Filters in topology

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

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.

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

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]

For any two families ${\displaystyle {\mathcal {C}}{\text{ and }}{\mathcal {F}},}$ declare that ${\displaystyle {\mathcal {C}}\leq {\mathcal {F}}}$ if and only if for every ${\displaystyle C\in {\mathcal {C}}}$ there exists some ${\displaystyle F\in {\mathcal {F}}{\text{ such that }}F\subseteq C,}$ in which case it is said that ${\displaystyle {\mathcal {C}}}$ is coarser than ${\displaystyle {\mathcal {F}}}$ and that ${\displaystyle {\mathcal {F}}}$ is finer than (or subordinate to) ${\displaystyle {\mathcal {C}}.}$[10][11][12] The notation ${\displaystyle {\mathcal {F}}\vdash {\mathcal {C}}{\text{ or }}{\mathcal {F}}\geq {\mathcal {C}}}$ may also be used in place of ${\displaystyle {\mathcal {C}}\leq {\mathcal {F}}.}$

If ${\displaystyle {\mathcal {C}}\leq {\mathcal {F}}}$ and ${\displaystyle {\mathcal {F}}\leq {\mathcal {C}}}$ then ${\displaystyle {\mathcal {C}}{\text{ and }}{\mathcal {F}}}$ are said to be equivalent (with respect to subordination).

Two families ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}}$ mesh,[8] written ${\displaystyle {\mathcal {B}}\#{\mathcal {C}},}$ if ${\displaystyle B\cap C\neq \varnothing {\text{ for all }}B\in {\mathcal {B}}{\text{ and }}C\in {\mathcal {C}}.}$

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 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.

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

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 ${\displaystyle {\mathcal {B}}}$ is a prefilter (although ${\displaystyle {\mathcal {B}}^{\uparrow X}}$ is always an upward closed filter subbase 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, the π–system) generated by ${\displaystyle {\mathcal {B}}}$ is principal, in which case ${\displaystyle {\mathcal {B}}\cup \{\ker {\mathcal {B}}\}}$ is the unique smallest prefilter containing ${\displaystyle {\mathcal {B}}.}$ Otherwise, in general, a ${\displaystyle \subseteq }$ –smallest prefilter containing ${\displaystyle {\mathcal {B}}}$ might not exist. For this reason, some authors may refer to the π–system generated by ${\displaystyle {\mathcal {B}}}$ as the prefilter generated by ${\displaystyle {\mathcal {B}}.}$ However, if a ${\displaystyle \subseteq }$ –smallest prefilter does exist (say it is denoted by ${\displaystyle \operatorname {minPre} {\mathcal {B}}}$) then contrary to usual expectations, it is not necessarily equal to "the prefilter generated by ${\displaystyle {\mathcal {B}}}$" (that is, ${\displaystyle \operatorname {minPre} {\mathcal {B}}\neq \pi ({\mathcal {B}})}$ is possible). And if the filter subbase ${\displaystyle {\mathcal {B}}}$ happens to also be a prefilter but not a π-system then unfortunately, "the prefilter generated by this prefilter" (meaning ${\displaystyle \pi ({\mathcal {B}})}$) will not be ${\displaystyle {\mathcal {B}}=\operatorname {minPre} {\mathcal {B}}}$ (that is, ${\displaystyle \pi ({\mathcal {B}})\neq {\mathcal {B}}}$ is possible even when ${\displaystyle {\mathcal {B}}}$ is a prefilter), which is why this article will prefer the accurate and unambiguous terminology of "the π–system generated by ${\displaystyle {\mathcal {B}}}$".
7. Subfilter of a filter ${\displaystyle {\mathcal {F}}}$ and that ${\displaystyle {\mathcal {F}}}$ is a superfilter of ${\displaystyle {\mathcal {B}}}$[17][25] if ${\displaystyle {\mathcal {B}}}$ is a filter and ${\displaystyle {\mathcal {B}}\subseteq {\mathcal {F}}}$ where for filters, ${\displaystyle {\mathcal {B}}\subseteq {\mathcal {F}}{\text{ if and only if }}{\mathcal {B}}\leq {\mathcal {F}}.}$
• Importantly, the expression "is a superfilter of" is for filters the analog of "is a subsequence of". So despite having the prefix "sub" in common, "is a subfilter of" is actually the reverse of "is a subsequence of." However, ${\displaystyle {\mathcal {B}}\leq {\mathcal {F}}}$ can also be written ${\displaystyle {\mathcal {F}}\vdash {\mathcal {B}}}$ which is described by saying "${\displaystyle {\mathcal {F}}}$ is subordinate to ${\displaystyle {\mathcal {B}}.}$" With this terminology, "is subordinate to" becomes for filters (and also for prefilters) the analog of "is a subsequence of,"[26] which makes this one situation where using the term "subordinate" and symbol ${\displaystyle \,\vdash \,}$ 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 ${\displaystyle X\neq \varnothing }$ whenever this assumption is needed.

