{"history":[],"article":{"id":"31c48492-c27f-4485-bdda-7d1b766241eb","title":"Filter (set theory)","slug":"filter-set-theory","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), a **filter** on a set X is a [family](https://bliptext.com/articles/family-of-sets) \\mathcal{B} of [subsets](https://bliptext.com/articles/subset) such that:\nA filter on a set may be thought of as representing a \"collection of large subsets\", one intuitive example being the [neighborhood filter](https://bliptext.com/articles/neighborhood-system).\nFilters appear in [order theory](https://bliptext.com/articles/order-theory), [model theory](https://bliptext.com/articles/model-theory), and [set theory](https://bliptext.com/articles/set-theory), but can also be found in [topology](https://bliptext.com/articles/topology), from which they originate.\nThe dual notion of a filter is an [ideal](https://bliptext.com/articles/ideal-set-theory).\nFilters were introduced by [Henri Cartan](https://bliptext.com/articles/henri-cartan) in [1937](https://bliptext.com/articles/1937) and as described in the article dedicated to [filters in topology](https://bliptext.com/articles/filters-in-topology), they were subsequently used by [Nicolas Bourbaki](https://bliptext.com/articles/nicolas-bourbaki) in their book [Topologie Générale](https://bliptext.com/articles/topologie-g-n-rale) as an alternative to the related notion of a [net](https://bliptext.com/articles/net-topology) developed in [1922](https://bliptext.com/articles/1922) by [E. H. Moore](https://bliptext.com/articles/e-h-moore) and [Herman L. Smith](https://bliptext.com/articles/herman-l-smith).\n[Order filters](https://bliptext.com/articles/filter-mathematics) are generalizations of filters from sets to arbitrary [partially ordered sets](https://bliptext.com/articles/partially-ordered-set).\nSpecifically, a filter on a set is just a proper order filter in the special case where the partially ordered set consists of the [power set](https://bliptext.com/articles/power-set) ordered by [set inclusion](https://bliptext.com/articles/set-inclusion).\n\n# Preliminaries, notation, and basic notions\nIn this article, upper case Roman letters like S and X denote sets (but not families unless indicated otherwise) and \\wp(X) will denote the [power set](https://bliptext.com/articles/power-set) of X. A subset of a [power set](https://bliptext.com/articles/power-set) is called (or simply, ) where it is if it is a subset of \\wp(X).\nFamilies of sets will be denoted by upper case calligraphy letters such as Whenever these assumptions are needed, then it should be assumed that X is non–empty and that etc. are families of sets over X.\nThe terms \"prefilter\" and \"filter base\" are synonyms and will be used interchangeably.\n**Warning about competing definitions and notation**\nThere are unfortunately several terms in the theory of filters that are defined differently by different authors.\nThese include some of the most important terms such as \"filter\".\nWhile different definitions of the same term usually have significant overlap, due to the very technical nature of filters (and point–set topology), these differences in definitions nevertheless often have important consequences.\nWhen reading mathematical literature, it is recommended that readers check how the terminology related to filters is defined by the author.\nFor this reason, this article will clearly state all definitions as they are used.\nUnfortunately, not all notation related to filters is well established and some notation varies greatly across the literature (for example, the notation for the set of all prefilters on a set) so in such cases this article uses whatever notation is most self describing or easily remembered.\nThe 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.\nTheir important properties are described later.\n**Sets operations**\nThe or in X of a [family of sets](https://bliptext.com/articles/family-of-sets) is\nand similarly the of \\mathcal{B} is\nThroughout, f is a map and S is a set.\n**Nets and their tails**\nA is a set I together with a [preorder](https://bliptext.com/articles/preorder), which will be denoted by \\,\\leq\\, (unless explicitly indicated otherwise), that makes (I, \\leq) into an ; this means that for all i, j \\in I, there exists some k \\in I such that For any indices the notation j \\geq i is defined to mean i \\leq j while i < j is defined to mean that i \\leq j holds but it is true that j \\leq i (if \\,\\leq\\, is [antisymmetric](https://bliptext.com/articles/antisymmetric-relation) then this is equivalent to ).\nA is a map from a non–empty directed set into X. The notation will be used to denote a net with domain I.\n**Warning about using strict comparison**\nIf is a net and i \\in I then it is possible for the set which is called, to be empty (for example, this happens if i is an [upper bound](https://bliptext.com/articles/upper-and-lower-bounds) of the [directed set](https://bliptext.com/articles/directed-set) I).\nIn this case, the family would contain the empty set, which would prevent it from being a prefilter (defined later).\nThis is the (important) reason for defining as rather than or even and it is for this reason that in general, when dealing with the prefilter of tails of a net, the strict inequality \\,<\\, may not be used interchangeably with the inequality \\,\\leq.\n\n# Filters and prefilters\nThe following is a list of properties that a family \\mathcal{B} of sets may possess and they form the defining properties of filters, prefilters, and filter subbases.\nWhenever it is necessary, it should be assumed that\nMany of the properties of \\mathcal{B} defined above and below, such as \"proper\" and \"directed downward,\" do not depend on X, so mentioning the set X is optional when using such terms.\nDefinitions involving being \"upward closed in X,\" such as that of \"filter on X,\" do depend on X so the set X should be mentioned if it is not clear from context.\nThere are no prefilters on (nor are there any nets valued in \\varnothing), which is why this article, like most authors, will automatically assume without comment that whenever this assumption is needed.\n\n## Basic examples\n**Named examples**\n

The singleton set is called the or It is the unique filter on X because it is a subset of every filter on X; however, it need not be a subset of every prefilter on X.
The dual ideal \\wp(X) is also called (despite not actually being a filter).\nIt is the only dual ideal on X that is not a filter on X.
If (X, \\tau) is a topological space and x \\in X, then the [neighborhood filter](https://bliptext.com/articles/neighbourhood-system) at x is a filter on X. By definition, a family is called a (resp. a ) at if and only if \\mathcal{B} is a prefilter (resp. \\mathcal{B} is a filter subbase) and the filter on X that \\mathcal{B} generates is equal to the [neighborhood filter](https://bliptext.com/articles/neighbourhood-system) The subfamily of open neighborhoods is a filter base for Both prefilters also form a [bases](https://bliptext.com/articles/base-topology) for topologies on X, with the topology generated \\tau(x) being [coarser](https://bliptext.com/articles/comparison-of-topologies) than \\tau.\nThis example immediately generalizes from neighborhoods of points to neighborhoods of non–empty subsets
\\mathcal{B} is an if for some sequence
\\mathcal{B} is an or a on X if \\mathcal{B} is a filter on X generated by some elementary prefilter.\nThe filter of tails generated by a sequence that is not eventually constant is necessarily an ultrafilter.\nEvery principal filter on a countable set is sequential as is every cofinite filter on a countably infinite set.\nThe intersection of finitely many sequential filters is again sequential.
The set \\mathcal{F} of all [cofinite subsets](https://bliptext.com/articles/cofiniteness) of X (meaning those sets whose complement in X is finite) is proper if and only if \\mathcal{F} is infinite (or equivalently, X is infinite), in which case \\mathcal{F} is a filter on X known as the[ or the ](https://bliptext.com/articles/fr-chet-filter) on X. If X is finite then \\mathcal{F} is equal to the dual ideal \\wp(X), which is not a filter.\nIf X is infinite then the family of complements of singleton sets is a filter subbase that generates the Fréchet filter on X. As with any family of sets over X that contains the kernel of the Fréchet filter on X is the empty set:
The intersection of all elements in any non–empty family is itself a filter on X called the or of which is why it may be denoted by Said differently, Because every filter on X has \\{X\\} as a subset, this intersection is never empty.\nBy definition, the infimum is the finest/largest (relative to ) filter contained as a subset of each member of \\mathbb{F}.\n
Let and let\nThe or of denoted by is the smallest (relative to \\subseteq) dual ideal on X containing every element of \\mathbb{F} as a subset; that is, it is the smallest (relative to \\subseteq) dual ideal on X containing as a subset.\nThis dual ideal is where is the π–system generated by As with any non–empty family of sets, is contained in filter on X if and only if it is a filter subbase, or equivalently, if and only if is a filter on X, in which case this family is the smallest (relative to \\subseteq) filter on X containing every element of \\mathbb{F} as a subset and necessarily\n
Let and let\nThe or of denoted by if it exists, is by definition the smallest (relative to \\subseteq) filter on X containing every element of \\mathbb{F} as a subset.\nIf it exists then necessarily (as defined above) and will also be equal to the intersection of all filters on X containing\nThis supremum of exists if and only if the dual ideal is a filter on X. The least upper bound of a family of filters \\mathbb{F} may fail to be a filter.\nIndeed, if X contains at least 2 distinct elements then there exist filters for which there does exist a filter that contains both\nIf is not a filter subbase then the supremum of does not exist and the same is true of its supremum in but their supremum in the set of all dual ideals on X will exist (it being the degenerate filter \\wp(X)).\n
Let be non−empty sets and for every i \\in I let be a dual ideal on X. If \\mathcal{I} is any dual ideal on I then is a dual ideal on X called or .
The [club filter](https://bliptext.com/articles/club-filter) of a [regular](https://bliptext.com/articles/regular-[cardinal](https://bliptext.com/articles/cardinal-number)) [uncountable](https://bliptext.com/articles/uncountable) [cardinal](https://bliptext.com/articles/cardinal-number) is the filter of all sets containing a [club subset](https://bliptext.com/articles/club-set) of \\kappa.\nIt is a \\kappa-complete filter closed under [diagonal intersection](https://bliptext.com/articles/diagonal-intersection).

