In mathematics, the Fréchet filter, also called the cofinite filter, on a set is a special subset of the set's power set. A member of this power set is in the Fréchet filter if and only if its complement in the power set is finite. This is of interest in topology, where filters originated, and relates to order and lattice theory because a set's power set is a partially ordered set (and more specifically, a lattice) under set inclusion.
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The Fréchet filter is named after the French mathematician Maurice Fréchet (1878-1973), who worked in topology. It is alternatively called a cofinite filter because its members are exactly the cofinite sets in a power set.
Definition
The Fréchet filter F on X is the set of all subsets A of X such that the complement of A in X is finite. That is,
This makes F a filter on the lattice (P(X), ⊆), the power set of X with set inclusion, since
- Intersection condition: if two sets are finitely complemented in X, then so is their intersection (since (A ∩ B)C = AC ∪ BC), where SC denotes the complement of a set S, and
- Upper-set condition: if a set is finitely complemented in X, then so are its supersets in X.
Properties
If the base set X is finite, then F = P(X) since every subset of X, and in particular every complement, is then finite. This case is sometimes excluded by definition or else called the improper filter on X. Allowing X to be finite creates a single exception to the Fréchet filter's being free and non-principal since a filter on a finite set cannot be free and a non-principal filter cannot contain any singletons as members.
If X is infinite, then every member of F is infinite since it is simply X minus finitely many of its members. Additionally, F is infinite since one of its subsets is the set of all {x}C, where x ∈ X.
The Fréchet filter is both free and non-principal, excepting the finite case mentioned above, and is included in every free filter. It is also the dual filter of the ideal of all finite subsets of (infinite) X.
The Fréchet filter is not necessarily an ultrafilter (or maximal proper filter). Consider P(N). The set of even numbers is the complement of the set of odd numbers. Since neither of these sets is finite, neither set is in the Fréchet filter on N. However, an ultrafilter is free if and only if it includes the Fréchet filter. The existence of free ultrafilters was established by Tarski in 1930, relying on a theorem equivalent to the axiom of choice and is used in the construction of the hyperreals in nonstandard analysis.
Examples
On the set N of natural numbers, the set B = { (n,∞) : n ∈ N} is a Fréchet filter base, i.e., the Fréchet filter on N consists of all supersets of elements of B.