Basic examples

Named examples

• The singleton set ${\displaystyle {\mathcal {B}}=\{X\}}$ is called the indiscrete or trivial filter on ${\displaystyle X.}$[27][28] It is the unique minimal filter on ${\displaystyle X}$ because it is a subset of every filter on ${\displaystyle X}$; however, it need not be a subset of every prefilter on ${\displaystyle X.}$
• The dual ideal ${\displaystyle \wp (X)}$ is also called the degenerate filter on ${\displaystyle X}$[20] (despite not actually being a filter). It is the only dual ideal on ${\displaystyle X}$ that is not a filter on ${\displaystyle X.}$
• If ${\displaystyle (X,\tau )}$ is a topological space and ${\displaystyle x\in X,}$ then the neighborhood filter ${\displaystyle {\mathcal {N}}(x)}$ at ${\displaystyle x}$ is a filter on ${\displaystyle X.}$ By definition, a family ${\displaystyle {\mathcal {B}}\subseteq \wp (X)}$ is called a neighborhood basis (resp. a neighborhood subbase) at ${\displaystyle x{\text{ for }}(X,\tau )}$ if and only if ${\displaystyle {\mathcal {B}}}$ is a prefilter (resp. ${\displaystyle {\mathcal {B}}}$ is a filter subbase) and the filter on ${\displaystyle X}$ that ${\displaystyle {\mathcal {B}}}$ generates is equal to the neighborhood filter ${\displaystyle {\mathcal {N}}(x).}$ The subfamily ${\displaystyle \tau (x)\subseteq {\mathcal {N}}(x)}$ of open neighborhoods is a filter base for ${\displaystyle {\mathcal {N}}(x).}$ Both prefilters ${\displaystyle {\mathcal {N}}(x){\text{ and }}\tau (x)}$ also form a bases for topologies on ${\displaystyle X,}$ with the topology generated ${\displaystyle \tau (x)}$ being coarser than ${\displaystyle \tau .}$ This example immediately generalizes from neighborhoods of points to neighborhoods of non–empty subsets ${\displaystyle S\subseteq X.}$
• ${\displaystyle {\mathcal {B}}}$ is an elementary prefilter[29] if ${\displaystyle {\mathcal {B}}=\operatorname {Tails} \left(x_{\bullet }\right)}$ for some sequence of points ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }.}$
• ${\displaystyle {\mathcal {B}}}$ is an elementary filter or a sequential filter on ${\displaystyle X}$[30] if ${\displaystyle {\mathcal {B}}}$ is a filter on ${\displaystyle X}$ generated by some elementary prefilter. The filter of tails generated by a sequence that is not eventually constant is necessarily not an ultrafilter.[31] Every principal filter on a countable set is sequential as is every cofinite filter on a countably infinite set.[20] The intersection of finitely many sequential filters is again sequential.[20]
• The set ${\displaystyle {\mathcal {F}}}$ of all cofinite subsets of ${\displaystyle X}$ (meaning those sets whose complement in ${\displaystyle X}$ is finite) is proper if and only if ${\displaystyle {\mathcal {F}}}$ is infinite (or equivalently, ${\displaystyle X}$ is infinite), in which case ${\displaystyle {\mathcal {F}}}$ is a filter on ${\displaystyle X}$ known as the Fréchet filter or the cofinite filter on ${\displaystyle X.}$[28][27] If ${\displaystyle X}$ is finite then ${\displaystyle {\mathcal {F}}}$ is equal to the dual ideal ${\displaystyle \wp (X),}$ which is not a filter. If ${\displaystyle X}$ is infinite then the family ${\displaystyle \{X\setminus \{x\}~:~x\in X\}}$ of complements of singleton sets is a filter subbase that generates the Fréchet filter on ${\displaystyle X.}$ As with any family of sets over ${\displaystyle X}$ that contains ${\displaystyle \{X\setminus \{x\}~:~x\in X\},}$ the kernel of the Fréchet filter on ${\displaystyle X}$ is the empty set: ${\displaystyle \ker {\mathcal {F}}=\varnothing .}$
