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filter (Definition)

Let $ X$ be a set. A filter on $ X$ is a set $ \mathbb{F}$ of subsets of $ X$ such that

  • $ X\in\mathbb{F}$
  • The intersection of any two elements of $ \mathbb{F}$ is an element of $ \mathbb{F}$.
  • $ \emptyset\notin\mathbb{F}$ (some authors do not include this axiom in the definition of filter)
  • If $ F\in\mathbb{F}$ and $ F\subset G\subset X$ then $ G\in\mathbb{F}$.
The first two axioms can be replaced by one:
  • Any finite intersection of elements of $ \mathbb{F}$ is an element of $ \mathbb{F}$.
with the usual understanding that the intersection of an empty family of subsets of $ X$ is the whole set $ X$.

A filter $ \mathbb{F}$ is said to be fixed or principal if there is $ F\in \mathbb{F}$ such that no proper subset of $ F$ belongs to $ \mathbb{F}$. In this case, $ \mathbb{F}$ consists of all subsets of $ X$ containing $ F$, and $ F$ is called a principal element of $ \mathbb{F}$. If $ \mathbb{F}$ is not principal, it is said to be non-principal or free.

If $ x$ is any point (or any subset) of any topological space $ X$, the set $ \mathcal{N}_x$ of neighbourhoods of $ x$ in $ X$ is a filter, called the neighbourhood filter of $ x$. If $ \mathbb{F}$ is any filter on the space $ X$, $ \mathbb{F}$ is said to converge to $ x$, and we write $ \mathbb{F}\to x$, if $ \mathcal{N}_x\subset\mathbb{F}$. If every neighbourhood of $ x$ meets every set of $ \mathbb{F}$, then $ x$ is called an accumulation point or cluster point of $ \mathbb{F}$.

Remarks: The notion of filter (due to H. Cartan) has a simplifying effect on various proofs in analysis and topology. Tychonoff'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 infinity, and the limit of a function at a point - are both special cases of the limit of a filter: the Fréchet filter and the neighbourhood filter respectively. The notion of a Cauchy sequence 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 converse holds then we say that the uniform space is complete.



"filter" is owned by Koro. [ full author list (3) | owner history (1) ]
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See Also: ultrafilter, kappa-complete, $\kappa$-complete, net, limit along a filter, upper set, order ideal

Also defines:  principal filter, nonprincipal filter, non-principal filter, free filter, fixed filter, neighbourhood filter, principal element, convergent filter
Keywords:  topology, set theory

Attachments:
examples of filters (Example) by Evandar
filter basis (Definition) by rspuzio
comparison of filters (Definition) by rspuzio
alternative characterization of filter (Theorem) by rspuzio
section filter (Definition) by CWoo
multiplicative filter (Example) by jocaps
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Cross-references: complete, converse, Cauchy filter, uniform space, Cauchy sequence, Fréchet filter, function, infinity, sequence, real, limit, Tychonoff's theorem, cluster point, accumulation point, converge, neighbourhoods, topological space, point, proper subset, finite, axiom, intersection, subsets
There are 30 references to this entry.

This is version 15 of filter, born on 2002-01-05, modified 2008-05-01.
Object id is 1342, canonical name is Filter.
Accessed 15226 times total.

Classification:
AMS MSC: 54A99 (General topology :: Generalities :: Miscellaneous)
 03E99 (Mathematical logic and foundations :: Set theory :: Miscellaneous)
Pending Errata and Addenda
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Discussion
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Filter over "something" by jocaps on 2007-04-16 08:10:03
When we refer to a filter, we may regard it as a filter over a subfamily of the powerset of X, and not necessarily over the powerset of X. See "multiplicative filters"
[ reply | up ]
WikiPedia defines differently by porton on 2006-09-03 03:33:27
WikiPedia defines principal filter differently:

http://en.wikipedia.org/wiki/Filter_(mathematics)
The smallest filter that contains a given element p is a principal filter and p is a principal element in this situation. The principal filter for p is just given by the set {x in P | p ≤ x} and is denoted by prefixing p with an upward arrow.

But PlanetMath defines:
A filter $ \mathbb{F}$ is said to be fixed or principal if the intersection of all elements of $\mathbb{F}$ is nonempty; otherwise, $\mathbb{F}$ is said to be free or non-principal.

Every filter principal in the sense of WikiPedia is principal in the sense of PlanetMath, but not vice verse.

We need to resolve this terminological issue.
--
Victor Porton - http://www.mathematics21.org
* Algebraic General Topology and Math Synthesis
* 21 Century Math Method (post axiomatic math logic)
* Category Theory - new concepts
[ reply | up ]
How can it replace the 2 first axioms by cvalente on 2006-04-23 18:00:59
"The first two axioms can be replaced by one:
Any finite intersection of elements of $ \mathbb{F}$ is an element of $ \mathbb{F}$."

How can this substitute the first axiom?
"$ X\in\mathbb{F}$"
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