Filters in topology

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Filters in topology, a subfield of mathematics, can be used to study topological spaces and define all basic topological notions such as convergence, continuity, compactness, and more. Filters, which are special families of subsets of some given set, also provide a common framework for defining various types of limits of functions such as limits from the left/right, to infinity, to a point or a set, and many others. Special types of filters called ultrafilters have many useful technical properties and they may often be used in place of arbitrary filters.

Filters have generalizations called prefilters (also known as filter bases) and filter subbases, all of which appear naturally and repeatedly throughout topology. Examples include neighborhood filters/bases/subbases and uniformities. 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 to generate. 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 certain preorder on families of sets, denoted by ${\displaystyle \,\leq ,\,}$ 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 ${\displaystyle {\mathcal {N}}\leq {\mathcal {B}},}$ where ${\displaystyle {\mathcal {N}}}$ is that point's neighborhood filter. Consequently, subordination also plays an important role in many concepts that are related to convergence, such as cluster points and limits of functions. In addition, the relation ${\displaystyle {\mathcal {S}}\geq {\mathcal {B}},}$ which denotes ${\displaystyle {\mathcal {B}}\leq {\mathcal {S}}}$ 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 ${\displaystyle \geq ,}$ which is called subordination, is for filters the analog of "is a subsequence of").

Filters were introduced by Henri Cartan in 1937^[1] and subsequently used by Bourbaki in their book Topologie Générale as an alternative to the similar notion of a net developed in 1922 by E. H. Moore and H. L. Smith. Filters can also be used to characterize the notions of sequence and net convergence. But unlike^{[note 1]} sequence and net convergence, filter convergence is defined entirely 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, the category of topological spaces can be equivalently 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 those given by Willard and by Kelley), then in general, this relationship does not extend to subordinate filters and subnets because as detailed 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 an AA–subnet.

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

Motivation[edit]

Archetypical example of a filter

The archetypical example of a filter is the neighborhood filter ${\displaystyle {\mathcal {N}}(x)}$ at a point ${\displaystyle x}$ in a topological space ${\displaystyle (X,\tau ),}$ which is the family of sets consisting of all neighborhoods of ${\displaystyle x.}$ By definition, a neighborhood of some given point ${\displaystyle x}$ is any subset ${\displaystyle B\subseteq X}$ whose topological interior contains this point; that is, such that ${\displaystyle x\in \operatorname {Int} _{X}B.}$ Importantly, neighborhoods are not required to be open sets; those are called open neighborhoods. Listed below are those fundamental properties of neighborhood filters that ultimately became the definition of a "filter." A filter 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:   ${\displaystyle \varnothing \not \in {\mathcal {B}}}$  –  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 contains a neighborhood of ${\displaystyle x}$ will necessarily be a neighborhood of ${\displaystyle x}$ (this follows from ${\displaystyle \operatorname {Int} _{X}B\subseteq \operatorname {Int} _{X}S}$ and the definition of "a neighborhood of ${\displaystyle x}$").

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

A sequence in ${\displaystyle X}$ is by definition a map ${\displaystyle \mathbb {N} \to X}$ from the natural numbers into the space ${\displaystyle X.}$ The original notion of convergence in a topological space was that of a sequence converging to some given point in a space, such as a metric space. With metrizable spaces (or more generally first–countable spaces or Fréchet–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 can not 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, which never 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 arbitrary directed set ${\displaystyle (I,\leq )}$ into the space ${\displaystyle X.}$ A sequence is just a net whose domain is ${\displaystyle I=\mathbb {N} }$ with the natural ordering. Nets have their own notion of convergence, which is a direct generalization of sequence convergence.

Filters generalize sequence convergence in a different way by considering only the values of a sequence. To see how this is done, consider a sequence ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }{\text{ in }}X,}$ 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 the image (sometimes called "the range") ${\displaystyle \operatorname {Im} x_{\bullet }:=\left\{x_{i}:i\in \mathbb {N} \right\}=\left\{x_{1},x_{2},\ldots \right\}}$ 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 the tails of the sequence ${\displaystyle x_{\bullet }}$:

$${\displaystyle {\begin{alignedat}{8}x_{\geq 1}=\;&\{&&x_{1},&&x_{2},&&x_{3},&&x_{4},&&\ldots &&\,\}\\[0.3ex]x_{\geq 2}=\;&\{&&x_{2},&&x_{3},&&x_{4},&&x_{5},&&\ldots &&\,\}\\[0.3ex]x_{\geq 3}=\;&\{&&x_{3},&&x_{4},&&x_{5},&&x_{6},&&\ldots &&\,\}\\[0.3ex]&&&&&&&\;\,\vdots &&&&&&\\[0.3ex]x_{\geq n}=\;&\{&&x_{n},\;\;\,&&x_{n+1},\;&&x_{n+2},\;&&x_{n+3},&&\ldots &&\,\}\\[0.3ex]&&&&&&&\;\,\vdots &&&&&&\\[0.3ex]\end{alignedat}}}$$

