is.