Filter on a set

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In mathematics, a filter on a set is a family of subsets which is closed under supersets and finite intersections. The concept originates in topology, where the neighborhoods of a point form a filter on the space. Filters were introduced by Henri Cartan in 1937[1][2] and, as described in the article dedicated to filters in topology, they were subsequently used by Nicolas Bourbaki in their book Topologie Générale as an alternative to the related notion of a net developed in 1922 by E. H. Moore and Herman L. Smith. They have also found applications in model theory and set theory.

Filters on a set were later generalized to order filters. Specifically, a filter on a set is a order filter on the power set of ordered by inclusion.

The notion dual to a filter is an ideal. Ultrafilters are a particularly important subclass of filters.

Definition

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Given a set  , a filter   on   is a set of subsets of   such that:

  •   is upwards-closed: If   are such that   and   then  ,
  •   is closed under finite intersections:  ,[a], and if   and   then  .

A proper (or non-degenerate) filter is a filter which is proper as a subset of the powerset   (i.e., the only improper filter is  , consisting of all possible subsets). By upwards-closure, a filter is proper if and only if it does not contain the empty set. Many authors adopt the convention that a filter must be proper by definition.

Given two filters   and   on the same set,   is said to be coarser than   (or a subfilter of  ) and   is said to be finer than   (or subordinate to  ) when  .

Examples

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  • The singleton set   is called the trivial or indiscrete filter on  .[3][4]
  • If   is a subset of  , the subsets of   which are supersets of   form a principal filter.
  • If   is a topological space and  , then the set of neighborhoods of   is a filter on  , the neighborhood filter of  .
  • Many examples arise from various "largeness" conditions:
    • If   is a set, the set of all cofinite subsets of   (i.e., those sets whose complement in   is finite) is a filter on  , the Fréchet filter.[4][3]
    • Similarly, if   is a set, the cocountable subsets of   (those whose complement is countable) form a filter which is finer than the Fréchet filter. More generally, for any cardinal  , the subsets whose complement has cardinal at most   form a filter.
    • If   is a complete measure space (e.g.,   with the Lebesgue measure), the conull subsets of  , i.e., the subsets whose complement has measure zero, form a filter on  . (For a non-complete measure space, one can take the subsets which, while not necessarily measurable, are contained in a measurable subset of measure zero.)
    • Similarly, if   is a measure space, the subsets whose complement is contained in a measurable subset of finite measure form a filter on  .
    • If   is a topological space, the comeager subsets of  , i.e., those whose complement is meager, form a filter on  .
    • The subsets of   which have a natural density of 1 form a filter on  .[5]
  • The club filter of a regular uncountable cardinal   is the filter of all sets containing a club subset of  .
  • If   is a family of filters on   and   is a filter on   then   is a filter on   called Kowalsky's filter.[6]

Principal and free filters

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The kernel of a filter   on   is the intersection of all the subsets of   in  .

A principal filter on   is a filter   which has a particularly simple form: it contains exactly the supersets of  , for some fixed subset  . When  , this yields the improper filter. In the particular case where   is a singleton, the filter is said to be discrete, i.e., a discrete filter on   is the set of subsets of   which contain  , for some fixed element   (the filter is also said to be "principal at  ").[3]

A filter   is principal if and only if the kernel of   is an element of  , and when this is the case,   consists of the supersets of its kernel. On a finite set, every filter is principal (since the intersection defining the kernel is finite).

A filter is said to be free when it has empty kernel, otherwise it is fixed (and if   is an element of the kernel, it is fixed by  ). A filter on a set   is free if and only if it contains the Fréchet filter on  .[7]

For every filter   on  , there exists a unique pair of filters   (the free part of  ) and   (the principal part of  ) on   such that   is free,   is principal, and  , and   and   do not mesh (defined below).[8]

A filter   is countably deep if the kernel of any countable subset of   belongs to  .[8]

Correspondence with order filters

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The concept of a filter on a set is a special case of the more general concept of a filter on a partially ordered set. By definition, a filter on a partially ordered set   is a subset   of   which is upwards-closed (if   and   then  ) and downwards-directed (every finite subset of   has a lower bound in  ). A filter on a set   is the same as a filter on the powerset   ordered by inclusion.[b]

Constructions of filters

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Intersection of filters

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If   is a family of filters on  , its intersection   is a filter on  . It consists of the subsets which can be written as   where   for each  .[9]

The intersection is a greatest lower bound operation in the set of filters on   partially ordered by inclusion. This endows the filters on   with a complete lattice structure.

Filter generated by a family of subsets

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Given a family of subsets  , there exists a minimum filter on   (in the sense of inclusion) which contains  . It can be constructed as the intersection (greatest lower bound) of all filters on   containing  . This filter   is called the filter generated by  , and   is said to be a filter subbase of  .

