Find some extraordinary subset of [0,1]

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SUMMARY

The discussion focuses on finding a subset A of the interval [0,1] that satisfies the condition A = cl(int A) and has a boundary (bd A) that does not have measure zero. The fat Cantor set is suggested as a potential candidate for modification to achieve this goal. Participants discuss the need to consider the boundary of the closure of the union of open intervals left out during the construction of the set, specifically referencing the fat Cantor set as a key component in the solution.

PREREQUISITES
  • Understanding of set theory concepts such as closure (cl) and interior (int).
  • Familiarity with the properties of the fat Cantor set.
  • Knowledge of measure theory, particularly the concept of measure zero.
  • Basic skills in constructing and manipulating subsets of real intervals.
NEXT STEPS
  • Research the properties and construction methods of the fat Cantor set.
  • Study the definitions and implications of closure and interior in topology.
  • Explore measure theory, focusing on sets with measure zero and their boundaries.
  • Investigate the relationship between open intervals and their closures in real analysis.
USEFUL FOR

Mathematicians, students studying real analysis, and anyone interested in advanced set theory and topology, particularly those exploring properties of subsets within the interval [0,1].

Maclaurin
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Homework Statement


I want to find a subset A of [0,1] such that A=cl(int A) and bd A does not have measure zero where cl means closure and int means interior.

Homework Equations





The Attempt at a Solution



I tried some sets, but failed.
 
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I tried that set but it was difficult for me to modify it. Can you give me more hint?
 
Well, you want the fat Cantor set to be the boundary of the set you want.

Every step you leave out some open intervals. Now let V be the union of the open intervals you leave out at the odd steps. Take \overline{V} to be your set.
 
I think bd of V is not the fat Cantor set... what should I do?
 
ah instead of bd of V, boundary of closure of V.
 

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