Editing Fréchet filter

In mathematics, the '''Fréchet filter''', also called the '''cofinite filter''', on a [[Set (mathematics)|set]] <math>X</math> is a certain collection of subsets of <math>X</math> (that is, it is a particular subset of the [[power set]] of <math>X</math>).
A subset <math>F</math> of <math>X</math> belongs to the Fréchet filter [[if and only if]] the [[Complement (set theory)|complement]] of <math>F</math> in <math>X</math> is finite.
Any such set <math>F</math> is said to be {{em|[[Cofiniteness|cofinite]] in <math>X</math>}}, which is why it is alternatively called the ''cofinite filter'' on <math>X</math>.

The Fréchet filter is of interest in [[topology]], where filters originated, and relates to [[Order theory|order]] and [[Lattice (order)|lattice theory]] because a set's power set is a [[partially ordered set]]  under [[Subset|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 <math>A</math> of a set <math>X</math> is said to be '''cofinite in <math>X</math>''' if its [[Complement (set theory)|complement]] in <math>X</math> (that is, the set <math>X \setminus A</math>) is [[Finite set|finite]].
If the empty set is allowed to be in a filter, the '''Fréchet filter on <math>X</math>''', denoted by <math>F</math> is the set of all cofinite subsets of <math>X</math>.
That is:<ref>{{cite web|url=https://mathworld.wolfram.com/CofiniteFilter.html|title=Cofinite filter|website=mathworld.wolfram.com}}</ref>
<math display=block>F = \{A \subseteq X : X \setminus A \; \text{ is finite }\}.</math>

If <math>X</math> is {{em|not}} a finite set, then every cofinite subset of <math>X</math> is necessarily not empty, so that in this case, it is not necessary to make the empty set assumption made before.

This makes <math>F</math> a {{em|[[Filter (mathematics)|filter]]}} on the lattice <math>(\wp(X), \subseteq),</math> the [[power set]] <math>\wp(X)</math> of <math>X</math> with set inclusion, given that <math>S^{\operatorname{C}}</math> denotes the complement of a set <math>S</math> in <math>X.</math> The following two conditions hold:
;Intersection condition: If two sets are finitely complemented in <math>X</math>, then so is their intersection, since <math>(A \cap B)^{\operatorname{C}} = A^{\operatorname{C}} \cup B^{\operatorname{C}},</math> and
;Upper-set condition: If a set is finitely complemented in <math>X</math>, then so are its supersets in <math>X</math>.

==Properties==

If the base set <math>X</math> is finite, then <math>F = \wp(X)</math> since every subset of <math>X</math>, and in particular every complement, is then finite.
This case is sometimes excluded by definition or else called the '''improper filter''' on <math>X.</math><ref>{{cite encyclopedia|author=Hodges, Wilfrid|title=Model Theory|encyclopedia=Encyclopedia of Mathematics and its Applications|publisher=Cambridge University Press|year=2008|page=265|ISBN=978-0-521-06636-5}}</ref> Allowing <math>X</math> to be finite creates a single exception to the Fréchet filter’s being [[Filter (mathematics)#Examples|free]] and [[Filter (mathematics)#General definition|non-principal]] since a filter on a finite set cannot be free and a non-principal filter cannot contain any singletons as members.

If <math>X</math> is infinite, then every member of <math>F</math> is infinite since it is simply <math>X</math> minus finitely many of its members.
Additionally, <math>F</math> is infinite since one of its subsets is the set of all <math>\{x\}^{\operatorname{C}},</math> where <math>x \in X.</math>

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 [[Duality (mathematics)|dual]] filter of the [[Ideal (order theory)|ideal]] of all finite subsets of (infinite) <math>X</math>.

The Fréchet filter is {{em|not}} necessarily an [[Ultrafilter (set theory)|ultrafilter]] (or maximal proper filter).
Consider the power set <math>\wp(\N),</math> where <math>\N</math> 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 <math>\N.</math>
However, an {{em|[[Ultrafilter (set theory)|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 [[Hyperreal number|hyperreals]] in [[Non-standard analysis|nonstandard analysis]].<ref>{{cite book|author1=Pinto, J. Sousa|author2=Hoskins, R.F.|title=Infinitesimal Methods for Mathematical Analysis|series=Mathematics and Applications Series|publisher=Horwood Publishing|year=2004|page=53|ISBN=978-1-898563-99-0}}</ref>

==Examples==
{{Expand section|date=January 2012}}

If <math>X</math> is a [[finite set]], assuming that the empty set can be in a filter, then the Fréchet filter on <math>X</math> consists of all the subsets of <math>X</math>.

On the set <math>\N</math> of [[natural number]]s, the set of infinite intervals
<math>B = \{(n, \infty) : n \in \N\}</math>
is a Fréchet [[filter base]], that is, the Fréchet filter on <math>\N</math> consists of all supersets of elements of <math>B</math>.{{citation needed|date=May 2020}}

==See also==

* {{annotated link|Boolean prime ideal theorem}}
* {{annotated link|Filter (mathematics)}}
* {{annotated link|Filter (set theory)}}
* {{annotated link|Filters in topology}}
* {{annotated link|Ultrafilter}}

==References==

{{reflist|group=note}}
{{reflist|25em}}

==External links==

* {{Mathworld|urlname=CofiniteFilter|title=Cofinite Filter}}
* J.B. Nation, [https://math.hawaii.edu/~jb/ ''Notes on Lattice Theory''], course notes, revised 2017.

{{Areas of mathematics|collapsed}}

{{DEFAULTSORT:Frechet Filter}}
[[Category:Order theory]]
[[Category:Topology]]