Filters were introduced by Henri Cartan in 1937[3][4] 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.
Families of sets will be denoted by upper case calligraphy letters such as
When reading mathematical literature, it is recommended that readers check how the terminology related to filters is defined by the author.
The theory of filters and prefilters is well developed and has a plethora of definitions and notations, many of which are now unceremoniously listed to prevent this article from becoming prolix and to allow for the easy look up of notation and definitions.
and it is for this reason that in general, when dealing with the prefilter of tails of a net, the strict inequality
The ultrafilter lemma The following important theorem is due to Alfred Tarski (1930).
The ultrafilter lemma implies the Axiom of choice for finite sets.
If only dealing with Hausdorff spaces, then most basic results (as encountered in introductory courses) in Topology (such as Tychonoff's theorem for compact Hausdorff spaces and the Alexander subbase theorem) and in functional analysis (such as the Hahn–Banach theorem) can be proven using only the ultrafilter lemma; the full strength of the axiom of choice might not be needed.
[6] The next theorem shows that every ultrafilter falls into one of two categories: either it is free or else it is a principal filter generated by a single point.
Stated in plain English, the prefilter of tails of a subsequence is always subordinate to that of the original sequence.
contains a unique representative (that is, element of the equivalence class) that is upward closed in
:[32] Missing from the above list is the word "filter" because this property is not preserved by equivalence.
Said differently, every equivalence class of prefilters contains exactly one representative that is a filter.
In this way, filters can be considered as just being distinguished elements of these equivalence classes of prefilters.
filter subbase, π–system, closed under finite unions, proper) if and only if this is true of
[10] This observation allows the results in this subsection to be applied to investigating the trace on a set.
is closely related to the downward closure of a family in a manner similar to how
of all dense open subsets of a topological space is a proper π–system and a prefilter.
is properly contained in, and not equivalent to, the prefilter consisting of all dense open subsets of
This section will describe the relationships between prefilters and nets in great detail because of how important these details are applying filters to topology − particularly in switching from utilizing nets to utilizing filters and vice verse − and because it to make it easier to understand later why subnets (with their most commonly used definitions) are not generally equivalent with "sub–prefilters".
However, in 1955 Bruns and Schmidt discovered[37] a construction that allows for the canonical net to have a domain that is both partially ordered and directed; this was independently rediscovered by Albert Wilansky in 1970.
[36] It begins with the construction of a strict partial order (meaning a transitive and irreflexive relation)
), there is typically nothing lost by assuming that the domain of the net associated with a prefilter is both directed and partially ordered.
is replaced with the positive rational numbers then the strict partial order
A subset is eventual if and only if its complement is not frequent (which is termed infrequent).
[38] Stephen Willard introduced his own variant of Kelley's definition of subnet in 1970.
to be order preserving while the definition of an AA–subnet does away entirely with any map between the two nets' domains and instead focuses entirely on
AA–subnets have a defining characterization that immediately shows that they are fully interchangeable with sub(ordinate)filters.
If "subnet" is defined to mean Willard–subnet or Kelley–subnet then nets and filters are not completely interchangeable because there exists a filter–sub(ordinate)filter relationships that cannot be expressed in terms of a net–subnet relationship between the two induced nets.
Despite the fact that AA–subnets do not have the problem that Willard and Kelley subnets have, they are not widely used or known about.