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Intopology,filters can be used to studytopological spaces and define basic topological notions such asconvergence,continuity,compactness, and more.Filters, which are specialfamilies ofsubsets of some given set, also provide a common framework for defining various types oflimits of functions such as limits from the left/right, to infinity, to a point or a set, and many others. Special types of filters calledultrafilters have many useful technical properties and they may often be used in place of arbitrary filters.

Filters have generalizations calledprefilters (also known asfilter bases) andfilter subbases, all of which appear naturally and repeatedly throughout topology. Examples includeneighborhood filters/bases/subbases anduniformities. Every filter is a prefilter and both are filter subbases. Every prefilter and filter subbase is contained in a unique smallest filter, which they are said togenerate. This establishes a relationship between filters and prefilters that may often be exploited to allow one to use whichever of these two notions is more technically convenient. There is a certainpreorder on families of sets (subordination), denoted by${\displaystyle \le ,}$ that helps to determine exactly when and how one notion (filter, prefilter, etc.) can or cannot be used in place of another. This preorder's importance is amplified by the fact that it also defines the notion of filter convergence, where by definition, a filter (or prefilter)${\displaystyle \mathcal{B}}$converges to a point if and only if${\displaystyle \mathcal{N}\le \mathcal{B},}$ where${\displaystyle \mathcal{N}}$ is that point'sneighborhood filter. Consequently, subordination also plays an important role in many concepts that are related to convergence, such ascluster points and limits of functions. In addition, therelation${\displaystyle \mathcal{S}\ge \mathcal{B},}$ which denotes${\displaystyle \mathcal{B}\le \mathcal{S}}$ and is expressed by saying that${\displaystyle \mathcal{S}}$is subordinate to${\displaystyle \mathcal{B},}$ also establishes a relationship in which${\displaystyle \mathcal{S}}$ is to${\displaystyle \mathcal{B}}$ as a subsequence is to a sequence (that is, the relation${\displaystyle \ge ,}$ which is calledsubordination, is for filters the analog of "is a subsequence of").

Filters were introduced byHenri Cartan in 1937^[1] and subsequently used byBourbaki in their bookTopologie Générale as an alternative to the similar notion of anet developed in 1922 byE. H. Moore andH. L. Smith. Filters can also be used to characterize the notions ofsequence andnet convergence. But unlike^{[note 1]} sequence and net convergence, filter convergence is definedentirely in terms of subsets of the topological space${\displaystyle X}$ and so it provides a notion of convergence that is completely intrinsic to the topological space; indeed, thecategory of topological spaces can beequivalently defined entirely in terms of filters. Every net induces a canonical filter and dually, every filter induces a canonical net, where this induced net (resp. induced filter) converges to a point if and only if the same is true of the original filter (resp. net). This characterization also holds for many other definitions such as cluster points. These relationships make it possible to switch between filters and nets, and they often also allow one to choose whichever of these two notions (filter or net) is more convenient for the problem at hand. However, assuming that "subnet" is defined using either of its most popular definitions (which are thosegiven by Willard andby Kelley), then in general, this relationship doesnot extend to subordinate filters and subnets because asdetailed below, there exist subordinate filters whose filter/subordinate-filter relationship cannot be described in terms of the corresponding net/subnet relationship; this issue can however be resolved by using a less commonly encountered definition of "subnet", which is that of anAA-subnet.

Thus filters/prefilters and this single preorder${\displaystyle \le }$ provide a framework that seamlessly ties together fundamental topological concepts such astopological spaces (via neighborhood filters),neighborhood bases,convergence,various limits of functions, continuity,compactness, sequences (viasequential filters), the filter equivalent of "subsequence" (subordination),uniform spaces, and more; concepts that otherwise seem relatively disparate and whose relationships are less clear.

Archetypical example of a filter

Thearchetypical example of a filter is theneighborhood filter${\displaystyle \mathcal{N}(x)}$ at a point${\displaystyle x}$ in a topological space${\displaystyle (X,\tau ),}$ which is thefamily of sets consisting of all neighborhoods of${\displaystyle x.}$ By definition, aneighborhood of some given point${\displaystyle x}$ is any subset${\displaystyle B\subseteq X}$ whosetopological interior contains this point; that is, such that${\displaystyle x\in {\mathrm{Int}}_{X}B.}$ Importantly, neighborhoods arenot required to be open sets; those are calledopen neighborhoods. Listed below are those fundamental properties of neighborhood filters that ultimately became the definition of a "filter." Afilter on${\displaystyle X}$ is a set${\displaystyle \mathcal{B}}$ of subsets of${\displaystyle X}$ that satisfies all of the following conditions:

1. Not empty:  ${\displaystyle X\in \mathcal{B}}$  –  just as${\displaystyle X\in \mathcal{N}(x),}$ since${\displaystyle X}$ is always a neighborhood of${\displaystyle x}$ (and of anything else that it contains);
2. Does not contain the empty set:  ${\displaystyle \varnothing \notin \mathcal{B}}$  –  just as no neighborhood of${\displaystyle x}$ is empty;
3. Closed under finite intersections:   If${\displaystyle B,C\in \mathcal{B}\text{ then }B\cap C\in \mathcal{B}}$  –  just as the intersection of any two neighborhoods of${\displaystyle x}$ is again a neighborhood of${\displaystyle x}$;
4. Upward closed:   If${\displaystyle B\in \mathcal{B}\text{ and }B\subseteq S\subseteq X}$ then${\displaystyle S\in \mathcal{B}}$  –  just as any subset of${\displaystyle X}$ that includes a neighborhood of${\displaystyle x}$ will necessarilybe a neighborhood of${\displaystyle x}$ (this follows from${\displaystyle {\mathrm{Int}}_{X}B\subseteq {\mathrm{Int}}_{X}S}$ and the definition of "a neighborhood of${\displaystyle x}$").

