Cantor–Bendixson Analysis of Countable Closed Sets

Abstract

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, n ∈ N) are shown to form an exhaustive enumeration, up to homeomorphism, of the countable closed bounded sets into ℵ ₁ many pairwise non-homeomorphic representative sets.