These sets completely determine this sequence's convergence (or non–convergence) because given any point, this sequence 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 ${\displaystyle x_{n},x_{n+1},\ldots .}$ This can be reworded as:

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

Or more briefly: every neighborhood must contain some tail ${\displaystyle x_{\geq n}}$ 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 of sets ${\displaystyle \{x_{\geq 1},x_{\geq 2},\ldots \}}$ in hand, the function ${\displaystyle x_{\bullet }:\mathbb {N} \to X}$ 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 its upward closure (which consists of all supersets of all tails). The same is true of other important families of sets such as any neighborhood 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 a filter base, also called a prefilter, which by definition is any family having the minimal properties necessary and sufficient for it to generate a filter via taking its upward 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 completely characterize 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, and especially ultrafilters, have many more uses outside of topology, such as in set 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 of hyperreal numbers.

Like sequences, nets are functions and so they have the advantages of functions. For example, like sequences, nets can be "plugged into" other functions, where "plugging in" is just function composition. Theorems related to functions and function composition may then be applied to nets. One example is the universal property of inverse 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 the Cartesian 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 a dense subspace ${\displaystyle S\subseteq X.}$^[5]

In contrast to nets, filters (and prefilters) are families of sets and so they have the advantages of sets. For example, if ${\displaystyle f}$ is surjective then the image ${\displaystyle f^{-1}({\mathcal {B}}):=\left\{f^{-1}(B)~:~B\in {\mathcal {B}}\right\}}$ 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 to pullback (unambiguously/without choice) 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 as closure or interior) may be applied to the sets that constitute the filter. Taking the closure of all the sets in a filter is sometimes useful in functional 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 of continuity's characterizations in terms of preimages of open/closed sets or in terms of the interior/closure operators. Special types of filters called ultrafilters 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 a proper class); this is because nets in ${\displaystyle X}$ can have domains of any cardinality. 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 ${\displaystyle \wp (\wp (X)).}$ Similar to a topology on ${\displaystyle X,}$ a filter on ${\displaystyle X}$ is "intrinsic to ${\displaystyle X}$" in the sense that both structures consist entirely 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).

Preliminaries, notation, and basic notions[edit]

In this article, upper case Roman letters like ${\displaystyle S{\text{ and }}X}$ denote sets (but not families unless indicated otherwise) and ${\displaystyle \wp (X)}$ will denote the power set of ${\displaystyle X.}$ A subset of a power set is called a family of sets (or simply, a family) where it is over ${\displaystyle X}$ if it is a subset of ${\displaystyle \wp (X).}$ Families of sets will be denoted by upper case calligraphy letters such as ${\displaystyle {\mathcal {B}},{\mathcal {C}},{\text{ and }}{\mathcal {F}}.}$ 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

The upward closure or isotonization in ${\displaystyle X}$^[6][7] of a family of sets ${\displaystyle {\mathcal {B}}\subseteq \wp (X)}$ is

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

and similarly the downward closure of ${\displaystyle {\mathcal {B}}}$ is ${\displaystyle {\mathcal {B}}^{\downarrow }:=\{S\subseteq B~:~B\in {\mathcal {B}}\,\}={\textstyle \bigcup \limits _{B\in {\mathcal {B}}}}\wp (B).}$

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]
${\displaystyle {\mathcal {B}}{\big \vert }_{S}:=\{B\cap S~:~B\in {\mathcal {B}}\}={\mathcal {B}}\,(\cap )\,\{S\}}$	Trace of ${\displaystyle {\mathcal {B}}{\text{ on }}S}$[8] or the 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 usual set subtraction)
${\displaystyle \wp (X)=\{S~:~S\subseteq X\}}$	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 the preimage of ${\displaystyle {\mathcal {B}}}$ under ${\displaystyle f}$
${\displaystyle f({\mathcal {B}})=\{f(B)~:~B\in {\mathcal {B}}\}}$[14]	Image of ${\displaystyle {\mathcal {B}}}$ under ${\displaystyle f}$
${\displaystyle \operatorname {image} f=f(\operatorname {domain} f)}$	Image (or range) of ${\displaystyle f}$