• The intersection of all elements in any non–empty family ${\displaystyle \mathbb {F} \subseteq \operatorname {Filters} (X)}$ is itself a filter on ${\displaystyle X}$ called the infimum or greatest lower bound of ${\displaystyle \mathbb {F} {\text{ in }}\operatorname {Filters} (X),}$ which is why it may be denoted by ${\displaystyle {\textstyle \bigwedge \limits _{{\mathcal {F}}\in \mathbb {F} }}{\mathcal {F}}.}$ Said differently, ${\displaystyle \ker \mathbb {F} ={\textstyle \bigcap \limits _{{\mathcal {F}}\in \mathbb {F} }}{\mathcal {F}}\in \operatorname {Filters} (X).}$ Because every filter on ${\displaystyle X}$ has ${\displaystyle \{X\}}$ as a subset, this intersection is never empty. By definition, the infimum is the finest/largest (relative to ${\displaystyle \,\subseteq \,{\text{ and }}\,\leq \,}$) filter contained as a subset of each member of ${\displaystyle \mathbb {F} .}$[28]
• If ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {F}}}$ are filters then their infimum in ${\displaystyle \operatorname {Filters} (X)}$ is the filter ${\displaystyle {\mathcal {B}}\,(\cup )\,{\mathcal {F}}.}$[9] If ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {F}}}$ are prefilters then ${\displaystyle {\mathcal {B}}\,(\cup )\,{\mathcal {F}}}$ is a prefilter that is coarser than both ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {F}}}$ (that is, ${\displaystyle {\mathcal {B}}\,(\cup )\,{\mathcal {F}}\leq {\mathcal {B}}{\text{ and }}{\mathcal {B}}\,(\cup )\,{\mathcal {F}}\leq {\mathcal {F}}}$); indeed, it is one of the finest such prefilters, meaning that if ${\displaystyle {\mathcal {S}}}$ is a prefilter such that ${\displaystyle {\mathcal {S}}\leq {\mathcal {B}}{\text{ and }}{\mathcal {S}}\leq {\mathcal {F}}}$ then necessarily ${\displaystyle {\mathcal {S}}\leq {\mathcal {B}}\,(\cup )\,{\mathcal {F}}.}$[9] More generally, if ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {F}}}$ are non−empty families and if ${\displaystyle \mathbb {S} :=\{{\mathcal {S}}\subseteq \wp (X)~:~{\mathcal {S}}\leq {\mathcal {B}}{\text{ and }}{\mathcal {S}}\leq {\mathcal {F}}\}}$ then ${\displaystyle {\mathcal {B}}\,(\cup )\,{\mathcal {F}}\in \mathbb {S} }$ and ${\displaystyle {\mathcal {B}}\,(\cup )\,{\mathcal {F}}}$ is a greatest element of ${\displaystyle (\mathbb {S} ,\leq ).}$[9]
• Let ${\displaystyle \varnothing \neq \mathbb {F} \subseteq \operatorname {DualIdeals} (X)}$ and let ${\displaystyle \cup \mathbb {F} ={\textstyle \bigcup \limits _{{\mathcal {F}}\in \mathbb {F} }}{\mathcal {F}}.}$ The supremum or least upper bound of ${\displaystyle \mathbb {F} {\text{ in }}\operatorname {DualIdeals} (X),}$ denoted by ${\displaystyle {\textstyle \bigvee \limits _{{\mathcal {F}}\in \mathbb {F} }}{\mathcal {F}},}$ is the smallest (relative to ${\displaystyle \subseteq }$) dual ideal on ${\displaystyle X}$ containing every element of ${\displaystyle \mathbb {F} }$ as a subset; that is, it is the smallest (relative to ${\displaystyle \subseteq }$) dual ideal on ${\displaystyle X}$ containing ${\displaystyle \cup \mathbb {F} }$ as a subset. This dual ideal is ${\displaystyle {\textstyle \bigvee \limits _{{\mathcal {F}}\in \mathbb {F} }}{\mathcal {F}}=\pi \left(\cup \mathbb {F} \right)^{\uparrow X},}$ where ${\displaystyle \pi \left(\cup \mathbb {F} \right):=\left\{F_{1}\cap \cdots \cap F_{n}~:~n\in \mathbb {N} {\text{ and every }}F_{i}{\text{ belongs to some }}{\mathcal {F}}\in \mathbb {F} \right\}}$ is the π–system generated by ${\displaystyle \cup \mathbb {F} .}$ As with any non–empty family of sets, ${\displaystyle \cup \mathbb {F} }$ is contained in some filter on ${\displaystyle X}$ if and only if it is a filter subbase, or equivalently, if and only if ${\displaystyle {\textstyle \bigvee \limits _{{\mathcal {F}}\in \mathbb {F} }}{\mathcal {F}}=\pi \left(\cup \mathbb {F} \right)^{\uparrow X}}$ is a filter on ${\displaystyle X,}$ in which case this family is the smallest (relative to ${\displaystyle \subseteq }$) filter on ${\displaystyle X}$ containing every element of ${\displaystyle \mathbb {F} }$ as a subset and necessarily ${\displaystyle \mathbb {F} \subseteq \operatorname {Filters} (X).}$
