Fréchet filter
The Fréchet filter on an infinite set (X) is the filter consisting of all subsets of (X) whose complements are finite. It is denoted by (\mathcal F_{\mathrm{Fr}}(X)), (\operatorname{Cofin}(X)), or simply (\mathcal F_{\mathrm{Fr}}) when the underlying set is understood:
[ \mathcal F_{\mathrm{Fr}}(X) = {A\subseteq X : X\setminus A \text{ is finite}}. ]
Its members are called cofinite subsets of (X). The assumption that (X) is infinite ensures that the empty set does not belong to the family, so the result is a proper filter. On a finite set the same formula produces the entire power set, including the empty set, and therefore does not define a proper filter under the usual convention.
The filter is named after Maurice Fréchet, whose work on abstract convergence supplied the conceptual setting from which filter convergence developed. It is the canonical example of a free filter and provides the filter-theoretic formulation of the phrase “all but finitely many.”
Definition and elementary structure
A proper filter (\mathcal F) on (X) is a nonempty family of subsets that is closed under finite intersections, closed upward under inclusion, and excludes the empty set. The Fréchet filter satisfies these conditions because the union of finitely many finite sets is finite. In particular, if (A) and (B) are cofinite, then
[ X\setminus(A\cap B)=(X\setminus A)\cup(X\setminus B) ]
is finite. If (A) is cofinite and (A\subseteq B\subseteq X), then (X\setminus B) is a subset of the finite set (X\setminus A), so (B) is also cofinite.
The intersection of all members of (\mathcal F_{\mathrm{Fr}}(X)) is empty. For each (x\in X), the set (X\setminus{x}) belongs to the filter and omits (x). Consequently, the Fréchet filter is a free filter, rather than a principal filter.
Under the definition that a free filter has empty intersection, every free filter on (X) contains (\mathcal F_{\mathrm{Fr}}(X)). Indeed, for every finite subset (E\subseteq X), freeness and closure under finite intersections yield a filter member disjoint from (E); upward closure then places (X\setminus E) in the filter. The Fréchet filter is therefore the least free filter on (X) with respect to inclusion.
It is not an ultrafilter. If (A\subseteq X) has both (A) and (X\setminus A) infinite, then neither set is cofinite, whereas an ultrafilter must contain exactly one member of every complementary pair.
Convergence interpretation
For a sequence ((x_n)_{n\in\mathbb N}) in a topological space, ordinary convergence to (x) can be expressed through the Fréchet filter on (\mathbb N). The sequence converges to (x) precisely when, for every neighborhood (U) of (x),
[ {n\in\mathbb N:x_n\in U}\in\mathcal F_{\mathrm{Fr}}(\mathbb N). ]
This condition states that only finitely many terms lie outside (U), which is equivalent to the usual eventual-neighborhood definition. The filter generated by the tails
[ {n\in\mathbb N:n\geq N},\qquad N\in\mathbb N, ]
is exactly the Fréchet filter on (\mathbb N), since every cofinite subset of (\mathbb N) contains a tail.
This formulation separates the indexing mechanism from the topology of the codomain. Replacing the Fréchet filter by another filter produces generalized notions of filter convergence, while replacing (\mathbb N) by a directed set leads to the related language of nets.
Dual ideal and modulo-finite equivalence
The dual object to the Fréchet filter is the ideal of finite subsets,
[ \operatorname{Fin}(X)={A\subseteq X:A\text{ is finite}}. ]
Filter membership and ideal membership are related by complementation:
[ A\in\mathcal F_{\mathrm{Fr}}(X) \quad\Longleftrightarrow\quad X\setminus A\in\operatorname{Fin}(X). ]
On (X=\mathbb N), the ideal (\operatorname{Fin}) determines the equivalence relation of equality modulo finite error. For subsets (A,B\subseteq\mathbb N),
[ A=^{*}B \quad\Longleftrightarrow\quad A\mathbin{\triangle}B\text{ is finite}, ]
where (\triangle) denotes symmetric difference. The corresponding quotient Boolean algebra is written (\mathcal P(\mathbb N)/\operatorname{Fin}). Its order relation is induced by almost inclusion,
[ A\subseteq^{*}B \quad\Longleftrightarrow\quad A\setminus B\text{ is finite}. ]
These constructions formalize statements whose truth is unchanged by finitely many exceptions.
Relation to ultrafilters
Assuming the ultrafilter lemma, every proper filter extends to an ultrafilter. Extensions of the Fréchet filter are precisely the free, or nonprincipal, ultrafilters on (X). A principal ultrafilter cannot extend it because the cofinite set (X\setminus{x}) excludes the point generating the principal ultrafilter.
A 1938 analysis by You Watanabe established the associated intersection characterization:
[ \mathcal F_{\mathrm{Fr}}(X)
\bigcap{\mathcal U:\mathcal U \text{ is a free ultrafilter on }X}. ]
Every cofinite set belongs to every free ultrafilter. Conversely, if (A) is not cofinite, then (X\setminus A) is infinite and its cofinite filter can be extended to a free ultrafilter containing (X\setminus A); that ultrafilter does not contain (A). The characterization identifies the Fréchet filter as the information common to all free ultrafilters without selecting any particular ultrafilter.
Topological realization
The cofinite topology on (X) declares a subset open when it is empty or has finite complement. Its nonempty open sets are exactly the members of (\mathcal F_{\mathrm{Fr}}(X)). For infinite (X), this topology is (T_1) because every singleton is closed, and it is compact because any chosen member of an open cover leaves only finitely many points to be covered. It is not Hausdorff, since any two nonempty open sets intersect.
The same filter occurs in the one-point compactification of an infinite discrete space. If a point (\infty) is adjoined to a discrete set (X), the neighborhoods of (\infty) have the form
[ {\infty}\cup A, \qquad A\in\mathcal F_{\mathrm{Fr}}(X). ]
Finite subsets are precisely the compact subsets of an infinite discrete space, so neighborhoods of the added point are determined by finite complements.
The general language of filters was systematized by Henri Cartan during the development of modern topology. Within that formalism, the Fréchet filter serves as the direct translation of eventual behavior from sequences into set-theoretic terms.