Topology notation

Denote the set of all topologies on a set ${\displaystyle X{\text{ by }}\operatorname {Top} (X).}$ Suppose ${\displaystyle \tau \in \operatorname {Top} (X),}$ ${\displaystyle S\subseteq X}$ 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 or prefilter[note 4] of open neighborhoods of ${\displaystyle S{\text{ in }}(X,\tau )}$
${\displaystyle \tau (x)=\{O\in \tau ~:~x\in O\}}$	Set or prefilter of open neighborhoods of ${\displaystyle x{\text{ in }}(X,\tau )}$
${\displaystyle {\mathcal {N}}_{\tau }(S)={\mathcal {N}}(S):=\tau (S)^{\uparrow X}}$	Set or filter[note 4] of neighborhoods of ${\displaystyle S{\text{ in }}(X,\tau )}$
${\displaystyle {\mathcal {N}}_{\tau }(x)={\mathcal {N}}(x):=\tau (x)^{\uparrow X}}$	Set or filter of neighborhoods of ${\displaystyle x{\text{ in }}(X,\tau )}$

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

Nets and their tails

A directed set is a set ${\displaystyle I}$ together with a preorder, which will be denoted by ${\displaystyle \,\leq \,}$ (unless explicitly indicated otherwise), that makes ${\displaystyle (I,\leq )}$ 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 ${\displaystyle i\leq k{\text{ and }}j\leq k.}$ For any indices ${\displaystyle i{\text{ and }}j,}$ the notation ${\displaystyle j\geq i}$ is defined to mean ${\displaystyle i\leq j}$ while ${\displaystyle i<j}$ is defined to mean that ${\displaystyle i\leq j}$ holds but it is not true that ${\displaystyle j\leq i}$ (if ${\displaystyle \,\leq \,}$ is antisymmetric then this is equivalent to ${\displaystyle i\leq j{\text{ and }}i\neq j}$).

A net 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
${\displaystyle I_{\geq i}=\{j\in I~:~j\geq i\}}$	Tail or section of ${\displaystyle I}$ starting at ${\displaystyle i\in I}$ where ${\displaystyle (I,\leq )}$ is a directed set.
${\displaystyle x_{\geq i}=\left\{x_{j}~:~j\geq i{\text{ and }}j\in I\right\}}$	Tail or section of ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}$ starting at ${\displaystyle i\in I}$
${\displaystyle \operatorname {Tails} \left(x_{\bullet }\right)=\left\{x_{\geq i}~:~i\in I\right\}}$	Set or prefilter of tails/sections of ${\displaystyle x_{\bullet }.}$ Also called the eventuality 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 ${\displaystyle \operatorname {Tails} \left(x_{\bullet }\right)}$ is also called the sequential filter base.[16]
${\displaystyle \operatorname {TailsFilter} \left(x_{\bullet }\right)=\operatorname {Tails} \left(x_{\bullet }\right)^{\uparrow X}}$	(Eventuality) filter of/generated by (tails of) ${\displaystyle x_{\bullet }}$[16]
${\displaystyle f\left(I_{\geq i}\right)=\{f(j)~:~j\geq i{\text{ and }}j\in I\}}$	Tail or section of a net ${\displaystyle f:I\to X}$ starting at ${\displaystyle i\in I}$[16] where ${\displaystyle (I,\leq )}$ 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 called the tail of ${\displaystyle x_{\bullet }}$ after ${\displaystyle i}$, to be empty (for example, this happens if ${\displaystyle i}$ is an upper bound of the directed 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 ${\displaystyle \operatorname {Tails} \left(x_{\bullet }\right)}$ as ${\displaystyle \left\{x_{\geq i}~:~i\in I\right\}}$ rather than ${\displaystyle \left\{x_{>i}~:~i\in I\right\}}$ or even ${\displaystyle \left\{x_{>i}~:~i\in I\right\}\cup \left\{x_{\geq i}~:~i\in I\right\}}$ and it is for this reason that in general, when dealing with the prefilter of tails of a net, the strict inequality ${\displaystyle \,<\,}$ may not be used interchangeably with the inequality ${\displaystyle \,\leq .}$

Filters and prefilters[edit]