\n**Other examples**\n

Let and let which makes \\mathcal{B} a prefilter and a filter subbase that is not closed under finite intersections.\nBecause \\mathcal{B} is a prefilter, the smallest prefilter containing \\mathcal{B} is The π–system generated by \\mathcal{B} is In particular, the smallest prefilter containing the filter subbase \\mathcal{B} is equal to the set of all finite intersections of sets in The filter on X generated by \\mathcal{B} is All three of the π–system \\mathcal{B} generates, and are examples of fixed, principal, ultra prefilters that are principal at the point is also an ultrafilter on X.
Let (X, \\tau) be a topological space, and define where \\mathcal{B} is necessarily finer than If \\mathcal{B} is non–empty (resp. non–degenerate, a filter subbase, a prefilter, closed under finite unions) then the same is true of If \\mathcal{B} is a filter on X then is a prefilter but not necessarily a filter on X although is a filter on X equivalent to
The set \\mathcal{B} of all dense open subsets of a (non–empty) topological space X is a proper π–system and so also a prefilter.\nIf the space is a [Baire space](https://bliptext.com/articles/baire-space), then the set of all countable intersections of dense open subsets is a π–system and a prefilter that is finer than If X = \\R^n (with ) then the set of all such that B has finite [Lebesgue measure](https://bliptext.com/articles/lebesgue-measure) is a proper π–system and free prefilter that is also a [proper subset](https://bliptext.com/articles/proper-subset) of The prefilters and \\mathcal{B} are equivalent and so generate the same filter on X. The prefilter is properly contained in, and not equivalent to, the prefilter consisting of all dense subsets of \\R.\nSince X is a [Baire space](https://bliptext.com/articles/baire-space), every countable intersection of sets in is dense in X (and also [comeagre](https://bliptext.com/articles/meagre-set) and non–meager) so the set of all countable intersections of elements of is a prefilter and π–system; it is also finer than, and not equivalent to,
A filter subbase with no smallest prefilter containing it: In general, if a filter subbase \\mathcal{S} is not a π–system then an intersection of n sets from \\mathcal{S} will usually require a description involving n variables that cannot be reduced down to only two (consider, for instance when ).\nThis example illustrates an atypical class of a filter subbases where all sets in both and its generated π–system can be described as sets of the form B_{r,s}, so that in particular, no more than two variables (specifically, ) are needed to describe the generated π–system.\nFor all let\nwhere always holds so no generality is lost by adding the assumption r \\leq s. For all real if is non-negative then\nFor every set R of positive reals, let\nLet X = \\R and suppose is not a singleton set.\nThen is a filter subbase but not a prefilter and is the π–system it generates, so that is the unique smallest filter in X = \\R containing However, is a filter on X (nor is it a prefilter because it is not directed downward, although it is a filter subbase) and is a proper subset of the filter\nIf are non−empty intervals then the filter subbases generate the same filter on X if and only if R = S.\n****I****f**** ****\\****m****a****t****h****c****a****l****{****C****}**** ****i****s**** ****a**** ****p****r****e****f****i****l****t****e****r**** ****s****a****t****i****s****f****y****i****n****g**** ****t****h****e****n**** ****f****o****r**** ****a****n****y**** ****t****h****e**** ****f****a****m****i****l****y**** **** ****i****s**** ****a****l****s****o**** ****a**** ****p****r****e****f****i****l****t****e****r**** ****s****a****t****i****s****f****y****i****n****g**** **** ****T****h****i****s**** ****s****h****o****w****s**** ****t****h****a****t**** ****t****h****e****r****e**** ****c****a****n****n****o****t**** ****e****x****i****s****t**** ****a**** ****m****i****n****i****m****a****l****/****l****e****a****s****t**** ****(****w****i****t****h**** ****r****e****s****p****e****c****t**** ****t****o**** ****\\****s****u****b****s****e****t****e****q****)**** ****p****r****e****f****i****l****t****e****r**** ****t****h****a****t**** ****b****o****t****h**** ****c****o****n****t****a****i****n****s**** **** ****a****n****d**** ****i****s**** ****a**** ****s****u****b****s****e****t**** ****o****f**** ****t****h****e**** ****π****–****s****y****s****t****e****m**** ****g****e****n****e****r****a****t****e****d**** ****b****y**** **** ****T****h****i****s**** ****r****e****m****a****i****n****s**** ****t****r****u****e**** ****e****v****e****n**** ****i****f**** ****t****h****e**** ****r****e****q****u****i****r****e****m****e****n****t**** ****t****h****a****t**** ****t****h****e**** ****p****r****e****f****i****l****t****e****r**** ****b****e**** ****a**** ****s****u****b****s****e****t**** ****o****f**** **** ****i****s**** ****r****e****m****o****v****e****d****;**** ****t****h****a****t**** ****i****s****,**** ****(****i****n**** ****s****h****a****r****p**** ****c****o****n****t****r****a****s****t**** ****t****o**** ****f****i****l****t****e****r****s****)**** ****t****h****e****r****e**** ****d****o****e****s**** **** ****e****x****i****s****t**** ****a**** ****m****i****n****i****m****a****l****/****l****e****a****s****t**** ****(****w****i****t****h**** ****r****e****s****p****e****c****t**** ****t****o**** ****\\****s****u****b****s****e****t****e****q****)**** ****f****i****l****t****e****r**** ****c****o****n****t****a****i****n****i****n****g**** ****t****h****e**** ****f****i****l****t****e****r**** ****s****u****b****b****a****s****e****\n

\n\n## Ultrafilters\nThere are many other characterizations of \"ultrafilter\" and \"ultra prefilter,\" which are listed in the article on [ultrafilters](https://bliptext.com/articles/ultrafilter-set-theory).\nImportant properties of ultrafilters are also described in that article.\nAny non–degenerate family that has a singleton set as an element is ultra, in which case it will then be an ultra prefilter if and only if it also has the finite intersection property.\nThe trivial filter is ultra if and only if X is a singleton set.\n**The ultrafilter lemma**\nThe following important theorem is due to [Alfred Tarski](https://bliptext.com/articles/alfred-tarski) (1930).\nA consequence of the ultrafilter lemma is that every filter is equal to the intersection of all ultrafilters containing it.\nAssuming the axioms of [Zermelo–Fraenkel (ZF)](https://bliptext.com/articles/zermelo-fraenkel-set-theory), the ultrafilter lemma follows from the [Axiom of choice](https://bliptext.com/articles/axiom-of-choice) (in particular from [Zorn's lemma](https://bliptext.com/articles/zorn-s-lemma)) but is strictly weaker than it.\nThe ultrafilter lemma implies the Axiom of choice for finite sets.\nIf dealing with [Hausdorff](https://bliptext.com/articles/hausdorff-space) spaces, then most basic results (as encountered in introductory courses) in Topology (such as [Tychonoff's theorem](https://bliptext.com/articles/tychonoff-s-theorem) for compact [Hausdorff](https://bliptext.com/articles/hausdorff-space) spaces and the [Alexander subbase theorem](https://bliptext.com/articles/alexander-subbase-theorem)) and in [functional analysis](https://bliptext.com/articles/functional-analysis) (such as the [Hahn–Banach theorem](https://bliptext.com/articles/hahn-banach-theorem)) can be proven using only the ultrafilter lemma; the full strength of the axiom of choice might not be needed.\n\n## Kernels\nThe kernel is useful in classifying properties of prefilters and other families of sets.\nIf then for any point\n**Properties of kernels **\nIf then and this set is also equal to the kernel of the π–system that is generated by In particular, if \\mathcal{B} is a filter subbase then the kernels of all of the following sets are equal:\nIf f is a map then and\nIf then while if \\mathcal{B} and \\mathcal{C} are equivalent then\nEquivalent families have equal kernels.\nTwo principal families are equivalent if and only if their kernels are equal; that is, if \\mathcal{B} and \\mathcal{C} are principal then they are equivalent if and only if\n\n### Classifying families by their kernels\nIf \\mathcal{B} is a principal filter on X then and\nwhere is also the smallest prefilter that generates\nFamily of examples: For any non–empty the family is free but it is a filter subbase if and only if no finite union of the form covers \\R, in which case the filter that it generates will also be free.\nIn particular, is a filter subbase if C is countable (for example, C = \\Q, \\Z, the primes), a [meager set](https://bliptext.com/articles/meager-set) in \\R, a set of finite measure, or a bounded subset of \\R.\nIf C is a singleton set then is a subbase for the Fréchet filter on \\R.\nFor every filter there exists a unique pair of dual ideals such that is free, is principal, and and do not mesh (that is, ).\nThe dual ideal is called of \\mathcal{F} while is called where at least one of these dual ideals is filter.\nIf \\mathcal{F} is principal then otherwise, and is a free (non–degenerate) filter.\n**Finite prefilters and finite sets**\nIf a filter subbase \\mathcal{B} is finite then it is fixed (that is, not free);\nthis is because is a finite intersection and the filter subbase \\mathcal{B} has the finite intersection property.\nA finite prefilter is necessarily principal, although it does not have to be closed under finite intersections.\nIf X is finite then all of the conclusions above hold for any\nIn particular, on a finite set X, there are no free filter subbases (and so no free prefilters), all prefilters are principal, and all filters on X are principal filters generated by their (non–empty) kernels.\nThe trivial filter \\{X\\} is always a finite filter on X and if X is infinite then it is the only finite filter because a non–trivial finite filter on a set X is possible if and only if X is finite.\nHowever, on any infinite set there are non–trivial filter subbases and prefilters that are finite (although they cannot be filters).\nIf X is a singleton set then the trivial filter \\{X\\} is the only proper subset of \\wp(X) and moreover, this set \\{X\\} is a principal ultra prefilter and any superset (where ) with the finite intersection property will also be a principal ultra prefilter (even if Y is infinite).\n\n### Characterizing fixed ultra prefilters\nIf a family of sets \\mathcal{B} is fixed (that is, ) then \\mathcal{B} is ultra if and only if some element of \\mathcal{B} is a singleton set, in which case \\mathcal{B} will necessarily be a prefilter.\nEvery principal prefilter is fixed, so a principal prefilter \\mathcal{B} is ultra if and only if is a singleton set.\nEvery filter on X that is principal at a single point is an ultrafilter, and if in addition X is finite, then there are no ultrafilters on X other than these.\nThe 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.\n\n## Finer/coarser, subordination, and meshing\nThe preorder \\,\\leq\\, that is defined below is of fundamental importance for the use of prefilters (and filters) in topology.\nFor instance, this preorder is used to define the prefilter equivalent of \"subsequence\", where \"\" can be interpreted as \"\\mathcal{F} is a subsequence of \\mathcal{C}\" (so \"subordinate to\" is the prefilter equivalent of \"subsequence of\").\nIt is also used to define prefilter convergence in a topological space.\nThe definition of \\mathcal{B} meshes with which is closely related to the preorder \\,\\leq, is used in Topology to define [cluster points](https://bliptext.com/articles/cluster-point).\nTwo families of sets and are, indicated by writing if If do not mesh then they are.\nIf then are said to if mesh, or equivalently, if the of which is the family\ndoes not contain the empty set, where the trace is also called the of\nExample: If is a [subsequence](https://bliptext.com/articles/subsequence) of then is subordinate to in symbols: and also\nStated in plain English, the prefilter of tails of a subsequence is always subordinate to that of the original sequence.\nTo see this, let be arbitrary (or equivalently, let i \\in \\N be arbitrary) and it remains to show that this set contains some\nFor the set to contain it is sufficient to have i \\leq i_n.\nSince are strictly increasing integers, there exists n \\in \\N such that i_n \\geq i, and so holds, as desired.\nConsequently,\nThe left hand side will be a subset of the right hand side if (for instance) every point of x_{\\bull} is unique (that is, when is injective) and is the even-indexed subsequence because under these conditions, every tail (for every n \\in \\N) of the subsequence will belong to the right hand side filter but not to the left hand side filter.\nFor another example, if \\mathcal{B} is any family then always holds and furthermore,\nAssume that are families of sets that satisfy Then and and also\nIf in addition to is a filter base and then \\mathcal{C} is a filter subbase and also mesh.\nMore generally, if both and if the intersection of any two elements of \\mathcal{F} is non–empty, then mesh.\nEvery filter subbase is coarser than both the π–system that it generates and the filter that it generates.\nIf are families such that the family \\mathcal{C} is ultra, and then \\mathcal{F} is necessarily ultra.\nIt follows that any family that is equivalent to an ultra family will necessarily ultra.\nIn particular, if \\mathcal{C} is a prefilter then either both \\mathcal{C} and the filter it generates are ultra or neither one is ultra.\nIf a filter subbase is ultra then it is necessarily a prefilter, in which case the filter that it generates will also be ultra.\nA filter subbase \\mathcal{B} that is not a prefilter cannot be ultra; but it is nevertheless still possible for the prefilter and filter generated by \\mathcal{B} to be ultra.\nIf is upward closed in X then\n**Relational properties of subordination**\nThe relation \\,\\leq\\, is [reflexive](https://bliptext.com/articles/reflexive-relation) and [transitive](https://bliptext.com/articles/transitive-relation), which makes it into a [preorder](https://bliptext.com/articles/preorder) on The relation is [antisymmetric](https://bliptext.com/articles/anti[symmetric](https://bliptext.com/articles/symmetric-relation)-relation) but if X has more than one point then it is [symmetric](https://bliptext.com/articles/symmetric-relation).\nFor any\nSo the set X has more than one point if and only if the relation is [symmetric](https://bliptext.com/articles/symmetric-relation).\nIf but while the converse does not hold in general, it does hold if \\mathcal{C} is upward closed (such as if \\mathcal{C} is a filter).\nTwo filters are equivalent if and only if they are equal, which makes the restriction of \\,\\leq\\, to [antisymmetric](https://bliptext.com/articles/antisymmetric-relation).\nBut in general, \\,\\leq\\, is [antisymmetric](https://bliptext.com/articles/antisymmetric-relation) on nor on \\wp(\\wp(X)); that is, does necessarily imply ; not even if both are prefilters.\nFor instance, if \\mathcal{B} is a prefilter but not a filter then\n\n### Equivalent families of sets\nThe preorder \\,\\leq\\, induces its canonical [equivalence relation](https://bliptext.com/articles/equivalence-relation) on where for all \\mathcal{B} is to \\mathcal{C} if any of the following equivalent conditions hold:\n

