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 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 under set inclusion. The Fréchet filter is named after the French mathematician Maurice Fréchet, 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 on is the set of all subsets of such that the complement of in is finite. That is: This makes a filter on the lattice, the power set of with set inclusion, given that denotes the complement of a set, the following two conditions hold:
Properties
If the base set is finite, then = since every subset of, and in particular every complement, is then finite. This case is sometimes excluded by definition or else called the improper filter on. Allowing to be finite creates a single exception to the Fréchet filter's being free and non-principal since a filter on a finite setcannot be free and a non-principal filter cannot contain any singletons as members. If is infinite, then every member of is infinite since it is simply minus finitely many of its members. Additionally, is infinite since one of its subsets is the set of all, where ∈. 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 . The Fréchet filter is not necessarily an ultrafilter. Consider, where is the natural numbers. 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. 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 of natural numbers, the set of infinite intervals = is a Fréchet filter base, i.e., the Fréchet filter on consists of all supersets of elements of.
