Cantor ternary set, construction

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SUMMARY

The Cantor set, denoted as $C_n$, is constructed by iteratively removing the open middle third from closed intervals. At each stage of this construction, the endpoints of the remaining intervals are preserved, ensuring they remain in the Cantor set. This process guarantees that all endpoints of the closed intervals that comprise $C_n$ are included in the set, as they are never removed during the iterations. The iterative nature of the construction solidifies the inclusion of these endpoints as part of the Cantor set.

PREREQUISITES
  • Understanding of the Cantor set construction process
  • Familiarity with closed and open intervals in real analysis
  • Basic knowledge of set theory and topology
  • Concept of convergence in sequences of sets
NEXT STEPS
  • Study the properties of the Cantor set in relation to measure theory
  • Explore the implications of the Cantor set in real analysis
  • Learn about the concept of nowhere dense sets in topology
  • Investigate the relationship between the Cantor set and fractals
USEFUL FOR

Mathematicians, students of real analysis, and anyone interested in set theory and topology will benefit from this discussion on the Cantor set and its construction.

Dustinsfl
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Explain why all the end points of the closed intervals that comprise $C_n$ are in the Cantor set $C_n$.

I understand why this is true but I don't know how to explain it.
 
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dwsmith said:
Explain why all the end points of the closed intervals that comprise $C_n$ are in the Cantor set $C_n$.

I understand why this is true but I don't know how to explain it.

Because at each step the open middle third of each remaining intervals is removed, which always leaves the end points of the intervals at the earlier stage in the set, and as end points of the new intervals (one of the end points the other for each interval is created at each step).

CB
 

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