• Let ${\displaystyle \varnothing \neq \mathbb {F} \subseteq \operatorname {Filters} (X)}$ and let ${\displaystyle \cup \mathbb {F} ={\textstyle \bigcup \limits _{{\mathcal {F}}\in \mathbb {F} }}{\mathcal {F}}.}$ The supremum or least upper bound of ${\displaystyle \mathbb {F} {\text{ in }}\operatorname {Filters} (X),}$ denoted by ${\displaystyle {\textstyle \bigvee \limits _{{\mathcal {F}}\in \mathbb {F} }}{\mathcal {F}}}$ if it exists, is by definition the smallest (relative to ${\displaystyle \subseteq }$) filter on ${\displaystyle X}$ containing every element of ${\displaystyle \mathbb {F} }$ as a subset. If it exists then necessarily ${\displaystyle {\textstyle \bigvee \limits _{{\mathcal {F}}\in \mathbb {F} }}{\mathcal {F}}=\pi \left(\cup \mathbb {F} \right)^{\uparrow X}}$[28] (as defined above) and ${\displaystyle {\textstyle \bigvee \limits _{{\mathcal {F}}\in \mathbb {F} }}{\mathcal {F}}}$ will also be equal to the intersection of all filters on ${\displaystyle X}$ containing ${\displaystyle \cup \mathbb {F} .}$ This supremum of ${\displaystyle \mathbb {F} {\text{ in }}\operatorname {Filters} (X)}$ exists if and only if the dual ideal ${\displaystyle \pi \left(\cup \mathbb {F} \right)^{\uparrow X}}$ is a filter on ${\displaystyle X.}$ The least upper bound of a family of filters ${\displaystyle \mathbb {F} }$ may fail to be a filter.[28] Indeed, if ${\displaystyle X}$ contains at least 2 distinct elements then there exist filters ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}{\text{ on }}X}$ for which there does not exist a filter ${\displaystyle {\mathcal {F}}{\text{ on }}X}$ that contains both ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}.}$ If ${\displaystyle \cup \mathbb {F} }$ is not a filter subbase then the supremum of ${\displaystyle \mathbb {F} {\text{ in }}\operatorname {Filters} (X)}$ does not exist and the same is true of its supremum in ${\displaystyle \operatorname {Prefilters} (X)}$ but their supremum in the set of all dual ideals on ${\displaystyle X}$ will exist (it being the degenerate filter ${\displaystyle \wp (X)}$).[20]
• If ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {F}}}$ are prefilters (resp. filters on ${\displaystyle X}$) then ${\displaystyle {\mathcal {B}}\,(\cap )\,{\mathcal {F}}}$ is a prefilter (resp. a filter) if and only if it is non–degenerate (or said differently, if and only if ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {F}}}$ mesh), in which case it is one of the coarsest prefilters (resp. the coarsest filter) on ${\displaystyle X}$ that is finer (with respect to ${\displaystyle \,\leq }$) than both ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {F}};}$ this means that if ${\displaystyle {\mathcal {S}}}$ is any prefilter (resp. any filter) such that ${\displaystyle {\mathcal {B}}\leq {\mathcal {S}}{\text{ and }}{\mathcal {F}}\leq {\mathcal {S}}}$ then necessarily ${\displaystyle {\mathcal {B}}\,(\cap )\,{\mathcal {F}}\leq {\mathcal {S}},}$[9] in which case it is denoted by ${\displaystyle {\mathcal {B}}\vee {\mathcal {F}}.}$[20]

