filter basis


A filter subbasis for a set S is a collectionMathworldPlanetmath 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 setMathworldPlanetmath such that, for every uB and every vB, there exists a wB such that wuv.

Given a filter basis B for a set S, the set of all supersetsMathworldPlanetmath 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 intersectionsDlmfMathworldPlanetmath 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 equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath 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