The upward closures of are equal.

\nTwo upward closed (in X) subsets of \\wp(X) are equivalent if and only if they are equal.\nIf then necessarily and \\mathcal{B} is equivalent to Every [equivalence class](https://bliptext.com/articles/equivalence-class) other than contains a unique representative (that is, element of the [equivalence class](https://bliptext.com/articles/equivalence-class)) that is upward closed in X.\n**Properties preserved between equivalent families**\nLet be arbitrary and let \\mathcal{F} be any family of sets.\nIf are equivalent (which implies that ) then for each of the statements/properties listed below, either it is true of or else it is false of :\n

Not empty
Proper (that is, \\varnothing is not an element)\n
Filter subbase
Prefilter\n
Free
Principal
Ultra
Is equal to the trivial filter \\{X\\}\n
Meshes with \\mathcal{F}
Is finer than \\mathcal{F}
Is coarser than \\mathcal{F}
Is equivalent to \\mathcal{F}

\nMissing from the above list is the word \"filter\" because this property is preserved by equivalence.\nHowever, if are filters on X, then they are equivalent if and only if they are equal; this characterization does extend to prefilters.\n**Equivalence of prefilters and filter subbases**\nIf \\mathcal{B} is a prefilter on X then the following families are always equivalent to each other:\n

\\mathcal{B};
the π–system generated by \\mathcal{B};
the filter on X generated by \\mathcal{B};

\nand moreover, these three families all generate the same filter on X (that is, the upward closures in X of these families are equal).\nIn particular, every prefilter is equivalent to the filter that it generates.\nBy transitivity, two prefilters are equivalent if and only if they generate the same filter.\nEvery prefilter is equivalent to exactly one filter on X, which is the filter that it generates (that is, the prefilter's upward closure).\nSaid differently, every equivalence class of prefilters contains exactly one representative that is a filter.\nIn this way, filters can be considered as just being distinguished elements of these equivalence classes of prefilters.\nA filter subbase that is also a prefilter can be equivalent to the prefilter (or filter) that it generates.\nIn contrast, every prefilter is equivalent to the filter that it generates.\nThis is why prefilters can, by and large, be used interchangeably with the filters that they generate while filter subbases cannot.\nEvery filter is both a [π–system](https://bliptext.com/articles/pi-system) and a [ring of sets](https://bliptext.com/articles/ring-of-sets).\n**Examples of determining equivalence/non–equivalence**\nExamples: Let X = \\R and let E be the set \\Z of integers (or the set \\N).\nDefine the sets\nAll three sets are filter subbases but none are filters on X and only \\mathcal{B} is prefilter (in fact, \\mathcal{B} is even free and closed under finite intersections).\nThe set is fixed while is free (unless E = \\N).\nThey satisfy but no two of these families are equivalent; moreover, no two of the filters generated by these three filter subbases are equivalent/equal.\nThis conclusion can be reached by showing that the π–systems that they generate are not equivalent.\nUnlike with every set in the π–system generated by contains \\Z as a subset, which is what prevents their generated π–systems (and hence their generated filters) from being equivalent.\nIf E was instead then all three families would be free and although the sets would remain equivalent to each other, their generated π–systems would be equivalent and consequently, they would generate the same filter on X; however, this common filter would still be strictly coarser than the filter generated by\n\n# Set theoretic properties and constructions\n\n## Trace and meshing\nIf \\mathcal{B} is a prefilter (resp. filter) on then the trace of which is the family is a prefilter (resp. a filter) if and only if mesh (that is, ), in which case the trace of is said to be.\nIf \\mathcal{B} is ultra and if mesh then the trace is ultra.\nIf \\mathcal{B} is an ultrafilter on X then the trace of is a filter on S if and only if\nFor example, suppose that \\mathcal{B} is a filter on is such that Then mesh and generates a filter on X that is strictly finer than\n**When prefilters mesh**\nGiven non–empty families the family\nsatisfies and\nIf is proper (resp. a prefilter, a filter subbase) then this is also true of both\nIn order to make any meaningful deductions about from needs to be proper (that is, which is the motivation for the definition of \"mesh\". In this case, is a prefilter (resp.\nfilter subbase) if and only if this is true of both Said differently, if are prefilters then they mesh if and only if is a prefilter. Generalizing gives a well known characterization of \"mesh\" entirely in terms of subordination (that is, \\,\\leq\\,):\nTwo prefilters (resp. filter subbases) mesh if and only if there exists a prefilter (resp. filter subbase) \\mathcal{F} such that and\nIf the least upper bound of two filters exists in then this least upper bound is equal to\n\n## Images and preimages under functions\nThroughout, will be maps between non–empty sets.\n**Images of prefilters**\nLet Many of the properties that \\mathcal{B} may have are preserved under images of maps; notable exceptions include being upward closed, being closed under finite intersections, and being a filter, which are not necessarily preserved.\nExplicitly, if one of the following properties is true of then it will necessarily also be true of (although possibly not on the codomain Z unless g is surjective):\n

Filter properties: ultra, ultrafilter, filter, prefilter, filter subbase, dual ideal, upward closed, proper/non–degenerate.
Ideal properties: ideal, closed under finite unions, downward closed, directed upward.

\nMoreover, if is a prefilter then so are both\nThe image under a map f : X \\to Y of an ultra set is again ultra and if \\mathcal{B} is an ultra prefilter then so is\nIf \\mathcal{B} is a filter then is a filter on the range g(Y), but it is a filter on the codomain Z if and only if g is surjective.\nOtherwise it is just a prefilter on Z and its upward closure must be taken in Z to obtain a filter.\nThe upward closure of is\nwhere if \\mathcal{B} is upward closed in Y (that is, a filter) then this simplifies to:\nIf then taking g to be the inclusion map X \\to Y shows that any prefilter (resp. ultra prefilter, filter subbase) on X is also a prefilter (resp. ultra prefilter, filter subbase) on Y.\n**Preimages of prefilters**\nLet\nUnder the assumption that f : X \\to Y is [surjective](https://bliptext.com/articles/surjective-function):\nis a prefilter (resp. filter subbase, π–system, closed under finite unions, proper) if and only if this is true of\nHowever, if \\mathcal{B} is an ultrafilter on Y then even if f is surjective (which would make a prefilter), it is nevertheless still possible for the prefilter to be neither ultra nor a filter on X (see this footnote for an example).\nIf f : X \\to Y is not surjective then denote the trace of by where in this case particular case the trace satisfies:\nand consequently also:\nThis last equality and the fact that the trace is a family of sets over f(X) means that to draw conclusions about the trace can be used in place of \\mathcal{B} and the can be used in place of For example:\nis a prefilter (resp. filter subbase, π–system, proper) if and only if this is true of\nIn this way, the case where f is not (necessarily) surjective can be reduced down to the case of a surjective function (which is a case that was described at the start of this subsection).\nEven if \\mathcal{B} is an ultrafilter on Y, if f is not surjective then it is nevertheless possible that which would make degenerate as well.\nThe next characterization shows that degeneracy is the only obstacle.\nIf \\mathcal{B} is a prefilter then the following are equivalent:\n

is a prefilter;
is a prefilter;
;
\\mathcal{B} meshes with f(X)

