# 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*).

Title | Cantor-Bendixson theorem |
---|---|

Canonical name | CantorBendixsonTheorem |

Date of creation | 2013-03-22 15:01:40 |

Last modified on | 2013-03-22 15:01:40 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 6 |

Author | CWoo (3771) |

Entry type | Theorem |

Classification | msc 03E15 |

Classification | msc 54H05 |

Synonym | CUP theorem |