Families ${\displaystyle {\mathcal {F}}}$ of sets over ${\displaystyle \Omega }$ v t e
Is necessarily true of ${\displaystyle {\mathcal {F}}\colon }$ or, is ${\displaystyle {\mathcal {F}}}$ closed under:	Directed by ${\displaystyle \,\supseteq }$	${\displaystyle A\cap B}$	${\displaystyle A\cup B}$	${\displaystyle B\setminus A}$	${\displaystyle \Omega \setminus A}$	${\displaystyle A_{1}\cap A_{2}\cap \cdots }$	${\displaystyle A_{1}\cup A_{2}\cup \cdots }$	${\displaystyle \Omega \in {\mathcal {F}}}$	${\displaystyle \varnothing \in {\mathcal {F}}}$	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 are disjoint			Never
Ring (Order theory)
Ring (Measure theory)										Never
δ-Ring										Never
𝜎-Ring										Never
Algebra (Field)										Never
𝜎-Algebra (𝜎-Field)										Never
Dual ideal
Filter				Never	Never				${\displaystyle \varnothing \not \in {\mathcal {F}}}$
Prefilter (Filter base)				Never	Never				${\displaystyle \varnothing \not \in {\mathcal {F}}}$
Filter subbase				Never	Never				${\displaystyle \varnothing \not \in {\mathcal {F}}}$
Open Topology							(even arbitrary ${\displaystyle \cup }$)			Never
Closed Topology						(even arbitrary ${\displaystyle \cap }$)				Never
Is necessarily true of ${\displaystyle {\mathcal {F}}\colon }$ or, is ${\displaystyle {\mathcal {F}}}$ closed under:	directed downward	finite intersections	finite unions	relative complements	complements in ${\displaystyle \Omega }$	countable intersections	countable unions	contains ${\displaystyle \Omega }$	contains ${\displaystyle \varnothing }$	Finite Intersection Property
Additionally, a semiring is a π-system where every complement ${\displaystyle B\setminus A}$ is equal to a finite disjoint union of sets in ${\displaystyle {\mathcal {F}}.}$ A semialgebra is a semiring where every complement ${\displaystyle \Omega \setminus A}$ is equal to a finite disjoint union of sets in ${\displaystyle {\mathcal {F}}.}$ ${\displaystyle A,B,A_{1},A_{2},\ldots }$ are arbitrary elements of ${\displaystyle {\mathcal {F}}}$ and it is assumed that ${\displaystyle {\mathcal {F}}\neq \varnothing .}$

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 ${\displaystyle {\mathcal {B}}\subseteq \wp (X).}$