\nand moreover, if is a prefilter then so is\nIf and if denotes the inclusion map then the trace of is equal to This observation allows the results in this subsection to be applied to investigating the trace on a set.\n**Bijections, injections, and surjections**\nAll properties involving filters are preserved under bijections.\nThis means that if is a bijection, then \\mathcal{B} is a prefilter (resp. ultra, ultra prefilter, filter on X, ultrafilter on X, filter subbase, π–system, ideal on X, etc.) if and only if the same is true of\nA map g : Y \\to Z is injective if and only if for all prefilters is equivalent to The image of an ultra family of sets under an injection is again ultra.\nThe map f : X \\to Y is a [surjection](https://bliptext.com/articles/surjective-map) if and only if whenever \\mathcal{B} is a prefilter on Y then the same is true of (this result does not require the ultrafilter lemma).\n\n### Subordination is preserved by images and preimages\nThe relation \\,\\leq\\, is preserved under both images and preimages of families of sets.\nThis means that for families\nMoreover, the following relations always hold for family of sets \\mathcal{C}:\nwhere equality will hold if f is surjective.\nFurthermore,\nIf then\nand where equality will hold if g is injective.\n\n## Products of prefilters\nSuppose is a family of one or more non–empty sets, whose product will be denoted by and for every index i \\in I, let\ndenote the canonical projection.\nLet be non−empty families, also indexed by I, such that for each i \\in I. The of the families is defined identically to how the basic open subsets of the [product topology](https://bliptext.com/articles/product-topology) are defined (had all of these been topologies).\nThat is, both the notations\ndenote the family of all [cylinder subsets](https://bliptext.com/articles/cylinder-set) such that S_i = X_i for all but finitely many i \\in I and where for any one of these finitely many exceptions (that is, for any i such that necessarily ).\nWhen every is a filter subbase then the family is a filter subbase for the filter on generated by If is a filter subbase then the filter on that it generates is called the.\nIf every is a prefilter on X_i then will be a prefilter on and moreover, this prefilter is equal to the coarsest prefilter such that\nfor every i \\in I. However, may fail to be a filter on even if every is a filter on X_i.\n\n## Set subtraction and some examples\n**Set subtracting away a subset of the kernel**\nIf \\mathcal{B} is a prefilter on then is a prefilter, where this latter set is a filter if and only if \\mathcal{B} is a filter and In particular, if \\mathcal{B} is a neighborhood basis at a point x in a topological space X having at least 2 points, then is a prefilter on X. This construction is used to define in terms of prefilter convergence.\n**Using duality between ideals and dual ideals**\nThere is a dual relation or which is defined to mean that every some Explicitly, this means that for every , there is some such that This relation is dual to \\,\\leq\\, in sense that if and only if The relation is closely related to the downward closure of a family in a manner similar to how \\,\\leq\\, is related to the upward closure family.\nFor an example that uses this duality, suppose f : X \\to Y is a map and Define\nwhich contains the empty set if and only if \\Xi does.\nIt is possible for \\Xi to be an ultrafilter and for \\Xi_f to be empty or not closed under finite intersections (see footnote for example).\nAlthough \\Xi_f does not preserve properties of filters very well, if \\Xi is downward closed (resp. closed under finite unions, an ideal) then this will also be true for \\Xi_f.\nUsing the duality between ideals and dual ideals allows for a construction of the following filter.\nSuppose \\mathcal{B} is a filter on Y and let be its dual in Y. If then \\Xi_f's dual will be a filter.\n**Other examples**\nExample: The set \\mathcal{B} of all dense open subsets of a topological space is a proper π–system and a prefilter.\nIf the space is a [Baire space](https://bliptext.com/articles/baire-space), then the set of all countable intersections of dense open subsets is a π–system and a prefilter that is finer than\nExample: The family of all dense open sets of X = \\R having finite Lebesgue measure is a proper π–system and a free prefilter.\nThe prefilter is properly contained in, and not equivalent to, the prefilter consisting of all dense open subsets of \\R.\nSince X is a [Baire space](https://bliptext.com/articles/baire-space), every countable intersection of sets in is dense in X (and also [comeagre](https://bliptext.com/articles/meagre-set) and non–meager) so the set of all countable intersections of elements of is a prefilter and π–system; it is also finer than, and not equivalent to,\n\n# Filters and nets\nThis section will describe the relationships between prefilters and nets in great detail because of how important these details are applying [filters to topology](https://bliptext.com/articles/filters-in-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\".\n\n## Nets to prefilters\nA [net](https://bliptext.com/articles/net-mathematics) is canonically associated with its prefilter of tails\nIf f : X \\to Y is a map and x_{\\bull} is a net in X then\n\n## Prefilters to nets\nA is a pair (S, s) consisting of a non–empty set S and an element s \\in S. For any family let\nDefine a canonical [preorder](https://bliptext.com/articles/preorder) \\,\\leq\\, on pointed sets by declaring\nIf even if so this preorder is not [antisymmetric](https://bliptext.com/articles/antisymmetric-relation) and given any family of sets is [partially ordered](https://bliptext.com/articles/partially-ordered-set) if and only if consists entirely of singleton sets.\nIf is a [maximal element](https://bliptext.com/articles/maximal-and-minimal-elements) of ; moreover, all [maximal element](https://bliptext.com/articles/maximal-and-minimal-elements)s are of this form.\nIf is a [greatest element](https://bliptext.com/articles/greatest-element-and-least-element) if and only if in which case is the set of all [greatest element](https://bliptext.com/articles/greatest-element-and-least-element)s.\nHowever, a greatest element (B, b) is a maximal element if and only if so there is at most one element that is both maximal and greatest.\nThere is a canonical map defined by\nAlthough is not, in general, a partially ordered set, it is a [directed set](https://bliptext.com/articles/directed-set) if (and only if) \\mathcal{B} is a prefilter.\nSo the most immediate choice for the definition of \"the net in X induced by a prefilter \\mathcal{B}\" is the assignment from into X.\nIf \\mathcal{B} is a prefilter on is a net in X and the prefilter associated with is \\mathcal{B}; that is:\nThis would not necessarily be true had been defined on a proper subset of For example, suppose X has at least two distinct elements, is the indiscrete filter, and x \\in X is arbitrary.\nHad instead been defined on the singleton set where the restriction of to D will temporarily be denote by then the prefilter of tails associated with would be the principal prefilter rather than the original filter ;\nthis means that the equality is, so unlike the prefilter \\mathcal{B} can be recovered from\nWorse still, while \\mathcal{B} is the unique filter on X, the prefilter instead generates a filter (that is, an ultrafilter) on X.\nHowever, if is a net in X then it is in general true that is equal to x_{\\bull} because, for example, the domain of x_{\\bull} may be of a completely different cardinality than that of (since unlike the domain of the domain of an arbitrary net in X could have cardinality).\n**Ultranets and ultra prefilters**\nA net is called an or in X if for every subset is [eventually](https://bliptext.com/articles/) in S or it is [eventually](https://bliptext.com/articles/) in ;\nthis happens if and only if is an ultra prefilter.\nA prefilter is an ultra prefilter if and only if is an ultranet in X.\n\n### Partially ordered net\nThe domain of the canonical net is in general not partially ordered.\nHowever, in 1955 Bruns and Schmidt discovered 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](https://bliptext.com/articles/albert-wilansky) in 1970.\nIt begins with the construction of a [strict partial order](https://bliptext.com/articles/strict-partial-order) (meaning a transitive and [irreflexive relation](https://bliptext.com/articles/irreflexive-relation)) \\,<\\, on a subset of that is similar to the [lexicographical order](https://bliptext.com/articles/lexicographic-order) on of the [strict partial order](https://bliptext.com/articles/strict-partial-order)s\nFor any in declare that i < j if and only if\nor equivalently, if and only if\nThe [non−strict partial order](https://bliptext.com/articles/partial-order) associated with \\,<, denoted by \\,\\leq, is defined by declaring that\nUnwinding these definitions gives the following characterization:\n\"i \\leq j if and only if and also\"\nwhich shows that \\,\\leq\\, is just the [lexicographical order](https://bliptext.com/articles/lexicographic-order) on induced by where X is partially ordered by equality \\,=.\\,\nBoth are [serial](https://bliptext.com/articles/serial-relation) and neither possesses a [greatest element](https://bliptext.com/articles/greatest-element-and-least-element) or a [maximal element](https://bliptext.com/articles/maximal-and-minimal-elements); this remains true if they are each restricted to the subset of defined by\nwhere it will henceforth be assumed that they are.\nDenote the assignment from this subset by:\nIf then just as with before, the tail of the starting at i_0 is equal to B_0.\nIf \\mathcal{B} is a prefilter on X then is a net in X whose domain is a partially ordered set and moreover,\nBecause the tails of are identical (since both are equal to the prefilter \\mathcal{B}), there is typically nothing lost by assuming that the domain of the net associated with a prefilter is both directed partially ordered.\nIf the set \\N is replaced with the positive rational numbers then the strict partial order < will also be a [dense order](https://bliptext.com/articles/dense-order).\n\n## Subordinate filters and subnets\nThe notion of \"\\mathcal{B} is subordinate to \\mathcal{C}\" (written ) is for filters and prefilters what \" is a [subsequence](https://bliptext.com/articles/subsequence) of \" is for sequences.\nFor example, if denotes the set of tails of x_{\\bull} and if denotes the set of tails of the subsequence (where ) then (that is, ) is true but is in general false.\n\n### Non–equivalence of subnets and subordinate filters\nA subset of a [preordered](https://bliptext.com/articles/preorder) space (I, \\leq) is or in I if for every i \\in I there exists some If contains a tail of I then R is said to be or ; explicitly, this means that there exists some (that is, ).\nAn eventual set is necessarily not empty.\nA subset is eventual if and only if its complement is not frequent (which is termed ).\nA map h : A \\to I between two preordered sets is if whenever\n[Subnets in the sense of Willard](https://bliptext.com/articles/) and [subnets in the sense of Kelley](https://bliptext.com/articles/) are the most commonly used definitions of \"[subnet](https://bliptext.com/articles/subnet-mathematics).\"\nThe first definition of a subnet was introduced by [John L. Kelley](https://bliptext.com/articles/john-l-kelley) in 1955.\nStephen Willard introduced his own variant of Kelley's definition of subnet in 1970.\nAA–subnets were introduced independently by Smiley (1957), Aarnes and Andenaes (1972), and Murdeshwar (1983); AA–subnets were studied in great detail by Aarnes and Andenaes but they are not often used.\nKelley did not require the map h 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 X − the nets' common codomain.\nEvery Willard–subnet is a Kelley–subnet and both are AA–subnets.\nIn particular, if is a Willard–subnet or a Kelley–subnet of then\n

Example: Let I = \\N and let x_{\\bull} be a constant sequence, say Let s_1 = 0 and A = \\{1\\} so that is a net on A. Then s_{\\bull} is an AA-subnet of x_{\\bull} because But s_{\\bull} is not a Willard-subnet of x_{\\bull} because there does not exist any map h : A \\to I whose image is a cofinal subset of I = \\N.\nNor is s_{\\bull} a Kelley-subnet of x_{\\bull} because if h : A \\to I is any map then is a cofinal subset of I = \\N but is not eventually in A.

