Problem 3. Consider the unit interval [0, 1], and let ξ be fixed real number with...

Problem 3. Consider the unit interval [0, 1], and let ξ be fixed real number with ξ ∈ (0, 1) (note that the case ξ = 1/3 corresponds to the regular Cantor set we learned in our lectures). In stage 1 of the construction, remove the centrally situated open interval in [0, 1] of length ξ. In stage 2 remove the centrally situated open intervals each of relative length ξ (i.e. if the interval has length a you remove an interval of length ξ × a), one in each of the remaining intervals after stage 1, and so on. Let C_ξ denote the set which remains after applying the above procedure indefinitely

(a) Prove that C_ξ is compact.

(b) Prove that C_ξ is totally disconnected and perfect.

(c) Atually, prove that the complement of C_ξ in [0, 1] is the union of open intervals of total length equal to 1.

(d) Show directly that m_∗(C_ξ) = 0.

Expert Solution

Colby Messinger answered 2 years ago

Problem 3. Consider the unit interval [0, 1], and let ξ be fixed real number with...

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Problem 3. Consider the unit interval [0, 1], and let ξ be fixed real number with...

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