A subset of a poset is called a filter if it is upward-closed and downward-directed; that is: 1. If in and , then ; 1. for some in , ; 1. if and , then for some , and .
Sometimes the term ‘filter’ is used for an upper set, that is any set satisfying axiom (1). (Ultimately this connects with the use of ‘ideal’ in monoid theory.)
In a lattice, one can use these alternative axioms: 1. If and in , then ; 1. ; 1. if and , then .
Here, (1) is equivalent to the previous version; the others, which here say that the lattice is closed under finite meets, are equivalent given (1). (These axioms look more like the axioms for an ideal of a ring.)
You can also interpret these axioms to say that, if you interpret as a function from to the set of truth values, that is a homomorphism of meet-semilattices.
A filter of subsets of a given set is a filter in the power set of . One also sees filters of open subsets, filters of compact subsets, etc.
A filter is proper if there exists an element of such that . A filter in a lattice is proper iff ; in particular, a filter of subsets of is proper iff . In constructive mathematics, however, one usually wants a stronger (but classically equivalent) notion: a filter of subsets of is proper if every element of is inhabited.
Filters are often assumed to be proper by default in analysis and topology, where proper filters correspond to nets. However, we will try to remember to include the adjective ‘proper’.
If the complement of a filter is an ideal, then we say that the filter is prime (and equivalently that the ideal is prime). A prime filter is necessarily proper; a proper filter in a lattice is prime iff, whenever , either or . In other words, must be a homomorphism of lattices.
In nonstandard analysis and model theory, a particularly important class of filters are the ultrafilters.
A subset of a lattice is a filterbase if it becomes a filter when closed under axiom (1). Equivalently, a filterbase is any downward-directed subset. Any subset of a meet-semilattice may be used as a filter subbase; form a filterbase by closing under finite meets.
A filterbase of sets is proper (that is, it generates a proper filter of sets) iff each set in is inhabited. A filter subbase of sets is proper iff it satisfies the finite intersection property (well known in topology from a criterion for compact spaces): every finite collection from the subfilter has inhabited intersection.
Every net defines an eventuality filter : let belong to if, for some index , for every , . (That is, is eventually in .) Note that is proper; conversely, any proper filter has a net whose eventuality filter is (as described at net). Everything below can be done for nets as well as filters, but filters often lead to a cleaner theory.
A convergence space is a set together with a relation from to , where is the set of filters on , satisfying some axioms. (If , we say that converges to .) The axioms are these: 1. If and , then ; 1. The principal ultrafilter at converges to ; 1. If and , then ; 1. If and , then .
In other words, the set of filters that converge to is a filter of subsets of the set of subfilters of the principal ultrafilter at . (But that is sort of a tongue twister.)
The morphisms of convergence spaces are the continuous functions; a function between convergence spaces is continuous if implies that , where is the filter generated by the filterbase . In this way, convergence spaces form a category .
Any topological space is a convergence space: if every neighbourhood of belongs to . A convergence space is topological if it comes from a topology on . The full subcategory of consisting of the topological convergence spaces is equivalent to the category Top of topological spaces. In this way, the definitions in this section all become theorems about topological spaces. However, not every convergence in analysis is topological; convergence in measure is a good counterexample.
Note that the improper filter (the power set of ) converges to every point. Conversely, a convergence space is Hausdorff if every proper filter converges to at most one point; then we have a partial function from to .
A convergence space is compact if every proper filter is contained in a convergent filter. Equivalently (assuming the Boolean prime ideal theorem), is compact iff every ultrafilter converges.
A Cauchy space is a set together with a collection of filters declared to be Cauchy filters. These must satisfy axioms: 1. If , is Cauchy, and is proper, then is Cauchy; 1. Every principal ultrafilter is Cauchy; 1. If and are Cauchy and is proper, then is Cauchy; 1. Every Cauchy filter is proper.
In other words, the set of Cauchy filters is a filter of subsets of the set of proper filters (another tongue twister).
The morphisms of Cauchy spaces are the Cauchy functions; a function between Cauchy spaces is Cauchy if is Cauchy whenever is. In this way, Cauchy spaces form a category .
Any metric space is a Cauchy space: is Cauchy if it has elements of arbitrarily small diameter. (In particular, a sequence is a Cauchy sequence iff its eventuality filter is Cauchy.) In this way, the category of metric spaces becomes a subcategory (but not a full one) of . (This can be generalised to uniform spaces; the inclusion is still not full.)
Every Cauchy space is a convergence space; if the intersection of with the principal ultrafilter at is Cauchy. Note that any convergent proper filter must be Cauchy. Conversely, if every Cauchy filter is convergent, then the Cauchy space is called complete.
A Cauchy space is totally bounded (or precompact) if every proper filter is contained in a Cauchy filter. Equivalently (assuming the Boolean prime ideal theorem), is totally bounded iff every ultrafilter is Cauchy.
A Cauchy space is compact (as a convergence space) if and only if it is both complete and totally bounded.
Revision on March 20, 2009 at 19:20:27 by Zoran Škoda See the history of this page for a list of all contributions to it.