Filter (mathematics)

Nicolas Bourbaki, in their book Topologie Générale, popularized filters as an alternative to E. H. Moore and Herman L. Smith's 1922 notion of a net; order filters generalize this notion from the specific case of a power set under inclusion to arbitrary partially ordered sets.

Nevertheless, the theory of power-set filters retains interest in its own right, in part for substantial applications in topology.

Fix a partially ordered set (poset) P. Intuitively, a filter F is a subset of P whose members are elements large enough to satisfy some criterion.

The thing in this case is slightly larger than x, but it still does not contain any other specific point of the line.

The above considerations motivate the upward closure requirement in the definition below: "large enough" objects can always be made larger.

To understand the other two conditions, reverse the roles and instead consider F as a "locating scheme" to find x.

In this interpretation, one searches in some space X, and expects F to describe those subsets of X that contain the goal.

A common use for a filter is to define properties that are satisfied by "generic" elements of some topological space.

[2] This application generalizes the "locating scheme" to find points that might be hard to write down explicitly.

Authors in set theory and mathematical logic often require all filters to be proper;[4] this article will eschew that convention.

A subset S of F is a base or basis for F if the upper set generated by S (i.e., the smallest upwards-closed set containing S) is equal to F. Since every filter is upwards-closed, every filter is a base for itself.

Moreover, if B ⊆ P is nonempty and downward directed, then B generates an upper set F that is a filter (for which B is a base).

Such sets are called prefilters, as well as the aforementioned filter base/basis, and F is said to be generated or spanned by B.

Given p ∈ P, the set {x : p ≤ x} is the smallest filter containing p, and sometimes written ↑ p. Such a filter is called a principal filter; p is said to be the principal element of F, or generate F. Suppose B and C are two prefilters on P, and, for each c ∈ C, there is a b ∈ B, such that b ≤ c. Then we say that B is finer than (or refines) C; likewise, C is coarser than (or coarsens) B. Refinement is a preorder on the set of prefilters.

Thus passage from prefilter to filter is an instance of passing from a preordering to associated partial ordering.

Historically, filters generalized to order-theoretic lattices before arbitrary partial orders.

In the case of lattices, downward direction can be written as closure under finite meets: for all x, y ∈ F, one has x ∧ y ∈ F.[3] A linear (ultra)filter is an (ultra)filter on the lattice of vector subspaces of a given vector space, ordered by inclusion.

For such posets, downward direction and upward closure reduce to:[4] A proper[7]/non-degenerate[8] filter is one that does not contain ∅, and these three conditions (including non-degeneracy) are Henri Cartan's original definition of a filter.

For every subset T of P(S), there is a smallest filter F containing T. As with prefilters, T is said to generate or span F; a base for F is the set U of all finite intersections of T. The set T is said to be a filter subbase when F (and thus U) is proper.

See the image at the top of this article for a simple example of filters on the finite poset P({1, 2, 3, 4}).

Given an ordinal a, a subset of a is called a club if it is closed in the order topology of a but has net-theoretic limit a.

The previous construction generalizes as follows: any club C is also a collection of dense subsets (in the ordinal topology) of a, and ♣(a) meets each element of C. Replacing C with an arbitrary collection C̃ of dense sets, there "typically" exists a filter meeting each element of C̃, called a generic filter.

Let P denote the set of partial orders of limited cardinality, modulo isomorphism.

Likewise, if I is the set of injective modules over some given commutative ring, of limited cardinality, modulo isomorphism, then a partial order on I is: Given any infinite cardinal κ, the modules in I that cannot be generated by fewer than κ elements form a filter.

The dual notion to a filter — that is, the concept obtained by reversing all ≤ and exchanging ∧ with ∨ — is an order ideal.

In general topology and analysis, filters are used to define convergence in a manner similar to the role of sequences in a metric space.

They unify the concept of a limit across the wide variety of arbitrary topological spaces.

A sequence is usually indexed by the natural numbers ℕ, which are a totally ordered set.

Nets generalize the notion of a sequence by replacing ℕ with an arbitrary directed set.

Any point x in the topological space X defines a neighborhood filter or system Nx: namely, the family of all sets containing x in their interior.