Recent Articles

Filter (mathematics)

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.

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

Filters of sets

An important special case of order filters are filters of sets, which are obtained by taking the powerset of a set S as a partial order, ordered by subset inclusion. Thus, a filter F on a set S is a set of subsets of 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)

Note that this definition is in absolute correspondence with the general notion introduced above, since the powerset clearly forms a lattice.

An important notion for filters of sets is that of a filter base. Given any subset T of P(S) such that the intersection of any finite subset of T is non-empty, there is a unique smallest filter F containing T, called the filter generated by T. If T is closed under finite intersections then F takes the simple form <math>F = \{ x : \exists y \in T \ y \subseteq x \}<math> and T is called a filter base for F

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 measure" if that term is construed rather loosely. Therefore the statement

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

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.

Filters in topology

Filters are used in topology and analysis. They are a good way of talking about convergence, in a manner similar to the role of sequences in a metric space.

Given a point x the set of all neighbourhoods of x is a filter, <math>N_x<math>. A (proper) filter which is a superset of <math>N_x<math> is said to converge to x, written <math>F \to x<math>. Note that if <math>F \to x<math> and <math>F \subseteq G<math> then <math>G \to x<math>.

Given a filter F on a set X and a function <math>f : X \to Y<math>, the set <math> \{ f(A) : A \in F \} <math> forms a filter base for a filter which, in a slight abuse of notation, we denote by <math>f(F)<math>.

The following useful results hold:

1. X is Hausdorff iff every filter on X has at most one limit.
2. f is continuous at x iff <math> F \to x <math> implies <math>f(F) \to f(x)<math>
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.

Filters in uniform spaces

Given a uniform space X, a filter F on X is said to be Cauchy if for every U in the entourage, there is an <math>A \in F<math> with for every <math>x, y \in A \ \ (x, y) \in U<math>. In a metric space this takes the form F is Cauchy if for every <math> \epsilon > 0 \ \ \exists A \in F \ \ \mathrm{diam}(A) < \epsilon <math>. X is said to be complete if every Cauchy filter converges.

Let <math>F \subseteq G , \ \ G \to x, \ F<math> Cauchy. Then <math>F \to x<math>. Thus every compact uniformity is complete. Further, a uniformity is compact iff it is complete and totally bounded.