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In mathematics, a filter is a special subset of a partially ordered set. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement 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. In Mathematics, given a set S, the power set (or powerset) of S, written \mathcal{P}(S P ( S) Filters appear in order and lattice theory, but can also be found in topology from where they originate. Order theory is a branch of Mathematics that studies various kinds of Binary relations that capture the intuitive notion of ordering providing a framework for saying In Mathematics, a lattice is a Partially ordered set (also called a poset) in which every pair of elements has a unique Supremum (the elements' Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of The dual notion of a filter is an ideal. In the mathematical area of Order theory, every Partially ordered set P gives rise to a dual (or opposite) partially ordered set which In mathematical Order theory, an ideal is a special subset of a Partially ordered set (poset

Filters were introduced by Henri Cartan in 1937[1][2] and subsequently used by Bourbaki in their book Topologie Générale as an alternative to the similar notion of a net developed in 1922 by E. H. Moore and H. Henri Paul Cartan ( July 8, 1904 &ndash August 13, 2008) was a son of Élie Cartan, and was as his father was a distinguished Nicolas Bourbaki is the collective Pseudonym under which a group of (mainly French) 20th-century Mathematicians wrote a series of books presenting an exposition Nicolas Bourbaki is the collective Pseudonym under which a group of (mainly French) 20th-century Mathematicians wrote a series of books presenting an exposition This article is about nets in Topological spaces and not about ε-nets in Metric spaces In Topology and related areas of Mathematics Eliakim Hastings Moore ( January 26, 1862, Marietta, Ohio – December 30, 1932, Chicago, Illinois L. Smith.

Contents

General definition

A non-empty subset F of a partially ordered set (P,≤) is a filter if the following conditions hold:

  1. For every x, y in F, there is some element z in F, such that z ≤ x and z ≤ y. In Mathematics, and more specifically Set theory, the empty set is the unique set having no ( Zero) members (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)
  3. A filter is proper if it is not equal to the whole set P. In Mathematics, an upper set, or upward set is a subset Y of a given Partially ordered set ( X,&le such that for all elements This is often taken as part of the definition of a filter.

While the above definition is the most general way to define a filter for arbitrary posets, it was originally defined for lattices only. In Mathematics, a lattice is a Partially ordered set (also called a poset) in which every pair of elements has a unique Supremum (the elements' 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, if and only if it is an upper set that is closed under finite meets (infima), i. ↔ In Mathematics the infimum of a Subset of some set is the Greatest element, not necessarily in the subset that is less than or equal to all elements of e. , for all x, y in F, we find that x ∧ y is also in F.

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: \uparrow p.

The dual notion of a filter, i. e. the concept obtained by reversing all ≤ and exchanging ∧ with ∨, 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. In mathematical Order theory, an ideal is a special subset of a Partially ordered set (poset There is a separate article on ultrafilters. In the mathematical field of Set theory, an ultrafilter on a set X is a collection of Subsets of X that is a filter, that

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. Define a filter F on S as 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 meets)
  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)

The first three properties imply that a filter on a set has the finite intersection property. In General topology, the finite intersection property is a property of a collection of subsets of a set X. Note that with this definition, a filter on a set is indeed a filter; in fact, it is a proper filter. Because of this, sometimes this is called a proper filter on a set; however, as long as the set context is clear, the shorter name is sufficient.

A filter base (or filter basis) 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

Given a filter base B, one may obtain a (proper) filter by including all sets of P(S) which contain a set of B. The resulting filter is said to be generated by or spanned by filter base B. Every filter is a fortiori a filter base, so the process of passing from filter base to filter may be viewed as a sort of completion.

If B and C are two filter bases on S, one says C is finer than B (or that C is a refinement of B) if for each B0 ∈ B, there is a C0 ∈ C such that C0 ⊆ B0.

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

Filters in model theory

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

m(A)=\left\{ \begin{matrix} \,1 & \mbox{if }A\in F \\ \,0 & \mbox{if }S\setminus A\in F \\ \,\mbox{undefined} & \mbox{otherwise} \end{matrix} \right.

is finitely additive — a "measure" if that term is construed rather loosely. In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with Therefore the statement

\left\{\,x\in S: \varphi(x)\,\right\}\in F

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 of which the ultrapower (defined below is a special case In Mathematics, model theory is the study of (classes of mathematical structures such as groups, Fields graphs or even models Mathematical logic is a subfield of Logic and Mathematics with close connections to Computer science and Philosophical logic.

