Let I be the interval [0, 1], Remove the open middle third segment and let A1 be the set...

Let I be the interval [0, 1], Remove the open middle third segment and let A1 be the set that remains. That is,

Then remove the open middle third segment from each of the two parts of A1and call the remaining set A2. Thus

Continue in this manner. That is, given Ak, remove the open middle third segment from each of the closed segments whose union is Ak, and call the remaining set Ak + 1. Note that A1 ⊇ A2⊇ A3 ⊇ … and that for each k ∈ ℕ, Ak is the union of 2k closed intervals each of length 3−k. The set C =  is called the Cantor set.

(a) Prove that C is compact.

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(b) Let x = 0. a1a2a3⋯ be the ternary (base 3) expansion of a number x in [0, 1]. Prove that x ∈ C iff x has a ternary expansion with an ∈ {0, 2} for all n ∈ ℕ.

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(c) Prove that C is uncountable.

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(d) Prove that C contains no intervals.

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(e) Prove that  but is not an endpoint of any of the intervals in any of the sets Ak (k ∈ ℕ).