filter
Let be a set. A filter on is a set of subsets of such that
- •
- •
The intersection
of any two elements of is an element of .
- •
(some authors do not include this axiom in the definition of filter)
- •
If and then .
The first two axioms can be replaced by one:
- •
Any finite intersection of elements of is an element of .
with the usual understanding that the intersection of an empty familyof subsets of is the whole set .
A filter is said to be fixedor principal if there is such that no proper subset of belongs to . In this case, consists of all subsets of containing , and is called a principal element of . If is not principal, it is said to be non-principal or free.
If is any point (or any subset) of any topological space ,the set of neighbourhoods of in is a filter,called the neighbourhood filter of .If is any filter on the space , is said to converge
to , and we write ,if .If every neighbourhood of meets every set of , then is called an accumulation point
or cluster point of .
Remarks: The notion of filter (due to H. Cartan) has a simplifying effect onvarious proofs in analysis and topology.Tychonoff
’s theorem would be one example.Also, the two kinds of limit that one sees in elementary realanalysis – the limit of a sequence at infinity
, and the limitof a function at a point – are both special cases of the limitof a filter: the Fréchet filter and the neighbourhood filterrespectively.The notion of a Cauchy sequence
can be extended with no difficultyto any uniform space (but not just a topological space),getting what is called a Cauchy filter; any convergent filter on auniform space is a Cauchy filter, and if the converse
holds thenwe say that the uniform space is complete
.
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 | Ultrafilter![]() |
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 |