\nAA–subnets have a defining characterization that immediately shows that they are fully interchangeable with sub(ordinate)filters.\nExplicitly, what is meant is that the following statement is true for AA–subnets:\nIf are prefilters then is an AA–subnet of\nIf \"AA–subnet\" is replaced by \"Willard–subnet\" or \"Kelley–subnet\" then the above statement becomes.\nIn particular, the problem is that the following statement is in general false:\n**** ****s****t****a****t****e****m****e****n****t****:**** ****I****f**** ****a****r****e**** ****p****r****e****f****i****l****t****e****r****s**** ****s****u****c****h**** ****t****h****a****t**** **** ****i****s**** ****a**** ****K****e****l****l****e****y****–****s****u****b****n****e****t**** ****o****f****\nSince every Willard–subnet is a Kelley–subnet, this statement remains false if the word \"Kelley–subnet\" is replaced with \"Willard–subnet\".\n

: For all n \\in \\N, let Let which is a proper π–system, and let where both families are prefilters on the [natural numbers](https://bliptext.com/articles/natural-number)\nBecause is to \\mathcal{B} as a subsequence is to a sequence.\nSo ideally, should be a subnet of Let be the domain of so I contains a cofinal subset that is order isomorphic to \\N and consequently contains neither a maximal nor greatest element.\nLet is both a maximal and greatest element of A. The directed set A also contains a subset that is order isomorphic to \\N (because it contains I, which contains such a subset) but no such subset can be cofinal in A because of the maximal element M. Consequently, any order–preserving map h : A \\to I must be eventually constant (with value h(M)) where h(M) is then a greatest element of the range h(A).\nBecause of this, there can be no order preserving map h : A \\to I that satisfies the conditions required for to be a Willard–subnet of (because the range of such a map h cannot be cofinal in I).\nSuppose for the sake of contradiction that there exists a map h : A \\to I such that is [eventually](https://bliptext.com/articles/) in A for all i \\in I. Because h(M) \\in I, there exist such that\nFor every i \\in I, because is eventually in A, it is necessary that\nIn particular, if then which by definition is equivalent to which is false.\nConsequently, is not a Kelley–subnet of