Generalizing sequence convergence by using sets − determining sequence convergence without the sequence

Asequence in${\displaystyle X}$ is by definition amap${\displaystyle \mathbb{N}\to X}$ from thenatural numbers into the space${\displaystyle X.}$ The original notion of convergence in atopological space was that of asequence converging to some given point in a space, such as ametric space. Withmetrizable spaces (or more generallyfirst-countable spaces orFréchet–Urysohn spaces), sequences usually suffices to characterize, or "describe", most topological properties, such as the closures of subsets or continuity of functions. But there are many spaces where sequences cannot be used to describe even basic topological properties like closure or continuity. This failure of sequences was the motivation for defining notions such as nets and filters, whichnever fail to characterize topological properties.

Nets directly generalize the notion of a sequence since nets are, by definition, maps${\displaystyle I\to X}$ from an arbitrarydirected set${\displaystyle (I,\le )}$ into the space${\displaystyle X.}$ A sequence is just a net whose domain is${\displaystyle I=\mathbb{N}}$ with the natural ordering. Nets havetheir own notion of convergence, which is a direct generalization of sequence convergence.

Filters generalize sequence convergence in a different way by consideringonly the values of a sequence. To see how this is done, consider a sequence${\displaystyle x_{\bullet }={\left(x_i\right)}_{i=1}^{\mathrm{\infty }}\text{ in }X,}$ which is by definition just a function${\displaystyle x_{\bullet }:\mathbb{N}\to X}$ whose value at${\displaystyle i\in \mathbb{N}}$ is denoted by${\displaystyle x_i}$ rather than by the usual parentheses notation${\displaystyle x_{\bullet }(i)}$ that is commonly used for arbitrary functions. Knowing only theimage (sometimes called "the range")${\displaystyle \mathrm{Im}{x}_{\bullet }:=\left\{x_i:i\in \mathbb{N}\right\}=\left\{x_1,x_2,\dots \right\}}$ of the sequence is not enough to characterize its convergence; multiple sets are needed. It turns out that the needed sets are the following,^{[note 2]} which are called thetails of the sequence${\displaystyle x_{\bullet }}$:

These sets completely determine this sequence's convergence (or non-convergence) because given any point, thissequence converges to it if and only if for every neighborhood${\displaystyle U}$ (of this point), there is some integer${\displaystyle n}$ such that${\displaystyle U}$ contains all of the points${\displaystyle x_n,x_{n+1},\dots .}$ This can be reworded as:

     every neighborhood${\displaystyle U}$ must contain some set of the form${\displaystyle \{x_n,x_{n+1},\dots \}}$ as a subset.

Or more briefly: every neighborhood must contain some tail${\displaystyle x_{\ge n}}$ as a subset. It is this characterization that can be used with the above family of tails to determine convergence (or non-convergence) of the sequence${\displaystyle x_{\bullet }:\mathbb{N}\to X.}$ Specifically, with the family ofsets${\displaystyle \{x_{\ge 1},x_{\ge 2},\dots \}}$ in hand, thefunction${\displaystyle x_{\bullet }:\mathbb{N}\to X}$ is no longer needed to determine convergence of this sequence (no matter what topology is placed on${\displaystyle X}$). By generalizing this observation, the notion of "convergence"can be extended from sequences/functions to families of sets.

The above set of tails of a sequence is in general not a filter but it does "generate" a filter via taking itsupward closure (which consists of all supersets of all tails). The same is true of other important families of sets such as anyneighborhood basis at a given point, which in general is also not a filter but does generate a filter via its upward closure (in particular, it generates the neighborhood filter at that point). The properties that these families share led to the notion of afilter base, also called aprefilter, which by definition is any family having the minimal properties necessary and sufficient for it to generate a filter via taking itsupward closure.

Nets versus filters − advantages and disadvantages

Filters and nets each have their own advantages and drawbacks and there's no reason to use one notion exclusively over the other.^{[note 3]} Depending on what is being proved, a proof may be made significantly easier by using one of these notions instead of the other.^[2] Both filters and nets can be used to completelycharacterize any given topology. Nets are direct generalizations of sequences and can often be used similarly to sequences, so the learning curve for nets is typically much less steep than that for filters. However, filters, andespecially ultrafilters, have many more uses outside of topology, such as inset theory,mathematical logic,model theory (ultraproducts, for example),abstract algebra,^[3]combinatorics,^[4]dynamics,^[4]order theory,generalized convergence spaces,Cauchy spaces, and in the definition and use ofhyperreal numbers.

Like sequences, nets arefunctions and so they have theadvantages offunctions. For example, like sequences, nets can be "plugged into" other functions, where "plugging in" is justfunction composition. Theorems related to functions and function composition may then be applied to nets. One example is the universal property ofinverse limits, which is defined in terms of composition of functions rather than sets and it is more readily applied to functions like nets than to sets like filters (a prominent example of an inverse limit is theCartesian product). Filters may be awkward to use in certain situations, such as when switching between a filter on a space${\displaystyle X}$ and a filter on adense subspace${\displaystyle S\subseteq X.}$^[5]

