In mathematics, a filter is a special subset of a partially ordered set. A frequently used special case is the situation that the ordered set under consideration is just the power set of some set, ordered by set inclusion. Filters appear in order and lattice theory, but can also be found in topology. The dual notion of a filter is an ideal. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, a partially ordered set (or poset for short) is a set equipped with a special binary relation which formalizes the intuitive concept of an ordering. ... In mathematics, given a set S, the power set of S, written P(S) or 2S, is the set of all subsets of S. In formal language, the existence of power set of any set is presupposed by the axiom of power set. ... Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of a mathematical ordering. ... See lattice for other mathematical as well as non-mathematical meanings of the term. ... Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with the study of topological spaces. ... In the mathematical area of order theory, every partially ordered set P gives rise to a dual (or opposite) partially ordered set which is often denoted by Pop. ... In mathematical order theory, an ideal is a special subset of a partially ordered set. ...

Filters were introduced by Henri Cartan in 1937 and subsequently used by Bourbaki in their book Topologie Générale. An equivalent notion called net was developed in 1922 by E. H. Moore and H. L. Smith. Henri Cartan (born July 8, 1904) is a son of Elie Cartan, and is, as his father was, a distinguished and influential mathematician. ... 1937 was a common year starting on Friday (link will take you to calendar). ... Nicolas Bourbaki is the pseudonym under which a group of mainly French 20th-century mathematicians wrote a series of books of exposition of modern advanced mathematics, beginning in 1935. ... Nicolas Bourbaki is the collective pseudonym under which a group of mainly French 20th-century mathematicians wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. ... In mathematics the term net has at least two meanings. ... 1922 was a common year starting on Sunday (see link for calendar). ... Eliakim Hastings Moore (born January 26, 1862, died December 30, 1932) was an American mathematician. ...

General definition

A non-empty subset F of a partially ordered set (P,≤) is a filter, if the following conditions hold: In set theory, a set is called non-empty (or nonempty) if it contains at least one element, and is therefore not the empty set. ...

1. For every x, y in F, there is some element z in F, such that z ≤ x and z ≤ y. (F is a filter base)
2. For every x in F and y in P, x ≤ y implies that y is in F. (F is an upper set)

A filter is proper if it is not equal to the whole set P.

While the above definition is the most general way to define a filter for arbitrary posets, it was originally defined for lattices only. In this case, the above definition can be characterized by the following equivalent statement: A non-empty subset F of a lattice (P,≤) is a filter, iff it is an upper set that is closed under finite meets (infima), i.e., for all x, y in F, we find that x ^ y is also in F. The term lattice derives from the shape of the Hasse diagrams that result from depicting these orders. ... ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Although P iff Q is most standard, common alternative phrases include P is necessary and sufficient for Q and P... In mathematics the infimum of a subset of some set is the greatest element, not necessarily in the subset, that is smaller than all other elements of the subset. ...

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.

The dual notion of a filter, i.e. the concept obtained by reversing all ≤ and exchanging ^ with v, is ideal. Because of this duality, the discussion of filters usually boils down to the discussion of ideals. Hence, most additional information on this topic (including the definition of maximal filters and prime filters) is to be found in the article on ideals. There is a separate article on ultrafilters. In mathematical order theory, an ideal is a special subset of a partially ordered set. ... In mathematics, especially in order theory, an ultrafilter is a subset of a partially ordered set (a poset) which is maximal among all proper filters. ...

Filter on a set

A special case of a filter is a filter defined on a set. Given a set S, a partial ordering ⊆ can be defined on the powerset P(S) by subset inclusion, turning (P(S),⊆) into a lattice. A filter F on S is then a subset of P(S) with the following properties:

1. S is in F. (F is non-empty)
2. The empty set is not in F. (F is proper)
3. If A and B are in F, then so is their intersection. (F is closed under finite joins)
4. If A is in F and A is a subset of B, then B is in F, for all subsets B of S. (F is an upper set)

A filter base is a subset B of P(S) with the following properties

1. The intersection of any two sets of B contains a set of B
2. B is non-empty and the empty set is not in B

A filter base B can be turned into a filter by including all sets of P(S) which contain a set of B.

