Closed sets in Cantor Space that are not Clopen

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SUMMARY

The discussion centers on the characterization of closed sets in Cantor space (C) that are not clopen. It is established that Cantor space is totally disconnected and has a basis of clopen sets. The participants identify that subsets containing finitely many points are examples of closed but not open sets within Cantor space. Additionally, subsets that do not contain any small copy of Cantor space are also classified as closed non-open sets.

PREREQUISITES
  • Understanding of Cantor space topology
  • Familiarity with the concepts of closed and open sets
  • Knowledge of totally disconnected spaces
  • Basic principles of set theory
NEXT STEPS
  • Research the properties of totally disconnected spaces in topology
  • Explore examples of closed sets in Cantor space
  • Study the concept of clopen sets and their significance in topology
  • Investigate the implications of subsets containing finitely many points in topological spaces
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Mathematicians, topology students, and researchers interested in set theory and the properties of Cantor space.

Bacle
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Hi,

Is there a characterization of subsets of the Cantor space C that are closed but not open? As a totally-disconnected set/space, C has a basis of clopen sets; but I'm just curious of what the closed non-open sets are.
 
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I guess the subsets which don't contain a 'part' of C ( i.e. a small copy of C) are among such sets.These include the subsets containing finitely many points; which are closed but not open.
 

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