Other examples

• Let ${\displaystyle X=\{p,1,2,3\}}$ and let ${\displaystyle {\mathcal {B}}=\{\{p\},\{p,1,2\},\{p,1,3\}\},}$ which makes ${\displaystyle {\mathcal {B}}}$ a prefilter and a filter subbase that is not closed under finite intersections. Because ${\displaystyle {\mathcal {B}}}$ is a prefilter, the smallest prefilter containing ${\displaystyle {\mathcal {B}}}$ is ${\displaystyle {\mathcal {B}}.}$ The π–system generated by ${\displaystyle {\mathcal {B}}}$ is ${\displaystyle \{\{p,1\}\}\cup {\mathcal {B}}.}$ In particular, the smallest prefilter containing the filter subbase ${\displaystyle {\mathcal {B}}}$ is not equal to the set of all finite intersections of sets in ${\displaystyle {\mathcal {B}}.}$ The filter on ${\displaystyle X}$ generated by ${\displaystyle {\mathcal {B}}}$ is ${\displaystyle {\mathcal {B}}^{\uparrow X}=\{S\subseteq X:p\in S\}=\{\{p\}\cup T~:~T\subseteq \{1,2,3\}\}.}$ All three of ${\displaystyle {\mathcal {B}},}$ the π–system ${\displaystyle {\mathcal {B}}}$ generates, and ${\displaystyle {\mathcal {B}}^{\uparrow X}}$ are examples of fixed, principal, ultra prefilters that are principal at the point ${\displaystyle p;{\mathcal {B}}^{\uparrow X}}$ is also an ultrafilter on ${\displaystyle X.}$
• Let ${\displaystyle (X,\tau )}$ be a topological space, ${\displaystyle {\mathcal {B}}\subseteq \wp (X),}$ and define ${\displaystyle {\overline {\mathcal {B}}}:=\left\{\operatorname {cl} _{X}B~:~B\in {\mathcal {B}}\right\},}$ where ${\displaystyle {\mathcal {B}}}$ is necessarily finer than ${\displaystyle {\overline {\mathcal {B}}}.}$[32] If ${\displaystyle {\mathcal {B}}}$ is non–empty (resp. non–degenerate, a filter subbase, a prefilter, closed under finite unions) then the same is true of ${\displaystyle {\overline {\mathcal {B}}}.}$ If ${\displaystyle {\mathcal {B}}}$ is a filter on ${\displaystyle X}$ then ${\displaystyle {\overline {\mathcal {B}}}}$ is a prefilter but not necessarily a filter on ${\displaystyle X}$ although ${\displaystyle \left({\overline {\mathcal {B}}}\right)^{\uparrow X}}$ is a filter on ${\displaystyle X}$ equivalent to ${\displaystyle {\overline {\mathcal {B}}}.}$
• The set ${\displaystyle {\mathcal {B}}}$ of all dense open subsets of a (non–empty) topological space ${\displaystyle X}$ is a proper π–system and so also a prefilter. If the space is a Baire space, then the set of all countable intersections of dense open subsets is a π–system and a prefilter that is finer than ${\displaystyle {\mathcal {B}}.}$ If ${\displaystyle X=\mathbb {R} ^{n}}$ (with ${\displaystyle 1\leq n\in \mathbb {N} }$) then the set ${\displaystyle {\mathcal {B}}_{\operatorname {LebFinite} }}$ of all ${\displaystyle B\in {\mathcal {B}}}$ such that ${\displaystyle B}$ has finite Lebesgue measure is a proper π–system and a free prefilter that is also a proper subset of ${\displaystyle {\mathcal {B}}.}$ The prefilters ${\displaystyle {\mathcal {B}}_{\operatorname {LebFinite} }}$ and ${\displaystyle {\mathcal {B}}}$ are equivalent and so generate the same filter on ${\displaystyle X.}$ The prefilter ${\displaystyle {\mathcal {B}}_{\operatorname {LebFinite} }}$ is properly contained in, and not equivalent to, the prefilter consisting of all dense open subsets of ${\displaystyle \mathbb {R} .}$ Since ${\displaystyle X}$ is a Baire space, every countable intersection of sets in ${\displaystyle {\mathcal {B}}_{\operatorname {LebFinite} }}$ is dense in ${\displaystyle X}$ (and also comeagre and non–meager) so the set of all countable intersections of elements of ${\displaystyle {\mathcal {B}}_{\operatorname {LebFinite} }}$ is a prefilter and π–system; it is also finer than, and not equivalent to, ${\displaystyle {\mathcal {B}}_{\operatorname {LebFinite} }.}$

