# filter basis

A *filter subbasis* for a set $S$ is a collection^{} of subsets of $S$ which has the finite intersection property.

A *filter basis* $B$ for a set $S$ is a non-empty collection of subsets of $S$ which does not contain the empty set^{} such that, for every $u\in B$ and every $v\in B$, there exists a $w\in B$ such that $w\subset u\cap v$.

Given a filter basis $B$ for a set $S$, the set of all supersets^{} of elements of $B$ forms a filter on the set $S$. This filter is known as the filter generated by the basis.

Given a filter subbasis $B$ for a set $S$, the set of all supersets of finite intersections^{} of elements of $B$ is a filter. This filter is known as the filter generated by the subbasis.

Two filter bases are said to be *equivalent ^{}* if they generate the same filter. Likewise, two filter subbases are said to be equivalent if they generate the same filter.

Note: Not every author requires that filters do not contain the empty set. Because every filter is a filter basis then accordingly some authors allow that a filter base can contain the empty set.

Title | filter basis |
---|---|

Canonical name | FilterBasis |

Date of creation | 2013-03-22 14:41:34 |

Last modified on | 2013-03-22 14:41:34 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 11 |

Author | rspuzio (6075) |

Entry type | Definition |

Classification | msc 03E99 |

Classification | msc 54A99 |

Synonym | filter base |

Defines | filter subbasis |

Defines | equivalent |