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 the branch of Mathematics that studies the properties of In Mathematics, a sequence is an ordered list of objects (or events In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined

In topology and related areas of mathematics, a filter is a generalization of a net. This article is about nets in Topological spaces and not about ε-nets in Metric spaces In Topology and related areas of Mathematics Both nets and filters provide very general contexts to unify the various notions of limit to arbitrary topological spaces. In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close" Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity.

A sequence is usually indexed by the natural numbers, which are a totally ordered set. In Mathematics, a sequence is an ordered list of objects (or events In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an In Mathematics and Set theory, a total order, linear order, simple order, or (non-strict ordering is a Binary relation Thus, limits in first-countable spaces can be described by sequences. In Topology, a branch of Mathematics, a first-countable space is a Topological space satisfying the "first Axiom of countability " However, if the space is not first-countable, nets or filters must be used. Nets generalize the notion of a sequence by requiring the index set simply be a directed set. In Mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive Filters can be thought of as sets built from multiple nets. Therefore, both the limit of a filter and the limit of a net is conceptually the same as the limit of a sequence.

An advantage to using filters is that many results can be shown without using the axiom of choice. In Mathematics, the axiom of choice, or AC, is an Axiom of Set theory.

Neighbourhood bases

Take a topological space T and a point x ∈ T.

Convergent filter bases

Take a topological space T and a point x ∈ T.

Clustering

Take a topological space T and a point x ∈ T.

Properties of a topological space

Take a topological space T.

Functions on topological spaces

Take topological spaces X and Y and subset E ⊆ X. Take a filter base B on E and a function f : E \to Y. The image of B under f is f[B] is the set \{ f(x) : x \in B \}. In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage The image f[B] forms a filter base on Y.

Cauchy filters

Take a metric space X with metric d. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined In Mathematics, a metric or distance function is a function which defines a Distance between elements of a set.

More generally, given a uniform space X, a filter F on X is called Cauchy filter if for every entourage U there is an A \in F with (x, y) \in U for every x, y \in A. In the Mathematical field of Topology, a uniform space is a set with a uniform structure. In the Mathematical field of Topology, a uniform space is a set with a uniform structure. In a metric space this agrees with the previous definition. X is said to be complete if every Cauchy filter converges. Conversely, on a uniform space every convergent filter is a Cauchy filter. Moreover, every cluster point of a Cauchy filter is a limit point.

A compact uniform space is complete: on a compact space each filter has a cluster point, and if the filter is Cauchy, such a cluster point is a limit point. Further, a uniformity is compact if and only if it is complete and totally bounded. In Topology and related branches of Mathematics, a totally bounded space is a space that can be covered by finitely many Subsets of any

Most generally, a Cauchy space is a set equipped with a class of filters declared to be Cauchy. In General topology and analysis, a Cauchy space is a generalization of Metric spaces and Uniform spaces for which the notion of Cauchy convergence These are required to have the following properties:

  1. for each x in X, the ultrafilter at x, U(x), is Cauchy. In the mathematical field of Set theory, an ultrafilter on a set X is a collection of Subsets of X that is a filter, that
  2. if F is a Cauchy filter, and F is a subset of a filter G, then G is Cauchy.
  3. if F and G are Cauchy filters and each member of F intersects each member of G, then F ∩ G is Cauchy.

The Cauchy filters on a uniform space have these properties, so every uniform space (hence every metric space) defines a Cauchy space.

See also

References

  1. ^ H. In Mathematics, a filtration is an Indexed set Si of Subobjects of a given Algebraic structure S, with the index This article is about nets in Topological spaces and not about ε-nets in Metric spaces In Topology and related areas of Mathematics Cartan, "Théorie des filtres". CR Acad. Paris, 205, (1937) 595–598.
  2. ^ H. Cartan, "Filtres et ultrafiltres" CR Acad. Paris, 205, (1937) 777–779.
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