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

1. Proper or nondegenerate if ${\displaystyle \varnothing \not \in {\mathcal {B}}.}$ Otherwise, if ${\displaystyle \varnothing \in {\mathcal {B}},}$ then it is called improper^[17] or degenerate.
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 of directedness, which explains the word "directed": A binary relation ${\displaystyle \,\preceq \,}$ on ${\displaystyle {\mathcal {B}}}$ is called (upward) directed if for any two ${\displaystyle A{\text{ and }}B,}$ there is some ${\displaystyle C}$ satisfying ${\displaystyle A\preceq C{\text{ and }}B\preceq C.}$ Using ${\displaystyle \,\supseteq \,}$ in place of ${\displaystyle \,\preceq \,}$ gives the definition of directed downward whereas using ${\displaystyle \,\subseteq \,}$ instead gives the definition of directed upward. Explicitly, ${\displaystyle {\mathcal {B}}}$ is directed 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 or Isotone in ${\displaystyle X}$^[6] if ${\displaystyle {\mathcal {B}}\subseteq \wp (X){\text{ and }}{\mathcal {B}}={\mathcal {B}}^{\uparrow X},}$ 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, ${\displaystyle {\mathcal {B}}}$ is downward closed if ${\displaystyle {\mathcal {B}}={\mathcal {B}}^{\downarrow }.}$ An upward (respectively, downward) closed set is also called an upper set or upset (resp. a lower set or down set).
   • The family ${\displaystyle {\mathcal {B}}^{\uparrow X},}$ which is the upward closure of ${\displaystyle {\mathcal {B}}{\text{ in }}X,}$ is the unique smallest (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^[17][18] if ${\displaystyle {\mathcal {B}}\neq \varnothing }$ is downward closed and closed under finite unions.
2. Dual ideal on ${\displaystyle X}$^[19] if ${\displaystyle {\mathcal {B}}\neq \varnothing }$ is upward closed in ${\displaystyle X}$ and also closed under finite intersections. Equivalently, ${\displaystyle {\mathcal {B}}\neq \varnothing }$ is a dual ideal if for all ${\displaystyle R,S\subseteq X,}$ ${\displaystyle R\cap S\in {\mathcal {B}}\;{\text{ if and only if }}\;R,S\in {\mathcal {B}}.}$^[20]
   • Explanation of the word "dual": A family ${\displaystyle {\mathcal {B}}}$ is a dual ideal (resp. an ideal) on ${\displaystyle X}$ if and only if the dual 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 "dual of an ideal". The dual of the dual is the original family, meaning ${\displaystyle X\setminus (X\setminus {\mathcal {B}})={\mathcal {B}}.}$^[17]
3. Filter on ${\displaystyle X}$^[19][8] if ${\displaystyle {\mathcal {B}}}$ is a proper dual ideal on ${\displaystyle X.}$ That is, a filter on ${\displaystyle X}$ is a non−empty subset of ${\displaystyle \wp (X)\setminus \{\varnothing \}}$ 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 a proper/non–degenerate dual ideal.^[21] 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 require non-degeneracy. This article uses Henri Cartan's original definition of "filter",^[1][22] which required non–degeneracy.
   • The power set ${\displaystyle \wp (X)}$ is the one and only dual ideal on ${\displaystyle X}$ that is not also a filter. Excluding ${\displaystyle \wp (X)}$ from the definition of "filter" in topology has the same benefit as excluding ${\displaystyle 1}$ from the definition of "prime number": it obviates 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 or filter base^[8][23] if ${\displaystyle {\mathcal {B}}\neq \varnothing }$ 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 is equivalent to some filter.^[9] A proper family ${\displaystyle {\mathcal {B}}\neq \varnothing }$ is a prefilter if and only if ${\displaystyle {\mathcal {B}}\,(\cap )\,{\mathcal {B}}\leq {\mathcal {B}}.}$^[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 called the filter generated by ${\displaystyle {\mathcal {B}}.}$ A filter ${\displaystyle {\mathcal {F}}}$ is said to be generated by a prefilter ${\displaystyle {\mathcal {B}}}$ if ${\displaystyle {\mathcal {F}}={\mathcal {B}}^{\uparrow X},}$ in which ${\displaystyle {\mathcal {B}}}$ is called a filter base for ${\displaystyle {\mathcal {F}}.}$
   • Unlike a filter, a prefilter is not necessarily closed under finite intersections.
5. π–system if ${\displaystyle {\mathcal {B}}\neq \varnothing }$ is closed under finite intersections. Every non–empty family ${\displaystyle {\mathcal {B}}}$ is contained in a unique smallest π–system called the π–system generated by ${\displaystyle {\mathcal {B}},}$ which is sometimes denoted by ${\displaystyle \pi ({\mathcal {B}}).}$ It is equal to the intersection of all π–systems containing ${\displaystyle {\mathcal {B}}}$ and also to the set of all possible finite intersections of sets from ${\displaystyle {\mathcal {B}}}$:
   $${\displaystyle \pi ({\mathcal {B}})=\left\{B_{1}\cap \cdots \cap B_{n}~:~n\geq 1{\text{ and }}B_{1},\ldots ,B_{n}\in {\mathcal {B}}\right\}.}$$

   
   • A π–system is a prefilter if and only if it is proper. Every filter is a proper π–system and every proper π–system is a prefilter but the converses do not hold in general.
   • A prefilter is equivalent to the π–system generated by it and both of these families generate the same filter on ${\displaystyle X.}$
6. Filter subbase^[8][24] and centered^[9] if ${\displaystyle {\mathcal {B}}\neq \varnothing }$ and ${\displaystyle {\mathcal {B}}}$ satisfies any of the following equivalent conditions:
   1. ${\displaystyle {\mathcal {B}}}$ has the finite 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 ${\displaystyle n\geq 1{\text{ and }}B_{1},\ldots ,B_{n}\in {\mathcal {B}}}$ then ${\displaystyle \varnothing \neq B_{1}\cap \cdots \cap B_{n}.}$
   2. The π–system generated by ${\displaystyle {\mathcal {B}}}$ is proper; that is, ${\displaystyle \varnothing \not \in \pi ({\mathcal {B}}).}$
   3. The π–system generated by ${\displaystyle {\mathcal {B}}}$ is a prefilter.
   4. ${\displaystyle {\mathcal {B}}}$ is a subset of some prefilter.
   5. ${\displaystyle {\mathcal {B}}}$ is a subset of some 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 the filter generated by ${\displaystyle {\mathcal {B}}}$, and ${\displaystyle {\mathcal {B}}}$ is said to be a filter subbase for this filter. This filter is equal to the intersection of all filters on ${\displaystyle X}$ that are supersets of ${\displaystyle {\mathcal {B}}.}$ The π–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, ${\displaystyle {\mathcal {B}}^{\uparrow X}={\mathcal {F}}_{\mathcal {B}}}$ if and only if