# ${F}_{\sigma}$ set

A subset of a topological space^{} is called a ${F}_{\sigma}$ set if it equals the union of a countable^{} collection^{} of closed sets^{}.

The complement of a ${F}_{\sigma}$ set is a ${G}_{\delta}$ set (http://planetmath.org/G_DeltaSet).

For instance, the $X$ set of all points $(x,y)$ in the plane such that either $y=0$ or $x/y$ is rational is an ${F}_{\sigma}$ set because it can be expressed as the union of a countable set of lines:

$$X=\{(x,0)\mid x\in \mathbb{R}\}\cup \bigcup _{r\in \mathbb{Q}}\{(ry,y)\mid y\in \mathbb{R}\}$$ |

Title | ${F}_{\sigma}$ set |
---|---|

Canonical name | FsigmaSet |

Date of creation | 2013-03-22 14:37:59 |

Last modified on | 2013-03-22 14:37:59 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 9 |

Author | rspuzio (6075) |

Entry type | Definition |

Classification | msc 54A05 |

Related topic | G_DeltaSet |

Related topic | G_deltaSet |

Related topic | PavedSet |

Related topic | PavedSpace |