Fréchet filter
In mathematics, the Fréchet filter, also called the cofinite filter, on a set  is a certain collection of subsets of  (that is, it is a particular subset of the power set of ). A subset  of  belongs to the Fréchet filter if and only if the complement of  in  is finite. Any such set  is said to be , which is why it is alternatively called the cofinite filter on . The Fréchet filter 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 (more specifically, it forms a lattice). The Fréchet filter is named after the French mathematician Maurice Fréchet (1878-1973), who worked in topology. Definition

A subset  of a set  is said to be cofinite in  if its complement in  (that is, the set ) is finite. If the empty set is allowed to be in a filter, the Fréchet filter on , denoted by  is the set of all cofinite subsets of . That is:

If  is  a finite set, then every cofinite subset of  is necessarily not empty, so that in this case, it is not necessary to make the empty set assumption made before. This makes  a  on the lattice  the power set  of  with set inclusion, given that  denotes the complement of a set  in  the following two conditions hold:
Intersection condition If two sets are finitely complemented in  then so is their intersection, since  and
Upper-set condition If a set is finitely complemented in  then so are its supersets in . 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 set cannot 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 (infinite) . The Fréchet filter is  necessarily an ultrafilter (or maximal proper filter). Consider the power set  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  (an any other non-degenerate filter) is free if and only if it includes the Fréchet filter. The ultrafilter lemma states that every non-degenerate filter is contained in some ultrafilter. 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

If  is a finite set, assuming that the empty set can be in a filter, then the Fréchet filter on  consists of all the subsets of . On the set  of natural numbers, the set of infinite intervals

is a Fréchet filter base, that is, the Fréchet filter on  consists of all supersets of elements of . See also

References

External links

 
 J.B. Nation, Notes on Lattice Theory, course notes, revised 2017. Order theory
Topology