In contrast to nets, filters (and prefilters) are families ofsets and so they have theadvantages ofsets. For example, if${\displaystyle f}$ is surjective then theimage${\displaystyle f^{-1}(\mathcal{B}):=\left\{f^{-1}(B):B\in \mathcal{B}\right\}}$ under${\displaystyle f^{-1}}$ of an arbitrary filter or prefilter${\displaystyle \mathcal{B}}$ is both easily defined and guaranteed to be a prefilter on${\displaystyle f}$'s domain, whereas it is less clear how topullback (unambiguously/withoutchoice) an arbitrary sequence (or net)${\displaystyle y_{\bullet }}$ so as to obtain a sequence or net in the domain (unless${\displaystyle f}$ is also injective and consequently a bijection, which is a stringent requirement). Similarly, the intersection of any collection of filters is once again a filter whereas it is not clear what this could mean for sequences or nets. Because filters are composed of subsets of the very topological space${\displaystyle X}$ that is under consideration, topological set operations (such asclosure orinterior) may be applied to the sets that constitute the filter. Taking the closure of all the sets in a filter is sometimes useful infunctional analysis for instance. Theorems and results about images or preimages of sets under a function may also be applied to the sets that constitute a filter; an example of such a result might be one ofcontinuity's characterizations in terms of preimages of open/closed sets or in terms of the interior/closure operators. Special types of filters calledultrafilters have many useful properties that can significantly help in proving results. One downside of nets is their dependence on the directed sets that constitute their domains, which in general may be entirely unrelated to the space${\displaystyle X.}$ In fact, the class of nets in a given set${\displaystyle X}$ is too large to even be a set (it is aproper class); this is because nets in${\displaystyle X}$ can have domains ofanycardinality. In contrast, the collection of all filters (and of all prefilters) on${\displaystyle X}$ is a set whose cardinality is no larger than that of${\displaystyle \mathrm{\wp }(\mathrm{\wp }(X)).}$ Similar to atopology on${\displaystyle X,}$ a filter on${\displaystyle X}$ is "intrinsic to${\displaystyle X}$" in the sense that both structures consistentirely of subsets of${\displaystyle X}$ and neither definition requires any set that cannot be constructed from${\displaystyle X}$ (such as${\displaystyle \mathbb{N}}$ or other directed sets, which sequences and nets require).

In this article, upper case Roman letters like${\displaystyle S}$ and${\displaystyle X}$ denote sets (but not families unless indicated otherwise) and${\displaystyle \mathrm{\wp }(X)}$ will denote thepower set of${\displaystyle X.}$ A subset of a power set is calledafamily of sets (or simply,a family) where it isover${\displaystyle X}$ if it is a subset of${\displaystyle \mathrm{\wp }(X).}$ Families of sets will be denoted by upper case calligraphy letters such as${\displaystyle \mathcal{B}}$,${\displaystyle \mathcal{C}}$, and${\displaystyle \mathcal{F}}$. Whenever these assumptions are needed, then it should be assumed that${\displaystyle X}$ is non-empty and that${\displaystyle \mathcal{B},\mathcal{F},}$ etc. are families of sets over${\displaystyle X.}$

The terms "prefilter" and "filter base" are synonyms and will be used interchangeably.

Warning about competing definitions and notation

There are unfortunately several terms in the theory of filters that are defined differently by different authors. These include some of the most important terms such as "filter." While 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. When reading mathematical literature, it is recommended that readers check how the terminology related to filters is defined by the author. For this reason, this article will clearly state all definitions as they are used. Unfortunately, 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.

The theory of filters and prefilters is well developed and has a plethora of definitions and notations, many of which are now unceremoniously listed to prevent this article from becoming prolix and to allow for the easy look up of notation and definitions. Their important properties are described later.

Sets operations

Theupward closure orisotonization in${\displaystyle X}$^[6][7] of afamily of sets${\displaystyle \mathcal{B}\subseteq \mathrm{\wp }(X)}$ is

${\displaystyle {\mathcal{B}}^{\uparrow X}:=\{S\subseteq X:B\subseteq S\text{ for some }B\in \mathcal{B}\}=\bigcup _{B\in \mathcal{B}}\{S:B\subseteq S\subseteq X\}}$

and similarly thedownward closure of${\displaystyle \mathcal{B}}$ is${\displaystyle {\mathcal{B}}^{\downarrow }:=\{S\subseteq B:B\in \mathcal{B}\}=\bigcup _{B\in \mathcal{B}}\mathrm{\wp }(B).}$

Notation and Definition	Name
${\displaystyle \ker \mathcal{B}=\bigcap _{B\in \mathcal{B}}B}$	Kernel of${\displaystyle \mathcal{B}}$[7]
${\displaystyle S\setminus \mathcal{B}:=\{S\setminus B:B\in \mathcal{B}\}=\{S\}(\setminus )\mathcal{B}}$	Dual of${\displaystyle \mathcal{B}\text{ in }S}$ where${\displaystyle S}$ is a set.[8]
${\displaystyle \mathcal{B}{|}_S:=\{B\cap S:B\in \mathcal{B}\}=\mathcal{B}(\cap )\{S\}}$	Trace of${\displaystyle \mathcal{B}\text{ on }S}$[8] orthe restriction of${\displaystyle \mathcal{B}\text{ to }S}$ where${\displaystyle S}$ is a set; sometimes denoted by${\displaystyle \mathcal{B}\cap S}$
${\displaystyle \mathcal{B}(\cap )\mathcal{C}=\{B\cap C:B\in \mathcal{B}\text{ and }C\in \mathcal{C}\}}$[9]	Elementwise (set)intersection (${\displaystyle \mathcal{B}\cap \mathcal{C}}$ will denote the usual intersection)
${\displaystyle \mathcal{B}(\cup )\mathcal{C}=\{B\cup C:B\in \mathcal{B}\text{ and }C\in \mathcal{C}\}}$[9]	Elementwise (set)union (${\displaystyle \mathcal{B}\cup \mathcal{C}}$ will denote the usual union)
${\displaystyle \mathcal{B}(\setminus )\mathcal{C}=\{B\setminus C:B\in \mathcal{B}\text{ and }C\in \mathcal{C}\}}$	Elementwise (set)subtraction (${\displaystyle \mathcal{B}\setminus \mathcal{C}}$ will denote the usualset subtraction)
${\displaystyle \mathrm{\wp }(X)=\{S:S\subseteq X\}}$	Power set of a set${\displaystyle X}$[7]

Throughout,${\displaystyle f}$ is a map.

