# Borel $\sigma $-algebra

For any topological space^{} $X$, the *Borel sigma algebra* of $X$ is the $\sigma $–algebra $\mathcal{B}$ generated by the open sets of $X$. In other words, the Borel sigma algebra is equal to the intersection of all sigma algebras $\mathcal{A}$ of $X$ having the property that every open set of $X$ is an element of $\mathcal{A}$.

An element of $\mathcal{B}$ is called a *Borel subset* of $X$, or a *Borel set*.

Title | Borel $\sigma $-algebra |
---|---|

Canonical name | Borelsigmaalgebra |

Date of creation | 2013-03-22 12:00:31 |

Last modified on | 2013-03-22 12:00:31 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 10 |

Author | djao (24) |

Entry type | Definition |

Classification | msc 28A05 |

Synonym | Borel $\sigma $ algebra |

Synonym | Borel sigma algebra |

Related topic | SigmaAlgebra |

Related topic | OuterRegular |

Related topic | LebesgueMeasure |

Defines | Borel subset |

Defines | Borel set |