The generated filter can also be described more explicitly:   is obtained by closing   under finite intersections, then upwards, i.e.,   consists of the subsets   such that   for some  .

Since these operations preserve the kernel, it follows that   is a proper filter if and only if   has the finite intersection property: the intersection of a finite subfamily of   is non-empty.

Two filters   and   on   mesh when   is proper.[citation needed]

Filter bases

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Let   be a filter on  . A filter base of   is a family of subsets   such that   is the upwards closure of  , i.e.,   consists of those subsets   for which   for some  .

This upwards closure is a filter if and only if   is downwards-directed, i.e.,   is non-empty and for all   there exists   such that  . When this is the case,   is also called a prefilter, and the upwards closure is also equal to the generated filter  . Hence, being a filter base of   is a stronger property than being a filter subbase of  .

Examples

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  • When   is a topological space and  , a filter base of the neighborhood filter of   is known as a neighborhood base for  , and similarly, a filter subbase of the neighborhood filter of   is known as a neighborhood subbase for  . The open neighborhoods of   always form a neighborhood base for  , by definition of the neighborhood filter. In  , the closed balls of positive radius around   also form a neighborhood base for  .
  • Let   be an infinite set and let   consist of the subsets of   which contain all points but one. Then   is a filter subbase of the Fréchet filter on  , which consists of the cofinite subsets. Its closure under finite intersections is the entire Fréchet filter, but there are smaller bases of the Fréchet filter which contain the subbase  , such as the one formed by the subsets of   which contain all points but a finite odd number. In fact, for every base of the Fréchet filter, removing any subset yields another base of the Fréchet filter.
  • If   is a topological space, the dense open subsets of   form a filter base on  , because they are closed under finite intersection. The filter they generate consists of the complements of nowhere dense subsets. On  , restricting to the null dense open subsets yields another filter base for the same filter.[citation needed]
  • Similarly, if   is a topological space, the countable intersections of dense open subsets form a filter base which generates the filter of comeager subsets.
  • Let   be a set and let   be a net with values in  , i.e., a family whose domain   is a directed set. The filter base of tails of   consists of the sets   for  ; it is downwards-closed by directedness of  . An elementary filter is a filter which has a filter base of this form. This example is fundamental in the application of filters in topology.[10]
  • Every π-system is a filter base.

Trace of a filter on a subset

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If   is a filter on   and  , the trace of   on   is  , which is a filter.

Image or preimage of a filter by a function

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Let   be a function.

When   is a family of subsets of  , its image by   is defined as

 

If   is a filter on   and   is additionally surjective, then   is a filter on  ; furthermore, if   is a filter base of   then   is a filter base of  . The kernels of   and   are linked by  .

Similarly, when   is a family of subsets of  , its preimage by   is defined as

 

If   is a filter on   and   is additionally injective, then   is a filter on   (since it is isomorphic to the trace of   on the image of  ); furthermore, if   is a filter base of   then   is a filter base of  , and unlike the image case, it also holds that if   is a filter subbase of   then   is a filter subbase of  . The kernels are linked by  .

Product of filters

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Given a family of sets   and a filter   on each  , the product filter   on the product set   is defined as the filter generated by the sets   for   and  , where   is the projection from the product set onto the  -th component. This construction is similar to the product topology.[4]

If each   is a filter base on  , a filter base of   is given by the sets   where   is a family such that   for all   and   for all but finitely many  .[4]

See also

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Notes

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  1. ^ The intersection of zero subsets of   is   itself.
  2. ^ It is immediate that a filter on   is an order filter on  . For the converse, let   be an order filter on  . It is upwards-closed by definition. We check closure under finite intersections. If   is a finite family of subsets from  , it has a lower bound in   by downwards-closure, which is some   such that  . Then  , hence   by upwards-closure.

Citations

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  1. ^ Cartan 1937a.
  2. ^ Cartan 1937b.
  3. ^ a b c Wilansky 2013, pp. 44–46.
  4. ^ a b c d Bourbaki 1987, pp. 57–68.
  5. ^ Jech 2006, pp. 73–89.
  6. ^ Schechter 1996, pp. 100–130.
  7. ^ Dolecki & Mynard 2016, pp. 33–35.
  8. ^ a b Dolecki & Mynard 2016, pp. 27–54.
  9. ^ Császár 1978, pp. 53–65.
  10. ^ Castillo, Jesus M. F.; Montalvo, Francisco (January 1990), "A Counterexample in Semimetric Spaces" (PDF), Extracta Mathematicae, 5 (1): 38–40

References

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