Notation and Definition	Name
${\displaystyle f^{-1}(\mathcal{B})=\left\{f^{-1}(B):B\in \mathcal{B}\right\}}$[13]	Image of${\displaystyle \mathcal{B}\text{ under }{f}^{-1},}$ or thepreimage of${\displaystyle \mathcal{B}}$ under${\displaystyle f}$
${\displaystyle f(\mathcal{B})=\{f(B):B\in \mathcal{B}\}}$[14]	Image of${\displaystyle \mathcal{B}}$ under${\displaystyle f}$
${\displaystyle \mathrm{image}f=f(\mathrm{domain}f)}$	Image (or range) of${\displaystyle f}$

Topology notation

Denote the set of all topologies on a set${\displaystyle X\text{ by }\mathrm{Top}(X).}$ Suppose${\displaystyle \tau \in \mathrm{Top}(X),}$${\displaystyle S\subseteq X}$ is any subset, and${\displaystyle x\in X}$ is any point.

Notation and Definition	Name
${\displaystyle \tau (S)=\{O\in \tau :S\subseteq O\}}$	Set orprefilter[note 4]ofopen neighborhoods of${\displaystyle S\text{ in }(X,\tau )}$
${\displaystyle \tau (x)=\{O\in \tau :x\in O\}}$	Set orprefilterof open neighborhoods of${\displaystyle x\text{ in }(X,\tau )}$
${\displaystyle {\mathcal{N}}_{\tau }(S)=\mathcal{N}(S):=\tau (S{)}^{\uparrow X}}$	Set orfilter[note 4]ofneighborhoods of${\displaystyle S\text{ in }(X,\tau )}$
${\displaystyle {\mathcal{N}}_{\tau }(x)=\mathcal{N}(x):=\tau (x{)}^{\uparrow X}}$	Set orfilter of neighborhoods of${\displaystyle x\text{ in }(X,\tau )}$

If${\displaystyle \varnothing \ne S\subseteq X}$ then${\displaystyle \tau (S)=\bigcap _{s\in S}\tau (s)\text{ and }{\mathcal{N}}_{\tau }(S)=\bigcap _{s\in S}{\mathcal{N}}_{\tau }(s).}$

Nets and their tails

Adirected set is a set${\displaystyle I}$ together with apreorder, which will be denoted by${\displaystyle \le }$ (unless explicitly indicated otherwise), that makes${\displaystyle (I,\le )}$ into an (upward)directed set;^[15] this means that for all${\displaystyle i,j\in I,}$ there exists some${\displaystyle k\in I}$ such that${\displaystyle i\le k\text{ and }j\le k.}$ For any indices${\displaystyle i\text{ and }j,}$ the notation${\displaystyle j\ge i}$ is defined to mean${\displaystyle i\le j}$ while is defined to mean that${\displaystyle i\le j}$ holds but it isnot true that${\displaystyle j\le i}$ (if${\displaystyle \le }$ isantisymmetric then this is equivalent to${\displaystyle i\le j\text{ and }i\ne j}$).

Anet in${\displaystyle X}$^[15] is a map from a non-empty directed set into${\displaystyle X.}$ The notation${\displaystyle x_{\bullet }={\left(x_i\right)}_{i\in I}}$ will be used to denote a net with domain${\displaystyle I.}$

Notation and Definition	Name
${\displaystyle I_{\ge i}=\{j\in I:j\ge i\}}$	Tail orsection of${\displaystyle I}$ starting at${\displaystyle i\in I}$ where${\displaystyle (I,\le )}$ is adirected set.
${\displaystyle x_{\ge i}=\left\{x_j:j\ge i\text{ and }j\in I\right\}}$	Tail orsection of${\displaystyle x_{\bullet }={\left(x_i\right)}_{i\in I}}$ starting at${\displaystyle i\in I}$
${\displaystyle \mathrm{Tails}\left(x_{\bullet }\right)=\left\{x_{\ge i}:i\in I\right\}}$	Set orprefilter of tails/sections of${\displaystyle x_{\bullet }.}$ Also called theeventuality filter base generated by (the tails of)${\displaystyle x_{\bullet }={\left(x_i\right)}_{i\in I}.}$ If${\displaystyle x_{\bullet }}$ is a sequence then${\displaystyle \mathrm{Tails}\left(x_{\bullet }\right)}$ is also called thesequential filter base[16] orelementary prefilter.[17]
${\displaystyle \mathrm{TailsFilter}\left(x_{\bullet }\right)=\mathrm{Tails}{\left(x_{\bullet }\right)}^{\uparrow X}}$	(Eventuality)filter of/generated by (tails of)${\displaystyle x_{\bullet }}$[16]
${\displaystyle f\left(I_{\ge i}\right)=\{f(j):j\ge i\text{ and }j\in I\}}$	Tail orsection of a net${\displaystyle f:I\to X}$ starting at${\displaystyle i\in I}$[16] where${\displaystyle (I,\le )}$ is a directed set.

Warning about using strict comparison

If${\displaystyle x_{\bullet }={\left(x_i\right)}_{i\in I}}$ is a net and${\displaystyle i\in I}$ then it is possible for the set${\displaystyle x_{>i}=\left\{x_j:j>i\text{ and }j\in I\right\},}$ which is calledthe tail of${\displaystyle x_{\bullet }}$after${\displaystyle i}$, to be empty (for example, this happens if${\displaystyle i}$ is anupper bound of thedirected set${\displaystyle I}$). In this case, the family${\displaystyle \left\{x_{>i}:i\in I\right\}}$ would contain the empty set, which would prevent it from being a prefilter (defined later). This is the (important) reason for defining${\displaystyle \mathrm{Tails}\left(x_{\bullet }\right)}$ as${\displaystyle \left\{x_{\ge i}:i\in I\right\}}$ rather than${\displaystyle \left\{x_{>i}:i\in I\right\}}$ or even${\displaystyle \left\{x_{>i}:i\in I\right\}\cup \left\{x_{\ge i}:i\in I\right\}}$ 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${\displaystyle \le .}$

