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# Cantor-Bendixson theorem

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

$X=C\cup P,$ |

where $C$ is countable and $P$ is a perfect set (hence this theorem is also known as the *CUP theorem*).

Synonym:

CUP theorem

Type of Math Object:

Theorem

Major Section:

Reference

## Mathematics Subject Classification

03E15*no label found*54H05

*no label found*

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## Comments

## Generalizations and proof

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?