Let X be a set. A filter on X is a set 𝔽 of subsets of X such that

  • β€’


  • β€’

    The intersectionMathworldPlanetmath of any two elements of 𝔽 is an element of 𝔽.

  • β€’

    βˆ…βˆ‰π”½ (some authors do not include this axiom in the definition of filter)

  • β€’

    If Fβˆˆπ”½ and FβŠ‚GβŠ‚X then Gβˆˆπ”½.

The first two axioms can be replaced by one:

  • β€’

    Any finite intersection of elements of 𝔽 is an element of 𝔽.

with the usual understanding that the intersection of an empty family of subsets of X is the whole set X.

A filter 𝔽 is said to be fixed or principal if there is Fβˆˆπ”½ such that no proper subsetMathworldPlanetmathPlanetmath of F belongs to 𝔽. In this case, 𝔽 consists of all subsets of X containing F, and F is called a principal element of 𝔽. If 𝔽 is not principal, it is said to be non-principal or free.

If x is any point (or any subset) of any topological spaceMathworldPlanetmath X, the set 𝒩x of neighbourhoods of x in X is a filter, called the neighbourhood filter of x. If 𝔽 is any filter on the space X, 𝔽 is said to convergePlanetmathPlanetmath to x, and we write 𝔽→x, if 𝒩xβŠ‚π”½. If every neighbourhood of x meets every set of 𝔽, then x is called an accumulation pointMathworldPlanetmathPlanetmath or cluster point of 𝔽.

Remarks: The notion of filter (due to H. Cartan) has a simplifying effect on various proofs in analysisMathworldPlanetmath and topology. TychonoffPlanetmathPlanetmath’s theorem would be one example. Also, the two kinds of limit that one sees in elementary real analysis – the limit of a sequence at infinityMathworldPlanetmath, and the limit of a function at a point – are both special cases of the limit of a filter: the FrΓ©chet filter and the neighbourhood filter respectively. The notion of a Cauchy sequencePlanetmathPlanetmath can be extended with no difficulty to any uniform space (but not just a topological space), getting what is called a Cauchy filter; any convergent filter on a uniform space is a Cauchy filter, and if the converseMathworldPlanetmath holds then we say that the uniform space is completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmath.

Title filter
Canonical name Filter
Date of creation 2013-03-22 12:09:06
Last modified on 2013-03-22 12:09:06
Owner Koro (127)
Last modified by Koro (127)
Numerical id 19
Author Koro (127)
Entry type Definition
Classification msc 03E99
Classification msc 54A99
Related topic UltrafilterMathworldPlanetmathPlanetmath
Related topic KappaComplete
Related topic KappaComplete2
Related topic Net
Related topic LimitAlongAFilter
Related topic UpperSet
Related topic OrderIdeal
Defines principal filter
Defines nonprincipal filter
Defines non-principal filter
Defines free filter
Defines fixed filter
Defines neighbourhood filter
Defines principal element
Defines convergent filter