Families${\displaystyle \mathcal{F}}$ of sets over${\displaystyle \mathrm{\Omega }}$ v t e
Is necessarily true of${\displaystyle \mathcal{F}:}$ or, is${\displaystyle \mathcal{F}}$closed under:	Directed by${\displaystyle \supseteq }$	${\displaystyle A\cap B}$	${\displaystyle A\cup B}$	${\displaystyle B\setminus A}$	${\displaystyle \mathrm{\Omega }\setminus A}$	${\displaystyle A_1\cap A_2\cap \cdots }$	${\displaystyle A_1\cup A_2\cup \cdots }$	${\displaystyle \mathrm{\Omega }\in \mathcal{F}}$	${\displaystyle \varnothing \in \mathcal{F}}$	F.I.P.
π-system
Semiring										Never
Semialgebra(Semifield)										Never
Monotone class						only if${\displaystyle A_i\searrow }$	only if${\displaystyle A_i\nearrow }$
𝜆-system(Dynkin System)				only if ${\displaystyle A\subseteq B}$			only if${\displaystyle A_i\nearrow }$ or they aredisjoint			Never
Ring(Order theory)
Ring(Measure theory)										Never
δ-Ring										Never
𝜎-Ring										Never
Algebra(Field)										Never
𝜎-Algebra(𝜎-Field)										Never
Filter
Proper filter				Never	Never				Never
Prefilter(Filter base)
Filter subbase
Open Topology							(even arbitrary${\displaystyle \cup }$)			Never
Closed Topology						(even arbitrary${\displaystyle \cap }$)				Never
Is necessarily true of${\displaystyle \mathcal{F}:}$ or, is${\displaystyle \mathcal{F}}$closed under:	directed downward	finite intersections	finite unions	relative complements	complements in${\displaystyle \mathrm{\Omega }}$	countable intersections	countable unions	contains${\displaystyle \mathrm{\Omega }}$	contains${\displaystyle \varnothing }$	Finite Intersection Property
Additionally, asemiring is aπ-system where every complement${\displaystyle B\setminus A}$ is equal to a finitedisjoint union of sets in${\displaystyle \mathcal{F}.}$ Asemialgebra is a semiring where every complement${\displaystyle \mathrm{\Omega }\setminus A}$ is equal to a finitedisjoint union of sets in${\displaystyle \mathcal{F}.}$ ${\displaystyle A,B,A_1,A_2,\dots }$ are arbitrary elements of${\displaystyle \mathcal{F}}$ and it is assumed that${\displaystyle \mathcal{F}\ne \varnothing .}$

The following is a list of properties that a family${\displaystyle \mathcal{B}}$ of sets may possess and they form the defining properties of filters, prefilters, and filter subbases. Whenever it is necessary, it should be assumed that${\displaystyle \mathcal{B}\subseteq \mathrm{\wp }(X).}$

The family of sets${\displaystyle \mathcal{B}}$ is:

1. Proper ornondegenerate if${\displaystyle \varnothing \notin \mathcal{B}.}$ Otherwise, if${\displaystyle \varnothing \in \mathcal{B},}$ then it is calledimproper^[18] ordegenerate.
2. Directed downward^[15] if whenever${\displaystyle A,B\in \mathcal{B}}$ then there exists some${\displaystyle C\in \mathcal{B}}$ such that${\displaystyle C\subseteq A\cap B.}$
   • This property can be characterized in terms ofdirectedness, which explains the word "directed": Abinary relation${\displaystyle ⪯}$ on${\displaystyle \mathcal{B}}$ is called(upward) directed if for any two${\displaystyle A\text{ and }B,}$ there is some${\displaystyle C}$ satisfying${\displaystyle A⪯C\text{ and }B⪯C.}$ Using${\displaystyle \supseteq }$ in place of${\displaystyle ⪯}$ gives the definition ofdirected downward whereas using${\displaystyle \subseteq }$ instead gives the definition ofdirected upward. Explicitly,${\displaystyle \mathcal{B}}$ isdirected downward (resp.directed upward) if and only if for all${\displaystyle A,B\in \mathcal{B},}$ there exists some "greater"${\displaystyle C\in \mathcal{B}}$ such that${\displaystyle A\supseteq C\text{ and }B\supseteq C}$ (resp. such that${\displaystyle A\subseteq C\text{ and }B\subseteq C}$) − where the "greater" element is always on the right hand side, − which can be rewritten as${\displaystyle A\cap B\supseteq C}$ (resp. as${\displaystyle A\cup B\subseteq C}$).
3. Closed under finite intersections (resp.unions) if the intersection (resp. union) of any two elements of${\displaystyle \mathcal{B}}$ is an element of${\displaystyle \mathcal{B}.}$
   • If${\displaystyle \mathcal{B}}$ is closed under finite intersections then${\displaystyle \mathcal{B}}$ is necessarily directed downward. The converse is generally false.
4. Upward closed orIsotone in${\displaystyle X}$^[6] if${\displaystyle \mathcal{B}\subseteq \mathrm{\wp }(X)\text{ and }\mathcal{B}={\mathcal{B}}^{\uparrow X},}$ or equivalently, if whenever${\displaystyle B\in \mathcal{B}}$ and some set${\displaystyle C}$ satisfies${\displaystyle B\subseteq C\subseteq X,\text{ then }C\in \mathcal{B}.}$ Similarly,${\displaystyle \mathcal{B}}$ isdownward closed if${\displaystyle \mathcal{B}={\mathcal{B}}^{\downarrow }.}$ An upward (respectively, downward) closed set is also called anupper set orupset (resp. alower set ordown set).
   • The family${\displaystyle {\mathcal{B}}^{\uparrow X},}$ which is the upward closure of${\displaystyle \mathcal{B}\text{ in }X,}$ is the uniquesmallest (with respect to${\displaystyle \subseteq }$) isotone family of sets over${\displaystyle X}$ having${\displaystyle \mathcal{B}}$ as a subset.