\nIf \"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.\nIn particular, the problem is that Kelley–subnets and Willard–subnets are fully interchangeable with subordinate filters.\nIf the notion of \"subnet\" is not used or if \"subnet\" is defined to mean AA–subnet, then this ceases to be a problem and so it becomes correct to say that nets and filters are interchangeable.\nDespite the fact that AA–subnets do not have the problem that Willard and Kelley subnets have, they are not widely used or known about.\n\n# Citations","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-26T09:54:41.598Z","updated_at":"2025-01-26T09:54:41.598Z","search_vector":"'filter':1 'set':2 'theori':3","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), a **filter** on a set X is a [family](https://bliptext.com/articles/family-of-sets) \\mathcal{B} of [subsets](https://bliptext.com/articles/subset) such that:\nA filter on a set may be thought of as representing a \"collection of large subsets\", one intuitive example being the [neighborhood filter](https://bliptext.com/articles/neighborhood-system).\nFilters appear in [order theory](https://bliptext.com/articles/order-theory), [model theory](https://bliptext.com/articles/model-theory), and [set theory](https://bliptext.com/articles/set-theory), but can also be found in [topology](https://bliptext.com/articles/topology), from which they originate.\nThe dual notion of a filter is an [ideal](https://bliptext.com/articles/ideal-set-theory).\nFilters were introduced by [Henri Cartan](https://bliptext.com/articles/henri-cartan) in [1937](https://bliptext.com/articles/1937) and as described in the article dedicated to [filters in topology](https://bliptext.com/articles/filters-in-topology), they were subsequently used by [Nicolas Bourbaki](https://bliptext.com/articles/nicolas-bourbaki) in their book [Topologie Générale](https://bliptext.com/articles/topologie-g-n-rale) as an alternative to the related notion of a [net](https://bliptext.com/articles/net-topology) developed in [1922](https://bliptext.com/articles/1922) by [E. H. Moore](https://bliptext.com/articles/e-h-moore) and [Herman L. Smith](https://bliptext.com/articles/herman-l-smith).\n[Order filters](https://bliptext.com/articles/filter-mathematics) are generalizations of filters from sets to arbitrary [partially ordered sets](https://bliptext.com/articles/partially-ordered-set).\nSpecifically, a filter on a set is just a proper order filter in the special case where the partially ordered set consists of the [power set](https://bliptext.com/articles/power-set) ordered by [set inclusion](https://bliptext.com/articles/set-inclusion).\n\n# Preliminaries, notation, and basic notions\nIn this article, upper case Roman letters like S and X denote sets (but not families unless indicated otherwise) and \\wp(X) will denote the [power set](https://bliptext.com/articles/power-set) of X. A subset of a [power set](https://bliptext.com/articles/power-set) is called (or simply, ) where it is if it is a subset of \\wp(X).\nFamilies of sets will be denoted by upper case calligraphy letters such as Whenever these assumptions are needed, then it should be assumed that X is non–empty and that etc. are families of sets over X.\nThe terms \"prefilter\" and \"filter base\" are synonyms and will be used interchangeably.\n**Warning about competing definitions and notation**\nThere are unfortunately several terms in the theory of filters that are defined differently by different authors.\nThese include some of the most important terms such as \"filter\".\nWhile different definitions of the same term usually have significant overlap, due to the very technical nature of filters (and point–set topology), these differences in definitions nevertheless often have important consequences.\nWhen reading mathematical literature, it is recommended that readers check how the terminology related to filters is defined by the author.\nFor this reason, this article will clearly state all definitions as they are used.\nUnfortunately, not all notation related to filters is well established and some notation varies greatly across the literature (for example, the notation for the set of all prefilters on a set) so in such cases this article uses whatever notation is most self describing or easily remembered.\nThe 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.\nTheir important properties are described later.\n**Sets operations**\nThe or in X of a [family of sets](https://bliptext.com/articles/family-of-sets) is\nand similarly the of \\mathcal{B} is\nThroughout, f is a map and S is a set.\n**Nets and their tails**\nA is a set I together with a [preorder](https://bliptext.com/articles/preorder), which will be denoted by \\,\\leq\\, (unless explicitly indicated otherwise), that makes (I, \\leq) into an ; this means that for all i, j \\in I, there exists some k \\in I such that For any indices the notation j \\geq i is defined to mean i \\leq j while i \n The singleton set is called the or It is the unique filter on X because it is a subset of every filter on X; however, it need not be a subset of every prefilter on X. \n The dual ideal \\wp(X) is also called (despite not actually being a filter).\nIt is the only dual ideal on X that is not a filter on X. \n If (X, \\tau) is a topological space and x \\in X, then the [neighborhood filter](https://bliptext.com/articles/neighbourhood-system) at x is a filter on X. By definition, a family is called a (resp. a ) at if and only if \\mathcal{B} is a prefilter (resp. \\mathcal{B} is a filter subbase) and the filter on X that \\mathcal{B} generates is equal to the [neighborhood filter](https://bliptext.com/articles/neighbourhood-system) The subfamily of open neighborhoods is a filter base for Both prefilters also form a [bases](https://bliptext.com/articles/base-topology) for topologies on X, with the topology generated \\tau(x) being [coarser](https://bliptext.com/articles/comparison-of-topologies) than \\tau.\nThis example immediately generalizes from neighborhoods of points to neighborhoods of non–empty subsets \n \\mathcal{B} is an if for some sequence \n \\mathcal{B} is an or a on X if \\mathcal{B} is a filter on X generated by some elementary prefilter.\nThe filter of tails generated by a sequence that is not eventually constant is necessarily an ultrafilter.\nEvery principal filter on a countable set is sequential as is every cofinite filter on a countably infinite set.\nThe intersection of finitely many sequential filters is again sequential. \n The set \\mathcal{F} of all [cofinite subsets](https://bliptext.com/articles/cofiniteness) of X (meaning those sets whose complement in X is finite) is proper if and only if \\mathcal{F} is infinite (or equivalently, X is infinite), in which case \\mathcal{F} is a filter on X known as the[ or the ](https://bliptext.com/articles/fr-chet-filter) on X. If X is finite then \\mathcal{F} is equal to the dual ideal \\wp(X), which is not a filter.\nIf X is infinite then the family of complements of singleton sets is a filter subbase that generates the Fréchet filter on X. As with any family of sets over X that contains the kernel of the Fréchet filter on X is the empty set: \n The intersection of all elements in any non–empty family is itself a filter on X called the or of which is why it may be denoted by Said differently, Because every filter on X has \\{X\\} as a subset, this intersection is never empty.\nBy definition, the infimum is the finest/largest (relative to ) filter contained as a subset of each member of \\mathbb{F}.\n Let and let\nThe or of denoted by is the smallest (relative to \\subseteq) dual ideal on X containing every element of \\mathbb{F} as a subset; that is, it is the smallest (relative to \\subseteq) dual ideal on X containing as a subset.\nThis dual ideal is where is the π–system generated by As with any non–empty family of sets, is contained in filter on X if and only if it is a filter subbase, or equivalently, if and only if is a filter on X, in which case this family is the smallest (relative to \\subseteq) filter on X containing every element of \\mathbb{F} as a subset and necessarily\n \n Let and let\nThe or of denoted by if it exists, is by definition the smallest (relative to \\subseteq) filter on X containing every element of \\mathbb{F} as a subset.\nIf it exists then necessarily (as defined above) and will also be equal to the intersection of all filters on X containing\nThis supremum of exists if and only if the dual ideal is a filter on X. The least upper bound of a family of filters \\mathbb{F} may fail to be a filter.\nIndeed, if X contains at least 2 distinct elements then there exist filters for which there does exist a filter that contains both\nIf is not a filter subbase then the supremum of does not exist and the same is true of its supremum in but their supremum in the set of all dual ideals on X will exist (it being the degenerate filter \\wp(X)).\n \n Let be non−empty sets and for every i \\in I let be a dual ideal on X. If \\mathcal{I} is any dual ideal on I then is a dual ideal on X called or . \n The [club filter](https://bliptext.com/articles/club-filter) of a [regular](https://bliptext.com/articles/regular-[cardinal](https://bliptext.com/articles/cardinal-number)) [uncountable](https://bliptext.com/articles/uncountable) [cardinal](https://bliptext.com/articles/cardinal-number) is the filter of all sets containing a [club subset](https://bliptext.com/articles/club-set) of \\kappa.\nIt is a \\kappa-complete filter closed under [diagonal intersection](https://bliptext.com/articles/diagonal-intersection). \n \n**Other examples**\n \n Let and let which makes \\mathcal{B} a prefilter and a filter subbase that is not closed under finite intersections.\nBecause \\mathcal{B} is a prefilter, the smallest prefilter containing \\mathcal{B} is The π–system generated by \\mathcal{B} is In particular, the smallest prefilter containing the filter subbase \\mathcal{B} is equal to the set of all finite intersections of sets in The filter on X generated by \\mathcal{B} is All three of the π–system \\mathcal{B} generates, and are examples of fixed, principal, ultra prefilters that are principal at the point is also an ultrafilter on X. \n Let (X, \\tau) be a topological space, and define where \\mathcal{B} is necessarily finer than If \\mathcal{B} is non–empty (resp. non–degenerate, a filter subbase, a prefilter, closed under finite unions) then the same is true of If \\mathcal{B} is a filter on X then is a prefilter but not necessarily a filter on X although is a filter on X equivalent to \n The set \\mathcal{B} of all dense open subsets of a (non–empty) topological space X is a proper π–system and so also a prefilter.\nIf the space is a [Baire space](https://bliptext.com/articles/baire-space), then the set of all countable intersections of dense open subsets is a π–system and a prefilter that is finer than If X = \\R^n (with ) then the set of all such that B has finite [Lebesgue measure](https://bliptext.com/articles/lebesgue-measure) is a proper π–system and free prefilter that is also a [proper subset](https://bliptext.com/articles/proper-subset) of The prefilters and \\mathcal{B} are equivalent and so generate the same filter on X. The prefilter is properly contained in, and not equivalent to, the prefilter consisting of all dense subsets of \\R.\nSince X is a [Baire space](https://bliptext.com/articles/baire-space), every countable intersection of sets in is dense in X (and also [comeagre](https://bliptext.com/articles/meagre-set) and non–meager) so the set of all countable intersections of elements of is a prefilter and π–system; it is also finer than, and not equivalent to, \n A filter subbase with no smallest prefilter containing it: In general, if a filter subbase \\mathcal{S} is not a π–system then an intersection of n sets from \\mathcal{S} will usually require a description involving n variables that cannot be reduced down to only two (consider, for instance when ).\nThis example illustrates an atypical class of a filter subbases where all sets in both and its generated π–system can be described as sets of the form B_{r,s}, so that in particular, no more than two variables (specifically, ) are needed to describe the generated π–system.\nFor all let\nwhere always holds so no generality is lost by adding the assumption r \\leq s. For all real if is non-negative then\nFor every set R of positive reals, let\nLet X = \\R and suppose is not a singleton set.\nThen is a filter subbase but not a prefilter and is the π–system it generates, so that is the unique smallest filter in X = \\R containing However, is a filter on X (nor is it a prefilter because it is not directed downward, although it is a filter subbase) and is a proper subset of the filter\nIf are non−empty intervals then the filter subbases generate the same filter on X if and only if R = S.\n****I****f**** ****\\****m****a****t****h****c****a****l****{****C****}**** ****i****s**** ****a**** ****p****r****e****f****i****l****t****e****r**** ****s****a****t****i****s****f****y****i****n****g**** ****t****h****e****n**** ****f****o****r**** ****a****n****y**** ****t****h****e**** ****f****a****m****i****l****y**** **** ****i****s**** ****a****l****s****o**** ****a**** ****p****r****e****f****i****l****t****e****r**** ****s****a****t****i****s****f****y****i****n****g**** **** ****T****h****i****s**** ****s****h****o****w****s**** ****t****h****a****t**** ****t****h****e****r****e**** ****c****a****n****n****o****t**** ****e****x****i****s****t**** ****a**** ****m****i****n****i****m****a****l****/****l****e****a****s****t**** ****(****w****i****t****h**** ****r****e****s****p****e****c****t**** ****t****o**** ****\\****s****u****b****s****e****t****e****q****)**** ****p****r****e****f****i****l****t****e****r**** ****t****h****a****t**** ****b****o****t****h**** ****c****o****n****t****a****i****n****s**** **** ****a****n****d**** ****i****s**** ****a**** ****s****u****b****s****e****t**** ****o****f**** ****t****h****e**** ****π****–****s****y****s****t****e****m**** ****g****e****n****e****r****a****t****e****d**** ****b****y**** **** ****T****h****i****s**** ****r****e****m****a****i****n****s**** ****t****r****u****e**** ****e****v****e****n**** ****i****f**** ****t****h****e**** ****r****e****q****u****i****r****e****m****e****n****t**** ****t****h****a****t**** ****t****h****e**** ****p****r****e****f****i****l****t****e****r**** ****b****e**** ****a**** ****s****u****b****s****e****t**** ****o****f**** **** ****i****s**** ****r****e****m****o****v****e****d****;**** ****t****h****a****t**** ****i****s****,**** ****(****i****n**** ****s****h****a****r****p**** ****c****o****n****t****r****a****s****t**** ****t****o**** ****f****i****l****t****e****r****s****)**** ****t****h****e****r****e**** ****d****o****e****s**** **** ****e****x****i****s****t**** ****a**** ****m****i****n****i****m****a****l****/****l****e****a****s****t**** ****(****w****i****t****h**** ****r****e****s****p****e****c****t**** ****t****o**** ****\\****s****u****b****s****e****t****e****q****)**** ****f****i****l****t****e****r**** ****c****o****n****t****a****i****n****i****n****g**** ****t****h****e**** ****f****i****l****t****e****r**** ****s****u****b****b****a****s****e****\n \n \n\n## Ultrafilters\nThere are many other characterizations of \"ultrafilter\" and \"ultra prefilter,\" which are listed in the article on [ultrafilters](https://bliptext.com/articles/ultrafilter-set-theory).