Given a subset T of P(S) we can ask whether there exists a smallest filter F containing T. Such a filter exists if and only if the finite intersection of subsets of T is non-empty. We call T a subbase of F and say F is generated by T. F can be constructed by taking all finite intersections of T which is then filter base for F.

Examples

• A simple example of a filter is the set of all subsets of S that include a particular subset C of S. Such a filter is called the principal filter generated by C.

• The Fréchet filter on an infinite set S is the set of all subsets of S that have finite complement.

• A uniform structure on a set X is a filter on X×X.

In mathematics, Fréchet filter is an important concept in order theory. ... In topology, one defines uniform spaces in order to study concepts such as uniform continuity, completeness and uniform convergence. ...

Filters in model theory

For any filter F on a set S, the set function defined by

is finitely additive -- a "measure" if that term is construed rather loosely. Therefore the statement In mathematics, a measure is a function that assigns a number, e. ...

can be considered somewhat analogous to the statement that φ holds "almost everywhere". That interpretation of membership in a filter is used (for motivation, although it is not needed for actual proofs) in the theory of ultraproducts in model theory, a branch of mathematical logic. An ultraproduct is a mathematical construction, which is used in abstract algebra to construct new fields from given ones, and in model theory, a branch of mathematical logic. ... In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the models which underlie mathematical systems. ... Mathematical logic is a discipline within mathematics, studying formal systems in relation to the way they encode intuitive concepts of proof and computation as part of the foundations of mathematics. ...

Filters in topology

In topology and analysis, filters are used to define convergence in a manner similar to the role of sequences in a metric space. Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with the study of topological spaces. ... This is a page about mathematics. ... In mathematics, a metric space is a set (or space) where a distance between points is defined. ...

Given a point x the set of all neighbourhoods of x is a filter, N_x. A (proper) filter which is a superset of N_x is said to converge to x, written . Note that if and then . In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. ...

Given a filter F on a set X and a function , the set forms a filter base for a filter which, in a slight abuse of notation, we denote by f(F).

The following useful results hold:

1. X is Hausdorff iff every filter on X has at most one limit (i.e., converges to at most one point x).
2. f is continuous at x iff implies
3. X is compact iff every filter on X is a subset of a convergent filter.
4. X is compact iff every ultrafilter on X converges.

The neighbourhood system for a non empty set A is a filter called the neighbourhood filter for A. Felix Hausdorff (November 8, 1868 - January 26, 1942) was a German mathematician who is considered to be one of the founders of modern topology and who contributed significantly to set theory and functional analysis. ... In topology and related areas of mathematics, the neighbourhood system or neighbourhood filter for a point x is the collection of all neighbourhoods for the point x. ...

Filters in uniform spaces

Given a uniform space X, a filter F on X is called Cauchy filter if for every U in the entourage, there is an with for every . In a metric space this takes the form F is Cauchy if for every . X is said to be complete if every Cauchy filter converges. In topology, one defines uniform spaces in order to study concepts such as uniform continuity, completeness and uniform convergence. ... In topology, one defines uniform spaces in order to study concepts such as uniform continuity, completeness and uniform convergence. ...

Let Cauchy. Then . Thus every compact uniformity is complete. Further, a uniformity is compact iff it is complete and totally bounded. In mathematics, a metric space is a set (or space) where a distance between points is defined. ...

See also

• Filtration (abstract algebra)
• Ideal (order theory)

In mathematics, a filtration is an indexed set Si of subobjects of a given algebraic structure S, with an index set I that is a totally ordered set, subject only to the condition that if i ≤ j in I then Si is contained in Sj. ... In mathematical order theory, an ideal is a special subset of a partially ordered set. ...

References

• Cartan, H. (1937) "Thèorie des filtres". CR Acad. Paris, 205, 595–598.
• Cartan, H. (1937) "Filtres et ultrafiltres" CR Acad. Paris, 205, 777–779

External links

An introductory account of the theory of filters in metric and topological spaces (http://www.efnet-math.org/~david/mathematics/filters.pdf)