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  3. problem 3. consider the unit interval [0, 1], and

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

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Advance Math

Problem 3. Consider the unit interval [0, 1], and let ? be fixedreal number with ? ? (0, 1) (note that the case ? = 1/3 correspondsto the regular Cantor set we learned in our lectures). In stage 1of the construction, remove the centrally situated open interval in[0, 1] of length ?. In stage 2 remove the centrally situated openintervals each of relative length ? (i.e. if the interval haslength a you remove an interval of length ? × a), one in each ofthe remaining intervals after stage 1, and so on. Let C?denote the set which remains after applying the above procedureindefinitely

(a) Prove that C? is compact.

(b) Prove that C? is totally disconnected andperfect.

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

(d) Show directly that m?(C?) = 0.

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