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Cantor-Bendixson theorem (Theorem)

Any closed subset $ X$ of the reals can be written as a disjoint union

$\displaystyle X = C \cup P, $
where $ C$ is countable and $ P$ is a perfect set (hence this theorem is also known as the CUP theorem).



"Cantor-Bendixson theorem" is owned by CWoo. [ owner history (1) ]
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Other names:  CUP theorem
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Cross-references: perfect set, countable, disjoint union, reals, closed subset

This is version 3 of Cantor-Bendixson theorem, born on 2005-02-10, modified 2005-02-12.
Object id is 6737, canonical name is CantorBendixsonTheorem.
Accessed 2653 times total.

Classification:
AMS MSC: 54H05 (General topology :: Connections with other structures, applications :: Descriptive set theory )
 03E15 (Mathematical logic and foundations :: Set theory :: Descriptive set theory)
Pending Errata and Addenda
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Generalizations and proof by igor on 2005-02-10 14:54:55
Looking at the proof, I think this theorem holds for any second countable T_1 topological space. Are there any other generalizations of this theorem?

Also, the proof that uses the Cantor-Bendixson rank of X requires an auxilliary result: a strictly decreasing chain of closed sets is at most countable. Kuratowski (Topology), attributes this result to Baire. Does anyone know if this theorem has a commonly accepted name?
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