In mathematics, the Fréchet filter (also called the cofinite filter) on a set $X$ is the collection of all subsets of $X$ whose complement in $X$ is finite. It is named after the French mathematician Maurice Fréchet (1878–1973), who made foundational contributions to general topology and metric spaces.
Definition
Let $X$ be a set. A subset $A \subseteq X$ is said to be cofinite in $X$ if its complement $X \setminus A$ is finite. The Fréchet filter on $X$ is defined as:
$$ \mathcal{F} = { A \subseteq X : X \setminus A \text{ is finite} }. $$
This collection $\mathcal{F}$ is a filter on the power set $\mathcal{P}(X)$ under set inclusion. It satisfies the two defining properties of a filter:
- Intersection condition: If two sets are cofinite, their intersection is also cofinite, since $(A \cap B)^c = A^c \cup B^c$ (a finite union of finite sets is finite).
- Upper-set condition: If a set is cofinite, any superset of it is also cofinite.
Properties
- Finite base set: If $X$ is finite, then every subset of $X$ is cofinite, so $\mathcal{F} = \mathcal{P}(X)$. This is sometimes called the improper filter and is often excluded by definition.
- Infinite base set: If $X$ is infinite, every member of the Fréchet filter is infinite. The filter itself is infinite.
- Free and non-principal: For infinite $X$, the Fréchet filter is both free (its intersection over all members is empty) and non-principal (it is not generated by a single element). It is contained in every free filter on $X$.
- Dual ideal: The Fréchet filter is the dual of the Fréchet ideal, which consists of all finite subsets of $X$.
- Not an ultrafilter: The Fréchet filter is not maximal (not an ultrafilter). For example, on $\mathbb{N}$, the set of even numbers and the set of odd numbers are both infinite and co-infinite; neither belongs to the Fréchet filter. However, an ultrafilter is free if and only if it contains the Fréchet filter. The existence of free ultrafilters (which extend the Fréchet filter) relies on the axiom of choice (via the ultrafilter lemma).
Example
On the set $\mathbb{N}$ of natural numbers, the collection of intervals
$$ B = { (n, \infty) : n \in \mathbb{N} } $$
forms a filter base for the Fréchet filter. That is, the Fréchet filter on $\mathbb{N}$ consists of all supersets of these intervals.
Significance
The Fréchet filter is a fundamental object in topology (where the theory of filters originated), order theory, and lattice theory. It plays a key role in the construction of hyperreal numbers in nonstandard analysis and in the study of convergence in general topological spaces.