Ultrafilters

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

A non–empty family ${\displaystyle {\mathcal {B}}\subseteq \wp (X)}$ of sets is/is an:
1. Ultra[8][33] if ${\displaystyle \varnothing \not \in {\mathcal {B}}}$ 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 ${\displaystyle S\subseteq {\textstyle \bigcup \limits _{B\in {\mathcal {B}}}}B}$ 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. For every set ${\displaystyle S}$ (not necessarily even a subset of ${\displaystyle X}$) there exists some set ${\displaystyle B\in {\mathcal {B}}}$ such that ${\displaystyle B\cap S{\text{ equals }}B{\text{ or }}\varnothing .}$
2. Ultra prefilter[8][33] 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. ${\displaystyle {\mathcal {B}}}$ is maximal in ${\displaystyle \operatorname {Prefilters} (X)}$ with respect to ${\displaystyle \,\leq ,\,}$ which means that
${\displaystyle {\text{For all }}{\mathcal {C}}\in \operatorname {Prefilters} (X),\;{\mathcal {B}}\leq {\mathcal {C}}\;{\text{ implies }}\;{\mathcal {C}}\leq {\mathcal {B}}.}$
2. ${\displaystyle {\text{For all }}{\mathcal {C}}\in \operatorname {Filters} (X),\;{\mathcal {B}}\leq {\mathcal {C}}\;{\text{ implies }}\;{\mathcal {C}}\leq {\mathcal {B}}.}$
• 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. ${\displaystyle {\mathcal {B}}}$ is equivalent 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 ${\displaystyle \,\leq \,}$ (as above).[17]
3. Ultrafilter on ${\displaystyle X}$[8][33] 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 ${\displaystyle S\subseteq X,S\in {\mathcal {B}}{\text{ or }}X\setminus S\in {\mathcal {B}}.}$[17]
3. ${\displaystyle {\mathcal {B}}\cup (X\setminus {\mathcal {B}})=\wp (X).}$ This condition can be restated as: ${\displaystyle \wp (X)}$ 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 a prime filter).
• This property extends to any finite union of two or more sets.
5. ${\displaystyle {\mathcal {B}}}$ is a maximal 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 ${\displaystyle {\mathcal {C}}\subseteq {\mathcal {B}}{\text{ or by }}{\mathcal {C}}\leq {\mathcal {B}}}$).
• If ${\displaystyle {\mathcal {C}}}$ is upward closed then ${\displaystyle {\mathcal {B}}\leq {\mathcal {C}}{\text{ if and only if }}{\mathcal {B}}\subseteq {\mathcal {C}}.}$ So this characterization of ultrafilters as maximal filters can be restated as:
${\displaystyle {\text{For all }}{\mathcal {C}}\in \operatorname {Filters} (X),\;{\mathcal {B}}\leq {\mathcal {C}}\;{\text{ implies }}\;{\mathcal {C}}\leq {\mathcal {B}}.}$
• Because subordination ${\displaystyle \,\geq \,}$ 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 by ultranets. The ultrafilter lemma is then the statement that every filter ("net") has some subordinate filter ("subnet") that is "maximally subordinate" ("maximally deep").

The ultrafilter lemma

The following important theorem is due to Alfred Tarski (1930).[34]

The ultrafilter lemma/principle/theorem[28] (Tarski) — Every filter on a set ${\displaystyle X}$ is a subset of some ultrafilter on ${\displaystyle X.}$

A consequence of the ultrafilter lemma is that every filter is equal to the intersection of all ultrafilters containing it.[28] Assuming the axioms of Zermelo–Fraenkel (ZF), the ultrafilter lemma follows from the Axiom of choice (in particular from Zorn's lemma) but is strictly weaker than it. The ultrafilter lemma implies the Axiom of choice for finite sets. If only dealing with Hausdorff spaces, then most basic results (as encountered in introductory courses) in Topology (such as Tychonoff's theorem for compact Hausdorff spaces and the Alexander subbase theorem) and in functional analysis (such as the Hahn–Banach theorem) can be proven using only the ultrafilter lemma; the full strength of the axiom of choice might not be needed.

Kernels

The kernel is useful in classifying properties of prefilters and other families of sets.

The kernel[6] of a family of sets ${\displaystyle {\mathcal {B}}}$ is the intersection of all sets that are elements of ${\displaystyle {\mathcal {B}}:}$
${\displaystyle \ker {\mathcal {B}}=\bigcap _{B\in {\mathcal {B}}}B}$

If ${\displaystyle {\mathcal {B}}\subseteq \wp (X)}$ then ${\displaystyle \ker \left({\mathcal {B}}^{\uparrow X}\right)=\ker {\mathcal {B}}}$ and this set is also equal to the kernel of the π–system that is generated by ${\displaystyle {\mathcal {B}}.}$ In particular, if ${\displaystyle {\mathcal {B}}}$ is a filter subbase then the kernels of all of the following sets are equal:

(1) ${\displaystyle {\mathcal {B}},}$ (2) the π–system generated by ${\displaystyle {\mathcal {B}},}$ and (3) the filter generated by ${\displaystyle {\mathcal {B}}.}$

If ${\displaystyle f}$ is a map then ${\displaystyle f(\ker {\mathcal {B}})\subseteq \ker f({\mathcal {B}}){\text{ and }}f^{-1}(\ker {\mathcal {B}})=\ker f^{-1}({\mathcal {B}}).}$ Equivalent families have equal kernels. Two principal families are equivalent if and only if their kernels are equal.

