If C is the Cantor set, C+C contains an open set.

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The discussion centers on the mathematical assertion that the sum of the Cantor set, denoted as C, with itself (C+C) results in the closed interval [0,2]. This conclusion is supported by the property that any number in base 3 between 0 and 2 can be expressed as a sum of two Cantor numbers. The method involves decomposing base 3 digits, where 0s and 2s remain unchanged, while 1s are represented as sums of 2s that total to 1. This process confirms that C+C indeed contains an open set.

PREREQUISITES
  • Understanding of the Cantor set and its properties
  • Familiarity with base 3 numeral system
  • Knowledge of Hamel bases in vector spaces
  • Basic concepts of real analysis and open sets
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  • Study the properties of the Cantor set in detail
  • Learn about Hamel bases and their applications in functional analysis
  • Explore the concept of open sets in topology
  • Investigate the implications of C+C in real analysis
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Mathematicians, students of real analysis, and anyone interested in set theory and the properties of the Cantor set.

WHOAguitarninja
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This is a statement my professor made in class some time ago (as a means to show that C contains a Hamel basis) that seemed fairly innocent, but it's bothered me for awhile. I did some searching online, and it seems that C+C=[0,2]. There it was again stated that this is fairly easy to show, but they neglected to give any insight as to how one might show it. Is there something simple I'm missing?
 
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my rough guess is that any base 3 number (between 0 and 2) can be decomposed into a sum of two cantor numbers. if the digit is a zero, or a two, leave it alone. if its a 1, break it onto a sum of 2's that add up to 1.

e.g. if its 0.12, then 0.0222...+0.02
 

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