Many of the properties of${\displaystyle \mathcal{B}}$ defined above and below, such as "proper" and "directed downward," do not depend on${\displaystyle X,}$ so mentioning the set${\displaystyle X}$ is optional when using such terms. Definitions involving being "upward closed in${\displaystyle X,}$" such as that of "filter on${\displaystyle X,}$" do depend on${\displaystyle X}$ so the set${\displaystyle X}$ should be mentioned if it is not clear from context.

A family${\displaystyle \mathcal{B}}$ is/is a(n):

1. Ideal^[18][19] if${\displaystyle \mathcal{B}\ne \varnothing }$ is downward closed and closed under finite unions.
2. Dual ideal on${\displaystyle X}$^[20] if${\displaystyle \mathcal{B}\ne \varnothing }$ is upward closed in${\displaystyle X}$ and also closed under finite intersections. Equivalently,${\displaystyle \mathcal{B}\ne \varnothing }$ is a dual ideal if for all${\displaystyle R,S\subseteq X,}$${\displaystyle R\cap S\in \mathcal{B}\text{ if and only if }R,S\in \mathcal{B}.}$^[21]
   • Explanation of the word "dual": A family${\displaystyle \mathcal{B}}$ is a dual ideal (resp. an ideal) on${\displaystyle X}$ if and only if thedual of${\displaystyle \mathcal{B}\text{ in }X,}$ which is the family$${\displaystyle X\setminus \mathcal{B}:=\{X\setminus B:B\in \mathcal{B}\},}$$ is an ideal (resp. a dual ideal) on${\displaystyle X.}$ In other words,dual ideal means "dualof an ideal". The dual of the dual is the original family, meaning${\displaystyle X\setminus (X\setminus \mathcal{B})=\mathcal{B}.}$^[18]
3. Filter on${\displaystyle X}$^[20][8] if${\displaystyle \mathcal{B}}$ is aproperdual ideal on${\displaystyle X.}$ That is, a filter on${\displaystyle X}$ is a non−empty subset of${\displaystyle \mathrm{\wp }(X)\setminus \{\varnothing \}}$ that is closed under finite intersections and upward closed in${\displaystyle X.}$ Equivalently, it is a prefilter that is upward closed in${\displaystyle X.}$ In words, a filter on${\displaystyle X}$ is a family of sets over${\displaystyle X}$ that (1) is not empty (or equivalently, it contains${\displaystyle X}$), (2) is closed under finite intersections, (3) is upward closed in${\displaystyle X,}$ and (4) does not have the empty set as an element.
   • Warning: Some authors, particularly algebrists, use "filter" to mean a dual ideal; others, particularly topologists, use "filter" to mean aproper/non-degenerate dual ideal.^[22] It is recommended that readers always check how "filter" is defined when reading mathematical literature. However, the definitions of "ultrafilter," "prefilter," and "filter subbase" always requirenon-degeneracy. This article usesHenri Cartan's original definition of "filter",^[1][23] which required non-degeneracy.
   • The power set${\displaystyle \mathrm{\wp }(X)}$ is the one and only dual ideal on${\displaystyle X}$ that is not also a filter. Excluding${\displaystyle \mathrm{\wp }(X)}$ from the definition of "filter" intopology has the same benefit asexcluding${\displaystyle 1}$ from the definition of "prime number": itobviates the need to specify "non-degenerate" (the analog of "non-unital" or "non-${\displaystyle 1}$") in many important results, thereby making their statements less awkward.
4. Prefilter orfilter base^[8][24] if${\displaystyle \mathcal{B}\ne \varnothing }$ is proper and directed downward. Equivalently,${\displaystyle \mathcal{B}}$ is called a prefilter if its upward closure${\displaystyle {\mathcal{B}}^{\uparrow X}}$ is a filter. It can also be defined as any family that isequivalent tosome filter.^[9] A proper family${\displaystyle \mathcal{B}\ne \varnothing }$ is a prefilter if and only if${\displaystyle \mathcal{B}(\cap )\mathcal{B}\le \mathcal{B}.}$^[9] A family is a prefilter if and only if the same is true of its upward closure.
   • If${\displaystyle \mathcal{B}}$ is a prefilter then its upward closure${\displaystyle {\mathcal{B}}^{\uparrow X}}$ is the unique smallest (relative to${\displaystyle \subseteq }$) filter on${\displaystyle X}$ containing${\displaystyle \mathcal{B}}$ and it is calledthe filter generated by${\displaystyle \mathcal{B}.}$ A filter${\displaystyle \mathcal{F}}$ is said to begenerated by a prefilter${\displaystyle \mathcal{B}}$ if${\displaystyle \mathcal{F}={\mathcal{B}}^{\uparrow X},}$ in which${\displaystyle \mathcal{B}}$ is called afilter base for${\displaystyle \mathcal{F}.}$
   • Unlike a filter, a prefilter isnot necessarily closed under finite intersections.
5. π-system if${\displaystyle \mathcal{B}\ne \varnothing }$ is closed under finite intersections. Every non-empty family${\displaystyle \mathcal{B}}$ is contained in a unique smallestπ-system calledtheπ-system generated by${\displaystyle \mathcal{B},}$ which is sometimes denoted by${\displaystyle \pi (\mathcal{B}).}$ It is equal to the intersection of allπ-systems containing${\displaystyle \mathcal{B}}$ and also to the set of all possible finite intersections of sets from${\displaystyle \mathcal{B}}$:$${\displaystyle \pi (\mathcal{B})=\left\{B_1\cap \cdots \cap B_n:n\ge 1\text{ and }{B}_1,\dots ,B_n\in \mathcal{B}\right\}.}$$
   • Aπ-system is a prefilter if and only if it is proper. Every filter is a properπ-system and every properπ-system is a prefilter but the converses do not hold in general.
