It is to be shown that by starting with the unit interval [ 0 , 1 ] and deleting the open interval 1 2 , 1 4 , 1 8 , etc . of each remaining sub-interval, the limiting set is a topological cantor, and the measure of this limiting set is greater than zero. Also, if possible, its exact value is to be found, or else, its lower bound is to be found. Concept Introduction: The Cantor set is created by starting with closed interval S 0 = [ 0 , 1 ] and removing its open middle third. The fractal properties of the Cantor set are has structure at arbitrarily small scales. is self-similar. The dimension of C is not an integer. has measure zero. consists of unaccountably many points.

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Chapter 11.3, Problem 10E

Interpretation Introduction

Interpretation:

It is to be shown that by starting with the unit interval [0,1] and deleting the open interval 12,14,18,etc. of each remaining sub-interval, the limiting set is a topological cantor, and the measure of this limiting set is greater than zero. Also, if possible, its exact value is to be found, or else, its lower bound is to be found.

Concept Introduction:

The Cantor set is created by starting with closed interval S0=[0,1] and removing its open middle third.

The fractal properties of the Cantor set are

has structure at arbitrarily small scales.

is self-similar.

The dimension of C is not an integer.

has measure zero.

consists of unaccountably many points.

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