In mathematics, the Fréchet filter, also called the cofinite filter, on a set
(that is, it is a particular subset of the power set of
belongs to the Fréchet filter if and only if the complement of
, 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.
If the empty set is allowed to be in a filter, the Fréchet filter on
is the set of all cofinite subsets of
is not 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.
a filter on the lattice
denotes the complement of a set
The following two conditions hold: If the base set
This case is sometimes excluded by definition or else called the improper filter on
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.
minus finitely many of its members.
is infinite since one of its subsets is the set of all
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 not necessarily an ultrafilter (or maximal proper filter).
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 (and 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.
is a finite set, assuming that the empty set can be in a filter, then the Fréchet filter on
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