   • A prefilter isequivalent to theπ-system generated by it and both of these families generate the same filter on${\displaystyle X.}$
6. Filter subbase^[8][25] andcentered^[9] if${\displaystyle \mathcal{B}\ne \varnothing }$ and${\displaystyle \mathcal{B}}$ satisfies any of the following equivalent conditions:
   1. ${\displaystyle \mathcal{B}}$ has thefinite intersection property, which means that the intersection of any finite family of (one or more) sets in${\displaystyle \mathcal{B}}$ is not empty; explicitly, this means that whenever${\displaystyle n\ge 1\text{ and }{B}_1,\dots ,B_n\in \mathcal{B}}$ then${\displaystyle \varnothing \ne B_1\cap \cdots \cap B_n.}$
   2. Theπ-system generated by${\displaystyle \mathcal{B}}$ is proper; that is,${\displaystyle \varnothing \notin \pi (\mathcal{B}).}$
   3. Theπ-system generated by${\displaystyle \mathcal{B}}$ is a prefilter.
   4. ${\displaystyle \mathcal{B}}$ is a subset ofsome prefilter.
   5. ${\displaystyle \mathcal{B}}$ is a subset ofsome filter.^[10]
   • Assume that${\displaystyle \mathcal{B}}$ is a filter subbase. Then there is a unique smallest (relative to${\displaystyle \subseteq }$) filter${\displaystyle {\mathcal{F}}_{\mathcal{B}}\text{ on }X}$ containing${\displaystyle \mathcal{B}}$ called thefilter generated by${\displaystyle \mathcal{B}}$, and${\displaystyle \mathcal{B}}$ is said tobe a filter subbase for this filter. This filter is equal to the intersection of all filters on${\displaystyle X}$ that are supersets of${\displaystyle \mathcal{B}.}$ Theπ-system generated by${\displaystyle \mathcal{B},}$ denoted by${\displaystyle \pi (\mathcal{B}),}$ will be a prefilter and a subset of${\displaystyle {\mathcal{F}}_{\mathcal{B}}.}$ Moreover, the filter generated by${\displaystyle \mathcal{B}}$ is equal to the upward closure of${\displaystyle \pi (\mathcal{B}),}$ meaning${\displaystyle \pi (\mathcal{B}{)}^{\uparrow X}={\mathcal{F}}_{\mathcal{B}}.}$^[9] However,${\displaystyle {\mathcal{B}}^{\uparrow X}={\mathcal{F}}_{\mathcal{B}}}$ ifand only if${\displaystyle \mathcal{B}}$ is a prefilter (although${\displaystyle {\mathcal{B}}^{\uparrow X}}$ is always an upward closed filtersubbase for${\displaystyle {\mathcal{F}}_{\mathcal{B}}}$).
   • A${\displaystyle \subseteq }$ -smallest (meaning smallest relative to${\displaystyle \subseteq }$ )prefilter containing a filter subbase${\displaystyle \mathcal{B}}$ will exist only under certain circumstances. It exists, for example, if the filter subbase${\displaystyle \mathcal{B}}$ happens to also be a prefilter. It also exists if the filter (or equivalently, theπ-system) generated by${\displaystyle \mathcal{B}}$ isprincipal, in which case${\displaystyle \mathcal{B}\cup \{\ker \mathcal{B}\}}$ is the unique smallest prefilter containing${\displaystyle \mathcal{B}.}$ Otherwise, in general, a${\displaystyle \subseteq }$ -smallestprefilter containing${\displaystyle \mathcal{B}}$ might not exist. For this reason, some authors may refer to theπ-system generated by${\displaystyle \mathcal{B}}$ asthe prefilter generated by${\displaystyle \mathcal{B}.}$ However, if a${\displaystyle \subseteq }$ -smallest prefilter does exist (say it is denoted by${\displaystyle \mathrm{minPre}\mathcal{B}}$) then contrary to usual expectations, it isnot necessarily equal to "the prefilter generated by${\displaystyle \mathcal{B}}$" (that is,${\displaystyle \mathrm{minPre}\mathcal{B}\ne \pi (\mathcal{B})}$ is possible). And if the filter subbase${\displaystyle \mathcal{B}}$ happens to also be a prefilter but not aπ-system then unfortunately, "the prefilter generated by this prefilter" (meaning${\displaystyle \pi (\mathcal{B})}$) will not be${\displaystyle \mathcal{B}=\mathrm{minPre}\mathcal{B}}$ (that is,${\displaystyle \pi (\mathcal{B})\ne \mathcal{B}}$ is possible even when${\displaystyle \mathcal{B}}$ is a prefilter), which is why this article will prefer the accurate and unambiguous terminology of "theπ-system generated by${\displaystyle \mathcal{B}}$".
7. Subfilter of a filter${\displaystyle \mathcal{F}}$ and that${\displaystyle \mathcal{F}}$ is asuperfilter of${\displaystyle \mathcal{B}}$^[18][26] if${\displaystyle \mathcal{B}}$ is a filter and${\displaystyle \mathcal{B}\subseteq \mathcal{F}}$ where for filters,${\displaystyle \mathcal{B}\subseteq \mathcal{F}\text{ if and only if }\mathcal{B}\le \mathcal{F}.}$
   • Importantly, the expression "is asuperfilter of" is for filters the analog of "is asubsequence of". So despite having the prefix "sub" in common, "is asubfilter of" is actually thereverse of "is asubsequence of." However,${\displaystyle \mathcal{B}\le \mathcal{F}}$ can also be written${\displaystyle \mathcal{F}⊢\mathcal{B}}$ which is described by saying "${\displaystyle \mathcal{F}}$ is subordinate to${\displaystyle \mathcal{B}.}$" With this terminology, "issubordinate to" becomes for filters (and also for prefilters) the analog of "is asubsequence of,"^[27] which makes this one situation where using the term "subordinate" and symbol${\displaystyle ⊢}$ may be helpful.

There are no prefilters on${\displaystyle X=\varnothing }$ (nor are there any nets valued in${\displaystyle \varnothing }$), which is why this article, like most authors, will automatically assume without comment that${\displaystyle X\ne \varnothing }$ whenever this assumption is needed.

