Set Theory pp 301-311 | Cite as

Cantor–Bendixson Analysis of Countable Closed Sets

  • Abhijit DasguptaEmail author


We devote this chapter to the Cantor, G. Bendixson, I. O.Cantor–Bendixson analysis of countable closed sets. We first prove the effective Cantor–Bendixson theorem which decomposes a closed set into an effectively countable set and a perfect set. We then obtain a full topological classification for the class of countable closed bounded subsets of R: The Cantor–Bendixson rank is shown to be a complete invariant for the relation of homeomorphism between these sets, and the countable ordinals ω ν n + 1 (ν < ω 1, nN) are shown to form an exhaustive enumeration, up to homeomorphism, of the countable closed bounded sets into 1 many pairwise non-homeomorphic representative sets.


  1. 9.
    J. W. Dauben. Georg Cantor: His Mathematics and Philosophy of the Infinite. Princeton University Press, 1990.Google Scholar
  2. 39.
    A. S. Kechris. Set theory and uniqueness for trigonometric series. preprint, 1997.Google Scholar

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© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Detroit MercyDetroitUSA

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