# club

If $\kappa $ is a cardinal then a set $C\subseteq \kappa $ is *closed* iff for any $S\subseteq C$ and $$, $sup(S\cap \alpha )=\alpha $ then $\alpha \in C$. (That is, if the limit of some sequence in $C$ is less than $\kappa $ then the limit is also in $C$.)

If $\kappa $ is a cardinal and $C\subseteq \kappa $ then $C$ is *unbounded ^{}* if, for any $$, there is some $\beta \in C$ such that $$.

If a set is both closed and unbounded then it is a *club* set.

Title | club |

Canonical name | Club |

Date of creation | 2013-03-22 12:53:01 |

Last modified on | 2013-03-22 12:53:01 |

Owner | Henry (455) |

Last modified by | Henry (455) |

Numerical id | 5 |

Author | Henry (455) |

Entry type | Definition |

Classification | msc 03E10 |

Defines | club |

Defines | closed |

Defines | unbounded |

Defines | closed unbounded |

Defines | closed set |

Defines | unbounded set |

Defines | closed unbounded set |

Defines | club set |