\nImportant properties of ultrafilters are also described in that article.\nAny non–degenerate family that has a singleton set as an element is ultra, in which case it will then be an ultra prefilter if and only if it also has the finite intersection property.\nThe trivial filter is ultra if and only if X is a singleton set.\n**The ultrafilter lemma**\nThe following important theorem is due to [Alfred Tarski](https://bliptext.com/articles/alfred-tarski) (1930).\nA consequence of the ultrafilter lemma is that every filter is equal to the intersection of all ultrafilters containing it.\nAssuming the axioms of [Zermelo–Fraenkel (ZF)](https://bliptext.com/articles/zermelo-fraenkel-set-theory), the ultrafilter lemma follows from the [Axiom of choice](https://bliptext.com/articles/axiom-of-choice) (in particular from [Zorn's lemma](https://bliptext.com/articles/zorn-s-lemma)) but is strictly weaker than it.\nThe ultrafilter lemma implies the Axiom of choice for finite sets.\nIf dealing with [Hausdorff](https://bliptext.com/articles/hausdorff-space) spaces, then most basic results (as encountered in introductory courses) in Topology (such as [Tychonoff's theorem](https://bliptext.com/articles/tychonoff-s-theorem) for compact [Hausdorff](https://bliptext.com/articles/hausdorff-space) spaces and the [Alexander subbase theorem](https://bliptext.com/articles/alexander-subbase-theorem)) and in [functional analysis](https://bliptext.com/articles/functional-analysis) (such as the [Hahn–Banach theorem](https://bliptext.com/articles/hahn-banach-theorem)) can be proven using only the ultrafilter lemma; the full strength of the axiom of choice might not be needed.\n\n## Kernels\nThe kernel is useful in classifying properties of prefilters and other families of sets.\nIf then for any point\n**Properties of kernels **\nIf then and this set is also equal to the kernel of the π–system that is generated by In particular, if \\mathcal{B} is a filter subbase then the kernels of all of the following sets are equal:\nIf f is a map then and\nIf then while if \\mathcal{B} and \\mathcal{C} are equivalent then\nEquivalent families have equal kernels.\nTwo principal families are equivalent if and only if their kernels are equal; that is, if \\mathcal{B} and \\mathcal{C} are principal then they are equivalent if and only if\n\n### Classifying families by their kernels\nIf \\mathcal{B} is a principal filter on X then and\nwhere is also the smallest prefilter that generates\nFamily of examples: For any non–empty the family is free but it is a filter subbase if and only if no finite union of the form covers \\R, in which case the filter that it generates will also be free.\nIn particular, is a filter subbase if C is countable (for example, C = \\Q, \\Z, the primes), a [meager set](https://bliptext.com/articles/meager-set) in \\R, a set of finite measure, or a bounded subset of \\R.\nIf C is a singleton set then is a subbase for the Fréchet filter on \\R.\nFor every filter there exists a unique pair of dual ideals such that is free, is principal, and and do not mesh (that is, ).\nThe dual ideal is called of \\mathcal{F} while is called where at least one of these dual ideals is filter.\nIf \\mathcal{F} is principal then otherwise, and is a free (non–degenerate) filter.\n**Finite prefilters and finite sets**\nIf a filter subbase \\mathcal{B} is finite then it is fixed (that is, not free);\nthis is because is a finite intersection and the filter subbase \\mathcal{B} has the finite intersection property.\nA finite prefilter is necessarily principal, although it does not have to be closed under finite intersections.\nIf X is finite then all of the conclusions above hold for any\nIn particular, on a finite set X, there are no free filter subbases (and so no free prefilters), all prefilters are principal, and all filters on X are principal filters generated by their (non–empty) kernels.\nThe trivial filter \\{X\\} is always a finite filter on X and if X is infinite then it is the only finite filter because a non–trivial finite filter on a set X is possible if and only if X is finite.\nHowever, on any infinite set there are non–trivial filter subbases and prefilters that are finite (although they cannot be filters).\nIf X is a singleton set then the trivial filter \\{X\\} is the only proper subset of \\wp(X) and moreover, this set \\{X\\} is a principal ultra prefilter and any superset (where ) with the finite intersection property will also be a principal ultra prefilter (even if Y is infinite).\n\n### Characterizing fixed ultra prefilters\nIf a family of sets \\mathcal{B} is fixed (that is, ) then \\mathcal{B} is ultra if and only if some element of \\mathcal{B} is a singleton set, in which case \\mathcal{B} will necessarily be a prefilter.\nEvery principal prefilter is fixed, so a principal prefilter \\mathcal{B} is ultra if and only if is a singleton set.\nEvery filter on X that is principal at a single point is an ultrafilter, and if in addition X is finite, then there are no ultrafilters on X other than these.\nThe 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.\n\n## Finer/coarser, subordination, and meshing\nThe preorder \\,\\leq\\, that is defined below is of fundamental importance for the use of prefilters (and filters) in topology.\nFor instance, this preorder is used to define the prefilter equivalent of \"subsequence\", where \"\" can be interpreted as \"\\mathcal{F} is a subsequence of \\mathcal{C}\" (so \"subordinate to\" is the prefilter equivalent of \"subsequence of\").\nIt is also used to define prefilter convergence in a topological space.\nThe definition of \\mathcal{B} meshes with which is closely related to the preorder \\,\\leq, is used in Topology to define [cluster points](https://bliptext.com/articles/cluster-point).\nTwo families of sets and are, indicated by writing if If do not mesh then they are.\nIf then are said to if mesh, or equivalently, if the of which is the family\ndoes not contain the empty set, where the trace is also called the of\nExample: If is a [subsequence](https://bliptext.com/articles/subsequence) of then is subordinate to in symbols: and also\nStated in plain English, the prefilter of tails of a subsequence is always subordinate to that of the original sequence.\nTo see this, let be arbitrary (or equivalently, let i \\in \\N be arbitrary) and it remains to show that this set contains some\nFor the set to contain it is sufficient to have i \\leq i_n.\nSince are strictly increasing integers, there exists n \\in \\N such that i_n \\geq i, and so holds, as desired.\nConsequently,\nThe left hand side will be a subset of the right hand side if (for instance) every point of x_{\\bull} is unique (that is, when is injective) and is the even-indexed subsequence because under these conditions, every tail (for every n \\in \\N) of the subsequence will belong to the right hand side filter but not to the left hand side filter.\nFor another example, if \\mathcal{B} is any family then always holds and furthermore,\nAssume that are families of sets that satisfy Then and and also\nIf in addition to is a filter base and then \\mathcal{C} is a filter subbase and also mesh.\nMore generally, if both and if the intersection of any two elements of \\mathcal{F} is non–empty, then mesh.\nEvery filter subbase is coarser than both the π–system that it generates and the filter that it generates.\nIf are families such that the family \\mathcal{C} is ultra, and then \\mathcal{F} is necessarily ultra.\nIt follows that any family that is equivalent to an ultra family will necessarily ultra.\nIn particular, if \\mathcal{C} is a prefilter then either both \\mathcal{C} and the filter it generates are ultra or neither one is ultra.\nIf a filter subbase is ultra then it is necessarily a prefilter, in which case the filter that it generates will also be ultra.\nA filter subbase \\mathcal{B} that is not a prefilter cannot be ultra; but it is nevertheless still possible for the prefilter and filter generated by \\mathcal{B} to be ultra.\nIf is upward closed in X then\n**Relational properties of subordination**\nThe relation \\,\\leq\\, is [reflexive](https://bliptext.com/articles/reflexive-relation) and [transitive](https://bliptext.com/articles/transitive-relation), which makes it into a [preorder](https://bliptext.com/articles/preorder) on The relation is [anti[symmetric](https://bliptext.com/articles/symmetric-relation)](https://bliptext.com/articles/anti[symmetric](https://bliptext.com/articles/symmetric-relation)-relation) but if X has more than one point then it is [symmetric](https://bliptext.com/articles/symmetric-relation).\nFor any\nSo the set X has more than one point if and only if the relation is [symmetric](https://bliptext.com/articles/symmetric-relation).\nIf but while the converse does not hold in general, it does hold if \\mathcal{C} is upward closed (such as if \\mathcal{C} is a filter).\nTwo filters are equivalent if and only if they are equal, which makes the restriction of \\,\\leq\\, to [antisymmetric](https://bliptext.com/articles/antisymmetric-relation).\nBut in general, \\,\\leq\\, is [antisymmetric](https://bliptext.com/articles/antisymmetric-relation) on nor on \\wp(\\wp(X)); that is, does necessarily imply ; not even if both are prefilters.\nFor instance, if \\mathcal{B} is a prefilter but not a filter then\n\n### Equivalent families of sets\nThe preorder \\,\\leq\\, induces its canonical [equivalence relation](https://bliptext.com/articles/equivalence-relation) on where for all \\mathcal{B} is to \\mathcal{C} if any of the following equivalent conditions hold:\n \n \n The upward closures of are equal. \n \nTwo upward closed (in X) subsets of \\wp(X) are equivalent if and only if they are equal.\nIf then necessarily and \\mathcal{B} is equivalent to Every [equivalence class](https://bliptext.com/articles/equivalence-class) other than contains a unique representative (that is, element of the [equivalence class](https://bliptext.com/articles/equivalence-class)) that is upward closed in X.\n**Properties preserved between equivalent families**\nLet be arbitrary and let \\mathcal{F} be any family of sets.\nIf are equivalent (which implies that ) then for each of the statements/properties listed below, either it is true of or else it is false of :\n \n Not empty \n Proper (that is, \\varnothing is not an element)\n Filter subbase \n Prefilter\n Free \n Principal \n Ultra \n Is equal to the trivial filter \\{X\\}\n Meshes with \\mathcal{F} \n Is finer than \\mathcal{F} \n Is coarser than \\mathcal{F} \n Is equivalent to \\mathcal{F} \n \nMissing from the above list is the word \"filter\" because this property is preserved by equivalence.\nHowever, if are filters on X, then they are equivalent if and only if they are equal; this characterization does extend to prefilters.\n**Equivalence of prefilters and filter subbases**\nIf \\mathcal{B} is a prefilter on X then the following families are always equivalent to each other:\n \n \\mathcal{B}; \n the π–system generated by \\mathcal{B}; \n the filter on X generated by \\mathcal{B}; \n \nand moreover, these three families all generate the same filter on X (that is, the upward closures in X of these families are equal).\nIn particular, every prefilter is equivalent to the filter that it generates.\nBy transitivity, two prefilters are equivalent if and only if they generate the same filter.\nEvery prefilter is equivalent to exactly one filter on X, which is the filter that it generates (that is, the prefilter's upward closure).\nSaid differently, every equivalence class of prefilters contains exactly one representative that is a filter.\nIn this way, filters can be considered as just being distinguished elements of these equivalence classes of prefilters.\nA filter subbase that is also a prefilter can be equivalent to the prefilter (or filter) that it generates.\nIn contrast, every prefilter is equivalent to the filter that it generates.\nThis is why prefilters can, by and large, be used interchangeably with the filters that they generate while filter subbases cannot.\nEvery filter is both a [π–system](https://bliptext.com/articles/pi-system) and a [ring of sets](https://bliptext.com/articles/ring-of-sets).\n**Examples of determining equivalence/non–equivalence**\nExamples: Let X = \\R and let E be the set \\Z of integers (or the set \\N).\nDefine the sets\nAll three sets are filter subbases but none are filters on X and only \\mathcal{B} is prefilter (in fact, \\mathcal{B} is even free and closed under finite intersections).\nThe set is fixed while is free (unless E = \\N).\nThey satisfy but no two of these families are equivalent; moreover, no two of the filters generated by these three filter subbases are equivalent/equal.\nThis conclusion can be reached by showing that the π–systems that they generate are not equivalent.\nUnlike with every set in the π–system generated by contains \\Z as a subset, which is what prevents their generated π–systems (and hence their generated filters) from being equivalent.\nIf E was instead then all three families would be free and although the sets would remain equivalent to each other, their generated π–systems would be equivalent and consequently, they would generate the same filter on X; however, this common filter would still be strictly coarser than the filter generated by\n\n# Set theoretic properties and constructions\n\n## Trace and meshing\nIf \\mathcal{B} is a prefilter (resp. filter) on then the trace of which is the family is a prefilter (resp. a filter) if and only if mesh (that is, ), in which case the trace of is said to be.\nIf \\mathcal{B} is ultra and if mesh then the trace is ultra.\nIf \\mathcal{B} is an ultrafilter on X then the trace of is a filter on S if and only if\nFor example, suppose that \\mathcal{B} is a filter on is such that Then mesh and generates a filter on X that is strictly finer than\n**When prefilters mesh**\nGiven non–empty families the family\nsatisfies and\nIf is proper (resp. a prefilter, a filter subbase) then this is also true of both\nIn order to make any meaningful deductions about from needs to be proper (that is, which is the motivation for the definition of \"mesh\". In this case, is a prefilter (resp.\nfilter subbase) if and only if this is true of both Said differently, if are prefilters then they mesh if and only if is a prefilter. Generalizing gives a well known characterization of \"mesh\" entirely in terms of subordination (that is, \\,\\leq\\,):\nTwo prefilters (resp. filter subbases) mesh if and only if there exists a prefilter (resp. filter subbase) \\mathcal{F} such that and\nIf the least upper bound of two filters exists in then this least upper bound is equal to\n\n## Images and preimages under functions\nThroughout, will be maps between non–empty sets.