Classifying families by their kernels
A family ${\displaystyle {\mathcal {B}}}$ of sets is:
1. Free[7] if ${\displaystyle \ker {\mathcal {B}}=\varnothing ,}$ or equivalently, if ${\displaystyle \{X\setminus \{x\}~:~x\in X\}\subseteq {\mathcal {B}}^{\uparrow X};}$ this can be restated as ${\displaystyle \{X\setminus \{x\}~:~x\in X\}\leq {\mathcal {B}}.}$
• A filter ${\displaystyle {\mathcal {F}}{\text{ on }}X}$ is free if and only if ${\displaystyle X}$ is infinite and ${\displaystyle {\mathcal {F}}}$ contains the Fréchet filter on ${\displaystyle X}$ as a subset.
2. Fixed if ${\displaystyle \ker {\mathcal {B}}\neq \varnothing }$ in which case, ${\displaystyle {\mathcal {B}}}$ is said to be fixed by any point ${\displaystyle x\in \ker {\mathcal {B}}.}$
• Any fixed family is necessarily a filter subbase.
3. Principal[7] if ${\displaystyle \ker {\mathcal {B}}\in {\mathcal {B}}.}$
• A proper principal family of sets is necessarily a prefilter.
4. Discrete or Principal at ${\displaystyle x\in X}$[27] if ${\displaystyle \{x\}=\ker {\mathcal {B}}\in {\mathcal {B}}.}$
• The principal filter at ${\displaystyle x{\text{ on }}X}$ is the filter ${\displaystyle \{x\}^{\uparrow X}.}$ A filter ${\displaystyle {\mathcal {F}}}$ is principal at ${\displaystyle x}$ if and only if ${\displaystyle {\mathcal {F}}=\{x\}^{\uparrow X}.}$
5. Countably deep if whenever ${\displaystyle {\mathcal {C}}\subseteq {\mathcal {B}}}$ is a countable subset then ${\displaystyle \ker {\mathcal {C}}\in {\mathcal {B}}.}$[20]

If ${\displaystyle {\mathcal {B}}}$ is a principal filter on ${\displaystyle X}$ then ${\displaystyle \varnothing \neq \ker {\mathcal {B}}\in {\mathcal {B}}}$ and ${\displaystyle {\mathcal {B}}=\{\ker {\mathcal {B}}\}^{\uparrow X}}$ and ${\displaystyle \{\ker {\mathcal {B}}\}}$ is also the smallest prefilter that generates ${\displaystyle {\mathcal {B}}.}$

Family of examples: For any non–empty ${\displaystyle C\subseteq \mathbb {R} ,}$ the family ${\displaystyle {\mathcal {B}}_{C}=\{\mathbb {R} \setminus (r+C)~:~r\in \mathbb {R} \}}$ is free but it is a filter subbase if and only if no finite union of the form ${\displaystyle \left(r_{1}+C\right)\cup \cdots \cup \left(r_{n}+C\right)}$ covers ${\displaystyle \mathbb {R} ,}$ in which case the filter that it generates will also be free. In particular, ${\displaystyle {\mathcal {B}}_{C}}$ is a filter subbase if ${\displaystyle C}$ is countable (for example, ${\displaystyle C=\mathbb {Q} ,\mathbb {Z} ,}$ the primes), a meager set in ${\displaystyle \mathbb {R} ,}$ a set of finite measure, or a bounded subset of ${\displaystyle \mathbb {R} .}$ If ${\displaystyle C}$ is a singleton set then ${\displaystyle {\mathcal {B}}_{C}}$ is a subbase for the Fréchet filter on ${\displaystyle \mathbb {R} .}$

Characterizing fixed ultra prefilters

If a family of sets ${\displaystyle {\mathcal {B}}}$ is fixed (that is, ${\displaystyle \ker {\mathcal {B}}\neq \varnothing }$) then ${\displaystyle {\mathcal {B}}}$ is ultra if and only if some element of ${\displaystyle {\mathcal {B}}}$ is a singleton set, in which case ${\displaystyle {\mathcal {B}}}$ will necessarily be a prefilter. Every principal prefilter is fixed, so a principal prefilter ${\displaystyle {\mathcal {B}}}$ is ultra if and only if ${\displaystyle \ker {\mathcal {B}}}$ is a singleton set.

Every filter on ${\displaystyle X}$ that is principal at a single point is an ultrafilter, and if in addition ${\displaystyle X}$ is finite, then there are no ultrafilters on ${\displaystyle X}$ other than these.[7]

The next theorem shows that every ultrafilter falls into one of two categories: either it is free or else it is a principal filter generated by a single point.