There are many other characterizations of "ultrafilter" and "ultra prefilter," which are listed in the article onultrafilters. Important properties of ultrafilters are also described in that article.

A non-empty family${\displaystyle \mathcal{B}\subseteq \mathrm{\wp }(X)}$ of sets is/is an:

8. Ultra^[8][28] if${\displaystyle \varnothing \notin \mathcal{B}}$ and any of the following equivalent conditions are satisfied:
   1. For every set${\displaystyle S\subseteq X}$ there exists some set${\displaystyle B\in \mathcal{B}}$ such that${\displaystyle B\subseteq S\text{ or }B\subseteq X\setminus S}$ (or equivalently, such that${\displaystyle B\cap S\text{ equals }B\text{ or }\varnothing }$).
   2. For every set${\displaystyle S\subseteq \bigcup _{B\in \mathcal{B}}B}$ there exists some set${\displaystyle B\in \mathcal{B}}$ such that${\displaystyle B\cap S\text{ equals }B\text{ or }\varnothing .}$
      • This characterization of "${\displaystyle \mathcal{B}}$ is ultra" does not depend on the set${\displaystyle X,}$ so mentioning the set${\displaystyle X}$ is optional when using the term "ultra."
   3. Forevery set${\displaystyle S}$ (not necessarily even a subset of${\displaystyle X}$) there exists some set${\displaystyle B\in \mathcal{B}}$ such that${\displaystyle B\cap S\text{ equals }B\text{ or }\varnothing .}$
9. Ultra prefilter^[8][28] if it is a prefilter that is also ultra. Equivalently, it is a filter subbase that is ultra. A prefilter${\displaystyle \mathcal{B}}$ is ultra if and only if it satisfies any of the following equivalent conditions:
   1. ${\displaystyle \mathcal{B}}$ ismaximal in${\displaystyle \mathrm{Prefilters}(X)}$ with respect to${\displaystyle \le ,}$ which means that$${\displaystyle \text{For all }\mathcal{C}\in \mathrm{Prefilters}(X),\mathcal{B}\le \mathcal{C}\text{ implies }\mathcal{C}\le \mathcal{B}.}$$
   2. ${\displaystyle \text{For all }\mathcal{C}\in \mathrm{Filters}(X),\mathcal{B}\le \mathcal{C}\text{ implies }\mathcal{C}\le \mathcal{B}.}$
      • Although this statement is identical to that given below for ultrafilters, here${\displaystyle \mathcal{B}}$ is merely assumed to be a prefilter; it need not be a filter.
   3. ${\displaystyle {\mathcal{B}}^{\uparrow X}}$ is ultra (and thus an ultrafilter).
   4. ${\displaystyle \mathcal{B}}$ isequivalent to some ultrafilter.
   • A filter subbase that is ultra is necessarily a prefilter. A filter subbase is ultra if and only if it is a maximal filter subbase with respect to${\displaystyle \le }$ (as above).^[18]
10. Ultrafilter on${\displaystyle X}$^[8][28] if it is a filter on${\displaystyle X}$ that is ultra. Equivalently, an ultrafilter on${\displaystyle X}$ is a filter${\displaystyle \mathcal{B}\text{ on }X}$ that satisfies any of the following equivalent conditions:
    1. ${\displaystyle \mathcal{B}}$ is generated by an ultra prefilter.
    2. For any${\displaystyle S\subseteq X,S\in \mathcal{B}\text{ or }X\setminus S\in \mathcal{B}.}$^[18]
    3. ${\displaystyle \mathcal{B}\cup (X\setminus \mathcal{B})=\mathrm{\wp }(X).}$ This condition can be restated as:${\displaystyle \mathrm{\wp }(X)}$ is partitioned by${\displaystyle \mathcal{B}}$ and its dual${\displaystyle X\setminus \mathcal{B}.}$
    4. For any${\displaystyle R,S\subseteq X,}$ if${\displaystyle R\cup S\in \mathcal{B}}$ then${\displaystyle R\in \mathcal{B}\text{ or }S\in \mathcal{B}}$ (a filter with this property is called aprime filter).
       • This property extends to any finite union of two or more sets.
    5. ${\displaystyle \mathcal{B}}$ is amaximal filter on${\displaystyle X}$; meaning that if${\displaystyle \mathcal{C}}$ is a filter on${\displaystyle X}$ such that${\displaystyle \mathcal{B}\subseteq \mathcal{C}}$ then necessarily${\displaystyle \mathcal{C}=\mathcal{B}}$ (this equality may be replaced by${\displaystyle \mathcal{C}\subseteq \mathcal{B}\text{ or by }\mathcal{C}\le \mathcal{B}}$).
       • If${\displaystyle \mathcal{C}}$ is upward closed then${\displaystyle \mathcal{B}\le \mathcal{C}\text{ if and only if }\mathcal{B}\subseteq \mathcal{C}.}$ So this characterization of ultrafilters as maximal filters can be restated as:$${\displaystyle \text{For all }\mathcal{C}\in \mathrm{Filters}(X),\mathcal{B}\le \mathcal{C}\text{ implies }\mathcal{C}\le \mathcal{B}.}$$
       • Because subordination${\displaystyle \ge }$ is for filters the analog of "is a subnet/subsequence of" (specifically, "subnet" should mean "AA-subnet," which is defined below), this characterization of an ultrafilter as being a "maximally subordinate filter" suggests that an ultrafilter can be interpreted as being analogous to some sort of "maximally deep net" (which could, for instance, mean that "when viewed only from${\displaystyle X}$" in some sense, it is indistinguishable from its subnets, as is the case with any net valued in a singleton set for example),^{[note 5]} which is an idea that is actually made rigorous byultranets. Theultrafilter lemma is then the statement that every filter ("net") has some subordinate filter ("subnet") that is "maximally subordinate" ("maximally deep").