\n**Images of prefilters**\nLet Many of the properties that \\mathcal{B} may have are preserved under images of maps; notable exceptions include being upward closed, being closed under finite intersections, and being a filter, which are not necessarily preserved.\nExplicitly, if one of the following properties is true of then it will necessarily also be true of (although possibly not on the codomain Z unless g is surjective):\n \n Filter properties: ultra, ultrafilter, filter, prefilter, filter subbase, dual ideal, upward closed, proper/non–degenerate. \n Ideal properties: ideal, closed under finite unions, downward closed, directed upward. \n \nMoreover, if is a prefilter then so are both\nThe image under a map f : X \\to Y of an ultra set is again ultra and if \\mathcal{B} is an ultra prefilter then so is\nIf \\mathcal{B} is a filter then is a filter on the range g(Y), but it is a filter on the codomain Z if and only if g is surjective.\nOtherwise it is just a prefilter on Z and its upward closure must be taken in Z to obtain a filter.\nThe upward closure of is\nwhere if \\mathcal{B} is upward closed in Y (that is, a filter) then this simplifies to:\nIf then taking g to be the inclusion map X \\to Y shows that any prefilter (resp. ultra prefilter, filter subbase) on X is also a prefilter (resp. ultra prefilter, filter subbase) on Y.\n**Preimages of prefilters**\nLet\nUnder the assumption that f : X \\to Y is [surjective](https://bliptext.com/articles/surjective-function):\nis a prefilter (resp. filter subbase, π–system, closed under finite unions, proper) if and only if this is true of\nHowever, if \\mathcal{B} is an ultrafilter on Y then even if f is surjective (which would make a prefilter), it is nevertheless still possible for the prefilter to be neither ultra nor a filter on X (see this footnote for an example).\nIf f : X \\to Y is not surjective then denote the trace of by where in this case particular case the trace satisfies:\nand consequently also:\nThis last equality and the fact that the trace is a family of sets over f(X) means that to draw conclusions about the trace can be used in place of \\mathcal{B} and the can be used in place of For example:\nis a prefilter (resp. filter subbase, π–system, proper) if and only if this is true of\nIn this way, the case where f is not (necessarily) surjective can be reduced down to the case of a surjective function (which is a case that was described at the start of this subsection).\nEven if \\mathcal{B} is an ultrafilter on Y, if f is not surjective then it is nevertheless possible that which would make degenerate as well.\nThe next characterization shows that degeneracy is the only obstacle.\nIf \\mathcal{B} is a prefilter then the following are equivalent:\n \n is a prefilter; \n is a prefilter; \n ; \n \\mathcal{B} meshes with f(X) \n \nand moreover, if is a prefilter then so is\nIf and if denotes the inclusion map then the trace of is equal to This observation allows the results in this subsection to be applied to investigating the trace on a set.\n**Bijections, injections, and surjections**\nAll properties involving filters are preserved under bijections.\nThis means that if is a bijection, then \\mathcal{B} is a prefilter (resp. ultra, ultra prefilter, filter on X, ultrafilter on X, filter subbase, π–system, ideal on X, etc.) if and only if the same is true of\nA map g : Y \\to Z is injective if and only if for all prefilters is equivalent to The image of an ultra family of sets under an injection is again ultra.\nThe map f : X \\to Y is a [surjection](https://bliptext.com/articles/surjective-map) if and only if whenever \\mathcal{B} is a prefilter on Y then the same is true of (this result does not require the ultrafilter lemma).\n\n### Subordination is preserved by images and preimages\nThe relation \\,\\leq\\, is preserved under both images and preimages of families of sets.\nThis means that for families\nMoreover, the following relations always hold for family of sets \\mathcal{C}:\nwhere equality will hold if f is surjective.\nFurthermore,\nIf then\nand where equality will hold if g is injective.\n\n## Products of prefilters\nSuppose is a family of one or more non–empty sets, whose product will be denoted by and for every index i \\in I, let\ndenote the canonical projection.\nLet be non−empty families, also indexed by I, such that for each i \\in I. The of the families is defined identically to how the basic open subsets of the [product topology](https://bliptext.com/articles/product-topology) are defined (had all of these been topologies).\nThat is, both the notations\ndenote the family of all [cylinder subsets](https://bliptext.com/articles/cylinder-set) such that S_i = X_i for all but finitely many i \\in I and where for any one of these finitely many exceptions (that is, for any i such that necessarily ).\nWhen every is a filter subbase then the family is a filter subbase for the filter on generated by If is a filter subbase then the filter on that it generates is called the.\nIf every is a prefilter on X_i then will be a prefilter on and moreover, this prefilter is equal to the coarsest prefilter such that\nfor every i \\in I. However, may fail to be a filter on even if every is a filter on X_i.\n\n## Set subtraction and some examples\n**Set subtracting away a subset of the kernel**\nIf \\mathcal{B} is a prefilter on then is a prefilter, where this latter set is a filter if and only if \\mathcal{B} is a filter and In particular, if \\mathcal{B} is a neighborhood basis at a point x in a topological space X having at least 2 points, then is a prefilter on X. This construction is used to define in terms of prefilter convergence.\n**Using duality between ideals and dual ideals**\nThere is a dual relation or which is defined to mean that every some Explicitly, this means that for every , there is some such that This relation is dual to \\,\\leq\\, in sense that if and only if The relation is closely related to the downward closure of a family in a manner similar to how \\,\\leq\\, is related to the upward closure family.\nFor an example that uses this duality, suppose f : X \\to Y is a map and Define\nwhich contains the empty set if and only if \\Xi does.\nIt is possible for \\Xi to be an ultrafilter and for \\Xi_f to be empty or not closed under finite intersections (see footnote for example).\nAlthough \\Xi_f does not preserve properties of filters very well, if \\Xi is downward closed (resp. closed under finite unions, an ideal) then this will also be true for \\Xi_f.\nUsing the duality between ideals and dual ideals allows for a construction of the following filter.\nSuppose \\mathcal{B} is a filter on Y and let be its dual in Y. If then \\Xi_f's dual will be a filter.\n**Other examples**\nExample: The set \\mathcal{B} of all dense open subsets of a topological space is a proper π–system and a prefilter.\nIf the space is a [Baire space](https://bliptext.com/articles/baire-space), then the set of all countable intersections of dense open subsets is a π–system and a prefilter that is finer than\nExample: The family of all dense open sets of X = \\R having finite Lebesgue measure is a proper π–system and a free prefilter.\nThe prefilter is properly contained in, and not equivalent to, the prefilter consisting of all dense open subsets of \\R.\nSince X is a [Baire space](https://bliptext.com/articles/baire-space), every countable intersection of sets in is dense in X (and also [comeagre](https://bliptext.com/articles/meagre-set) and non–meager) so the set of all countable intersections of elements of is a prefilter and π–system; it is also finer than, and not equivalent to,\n\n# Filters and nets\nThis section will describe the relationships between prefilters and nets in great detail because of how important these details are applying [filters to topology](https://bliptext.com/articles/filters-in-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\".\n\n## Nets to prefilters\nA [net](https://bliptext.com/articles/net-mathematics) is canonically associated with its prefilter of tails\nIf f : X \\to Y is a map and x_{\\bull} is a net in X then\n\n## Prefilters to nets\nA is a pair (S, s) consisting of a non–empty set S and an element s \\in S. For any family let\nDefine a canonical [preorder](https://bliptext.com/articles/preorder) \\,\\leq\\, on pointed sets by declaring\nIf even if so this preorder is not [antisymmetric](https://bliptext.com/articles/antisymmetric-relation) and given any family of sets is [partially ordered](https://bliptext.com/articles/partially-ordered-set) if and only if consists entirely of singleton sets.\nIf is a [maximal element](https://bliptext.com/articles/maximal-and-minimal-elements) of ; moreover, all [maximal element](https://bliptext.com/articles/maximal-and-minimal-elements)s are of this form.\nIf is a [greatest element](https://bliptext.com/articles/greatest-element-and-least-element) if and only if in which case is the set of all [greatest element](https://bliptext.com/articles/greatest-element-and-least-element)s.\nHowever, a greatest element (B, b) is a maximal element if and only if so there is at most one element that is both maximal and greatest.\nThere is a canonical map defined by\nAlthough is not, in general, a partially ordered set, it is a [directed set](https://bliptext.com/articles/directed-set) if (and only if) \\mathcal{B} is a prefilter.\nSo the most immediate choice for the definition of \"the net in X induced by a prefilter \\mathcal{B}\" is the assignment from into X.\nIf \\mathcal{B} is a prefilter on is a net in X and the prefilter associated with is \\mathcal{B}; that is:\nThis would not necessarily be true had been defined on a proper subset of For example, suppose X has at least two distinct elements, is the indiscrete filter, and x \\in X is arbitrary.\nHad instead been defined on the singleton set where the restriction of to D will temporarily be denote by then the prefilter of tails associated with would be the principal prefilter rather than the original filter ;\nthis means that the equality is, so unlike the prefilter \\mathcal{B} can be recovered from\nWorse still, while \\mathcal{B} is the unique filter on X, the prefilter instead generates a filter (that is, an ultrafilter) on X.\nHowever, if is a net in X then it is in general true that is equal to x_{\\bull} because, for example, the domain of x_{\\bull} may be of a completely different cardinality than that of (since unlike the domain of the domain of an arbitrary net in X could have cardinality).\n**Ultranets and ultra prefilters**\nA net is called an or in X if for every subset is [eventually](https://bliptext.com/articles/) in S or it is [eventually](https://bliptext.com/articles/) in ;\nthis happens if and only if is an ultra prefilter.\nA prefilter is an ultra prefilter if and only if is an ultranet in X.\n\n### Partially ordered net\nThe domain of the canonical net is in general not partially ordered.\nHowever, in 1955 Bruns and Schmidt discovered 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](https://bliptext.com/articles/albert-wilansky) in 1970.\nIt begins with the construction of a [strict partial order](https://bliptext.com/articles/strict-partial-order) (meaning a transitive and [irreflexive relation](https://bliptext.com/articles/irreflexive-relation)) \\, \n Example: Let I = \\N and let x_{\\bull} be a constant sequence, say Let s_1 = 0 and A = \\{1\\} so that is a net on A. Then s_{\\bull} is an AA-subnet of x_{\\bull} because But s_{\\bull} is not a Willard-subnet of x_{\\bull} because there does not exist any map h : A \\to I whose image is a cofinal subset of I = \\N.\nNor is s_{\\bull} a Kelley-subnet of x_{\\bull} because if h : A \\to I is any map then is a cofinal subset of I = \\N but is not eventually in A. \n \nAA–subnets have a defining characterization that immediately shows that they are fully interchangeable with sub(ordinate)filters.\nExplicitly, what is meant is that the following statement is true for AA–subnets:\nIf are prefilters then is an AA–subnet of\nIf \"AA–subnet\" is replaced by \"Willard–subnet\" or \"Kelley–subnet\" then the above statement becomes.\nIn particular, the problem is that the following statement is in general false:\n**** ****s****t****a****t****e****m****e****n****t****:**** ****I****f**** ****a****r****e**** ****p****r****e****f****i****l****t****e****r****s**** ****s****u****c****h**** ****t****h****a****t**** **** ****i****s**** ****a**** ****K****e****l****l****e****y****–****s****u****b****n****e****t**** ****o****f****\nSince every Willard–subnet is a Kelley–subnet, this statement remains false if the word \"Kelley–subnet\" is replaced with \"Willard–subnet\".\n \n : For all n \\in \\N, let Let which is a proper π–system, and let where both families are prefilters on the [natural numbers](https://bliptext.com/articles/natural-number)\nBecause is to \\mathcal{B} as a subsequence is to a sequence.\nSo ideally, should be a subnet of Let be the domain of so I contains a cofinal subset that is order isomorphic to \\N and consequently contains neither a maximal nor greatest element.\nLet is both a maximal and greatest element of A. The directed set A also contains a subset that is order isomorphic to \\N (because it contains I, which contains such a subset) but no such subset can be cofinal in A because of the maximal element M. Consequently, any order–preserving map h : A \\to I must be eventually constant (with value h(M)) where h(M) is then a greatest element of the range h(A).\nBecause of this, there can be no order preserving map h : A \\to I that satisfies the conditions required for to be a Willard–subnet of (because the range of such a map h cannot be cofinal in I).\nSuppose for the sake of contradiction that there exists a map h : A \\to I such that is [eventually](https://bliptext.com/articles/) in A for all i \\in I. Because h(M) \\in I, there exist such that\nFor every i \\in I, because is eventually in A, it is necessary that\nIn particular, if then which by definition is equivalent to which is false.\nConsequently, is not a Kelley–subnet of \n \nIf \"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.\nIn particular, the problem is that Kelley–subnets and Willard–subnets are fully interchangeable with subordinate filters.\nIf the notion of \"subnet\" is not used or if \"subnet\" is defined to mean AA–subnet, then this ceases to be a problem and so it becomes correct to say that nets and filters are interchangeable.\nDespite the fact that AA–subnets do not have the problem that Willard and Kelley subnets have, they are not widely used or known about.\n\n# Citations","revisionCount":0},"hasMore":false,"nextCursor":null}

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