Proposition — If ${\displaystyle {\mathcal {F}}}$ is an ultrafilter on ${\displaystyle X}$ then the following are equivalent:

1. ${\displaystyle {\mathcal {F}}}$ is fixed, or equivalently, not free, meaning ${\displaystyle \ker {\mathcal {F}}\neq \varnothing .}$
2. ${\displaystyle {\mathcal {F}}}$ is principal, meaning ${\displaystyle \ker {\mathcal {F}}\in {\mathcal {F}}.}$
3. Some element of ${\displaystyle {\mathcal {F}}}$ is a finite set.
4. Some element of ${\displaystyle {\mathcal {F}}}$ is a singleton set.
5. ${\displaystyle {\mathcal {F}}}$ is principal at some point of ${\displaystyle X,}$ which means ${\displaystyle \ker {\mathcal {F}}=\{x\}\in {\mathcal {F}}}$ for some ${\displaystyle x\in X.}$
6. ${\displaystyle {\mathcal {F}}}$ does not contain the Fréchet filter on ${\displaystyle X.}$
7. ${\displaystyle {\mathcal {F}}}$ is sequential.[20]

Finer/coarser, subordination, and meshing

The preorder ${\displaystyle \,\leq \,}$ that is defined below is of fundamental importance for the use of prefilters (and filters) in topology. For instance, this preorder is used to define the prefilter equivalent of "subsequence",[26] where "${\displaystyle {\mathcal {F}}\geq {\mathcal {C}}}$" can be interpreted as "${\displaystyle {\mathcal {F}}}$ is a subsequence of ${\displaystyle {\mathcal {C}}}$" (so "subordinate to" is the prefilter equivalent of "subsequence of"). It is also used to define prefilter convergence in a topological space. The definition of ${\displaystyle {\mathcal {B}}}$ meshes with ${\displaystyle {\mathcal {C}},}$ which is closely related to the preorder ${\displaystyle \,\leq ,}$ is used in topology to define cluster points.

Two families of sets ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}}$ mesh[8] and are compatible, indicated by writing ${\displaystyle {\mathcal {B}}\#{\mathcal {C}},}$ if ${\displaystyle B\cap C\neq \varnothing {\text{ for all }}B\in {\mathcal {B}}{\text{ and }}C\in {\mathcal {C}}.}$ If ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}}$ do not mesh then they are dissociated. If ${\displaystyle S\subseteq X{\text{ and }}{\mathcal {B}}\subseteq \wp (X)}$ then ${\displaystyle {\mathcal {B}}{\text{ and }}S}$ are said to mesh if ${\displaystyle {\mathcal {B}}{\text{ and }}\{S\}}$ mesh, or equivalently, if the trace of ${\displaystyle {\mathcal {B}}{\text{ on }}S,}$ which is the family

${\displaystyle {\mathcal {B}}{\big \vert }_{S}=\{B\cap S~:~B\in {\mathcal {B}}\},}$
does not contain the empty set, where the trace is also called the restriction of ${\displaystyle {\mathcal {B}}{\text{ to }}S.}$

Declare that ${\displaystyle {\mathcal {C}}\leq {\mathcal {F}},{\mathcal {F}}\geq {\mathcal {C}},{\text{ and }}{\mathcal {F}}\vdash {\mathcal {C}},}$ stated as ${\displaystyle {\mathcal {C}}}$ is coarser than ${\displaystyle {\mathcal {F}}}$ and ${\displaystyle {\mathcal {F}}}$ is finer than (or subordinate to) ${\displaystyle {\mathcal {C}},}$[28][11][12][9][20] if any of the following equivalent conditions hold:
1. Definition: Every ${\displaystyle C\in {\mathcal {C}}}$ contains some ${\displaystyle F\in {\mathcal {F}}.}$ Explicitly, this means that for every ${\displaystyle C\in {\mathcal {C}},}$ there is some ${\displaystyle F\in {\mathcal {F}}}$ such that ${\displaystyle F\subseteq C}$ (thus ${\displaystyle {\mathcal {C}}\ni C\supseteq F\in {\mathcal {F}}}$ holds).
• Said more briefly in plain English, ${\displaystyle {\mathcal {C}}\leq {\mathcal {F}}}$ if every set in ${\displaystyle {\mathcal {C}}}$ is larger than some set in ${\displaystyle {\mathcal {F}}.}$ Here, a "larger set" means a superset.
2. ${\displaystyle \{C\}\leq {\mathcal {F}}{\text{ for every }}C\in {\mathcal {C}}.}$
• In words, ${\displaystyle \{C\}\leq {\mathcal {F}}}$ states exactly that