Filters in topology

Filters in topology, a subfield of mathematics, can be used to study topological spaces and define all basic topological notions such a 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 notion is more technically convenient. A preorder ${\displaystyle \,\leq \,}$ on families of sets 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 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][2] 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. 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.

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. The fundamental properties shared by neighborhood filters, which are listed below, 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 ${\displaystyle x_{\bullet }}$:

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.

It is the above 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 these sets 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 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. 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 only.

Nets vs. 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.^[3] 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,^[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 preimage or pullback ${\displaystyle f^{-1}({\mathcal {B}}):=\left\{f^{-1}(B)~:~B\in {\mathcal {B}}\right\}}$ of an arbitrary filter or prefilter ${\displaystyle {\mathcal {B}}}$ is both easily defined and guaranteed to be a prefilter, whereas it is less clear how to define the pullback of an arbitrary sequence (or net) ${\displaystyle x_{\bullet }}$ so that it is once again a sequence or net (unless ${\displaystyle f}$ is also injective and consequently a bijection, which is a stringent requirement). 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 that are used in this article. 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

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

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

Notation and Definition	Assumptions	Name
${\displaystyle \ker {\mathcal {B}}=\bigcap _{B\in {\mathcal {B}}}B}$		Kernel of ${\displaystyle {\mathcal {B}}}$[7]
${\displaystyle S\setminus {\mathcal {B}}:=\{S\setminus B~:~B\in {\mathcal {B}}\}=\{S\}\,(\setminus )\,{\mathcal {B}}}$	${\displaystyle S}$ is a set	Dual of ${\displaystyle {\mathcal {B}}{\text{ in }}S}$[8]
${\displaystyle {\mathcal {B}}{\big \vert }_{S}:=\{B\cap S~:~B\in {\mathcal {B}}\}={\mathcal {B}}\,(\cap )\,\{S\}}$	${\displaystyle S}$ is a set	Trace of ${\displaystyle {\mathcal {B}}{\text{ on }}S}$[8] or the restriction of ${\displaystyle {\mathcal {B}}{\text{ to }}S}$; 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)
${\displaystyle {\mathcal {B}}^{\#X}={\mathcal {B}}^{\#}=\{S\subseteq X~:~S\cap B\neq \varnothing {\text{ for all }}B\in {\mathcal {B}}\}}$		Grill of ${\displaystyle {\mathcal {B}}{\text{ in }}X}$[10]
${\displaystyle \wp (X)=\{S~:~S\subseteq X\}}$		Power set of a set ${\displaystyle X}$[7]

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

Notation and Definition	Assumptions	Name
${\displaystyle f^{-1}({\mathcal {B}})=\left\{f^{-1}(B)~:~B\in {\mathcal {B}}\right\}}$		Preimage of ${\displaystyle {\mathcal {B}}{\text{ under }}f}$[14]
${\displaystyle f^{-1}(S)=\{x\in \operatorname {domain} f~:~f(x)\in S\}}$	${\displaystyle S}$ is an arbitrary set.	Preimage of ${\displaystyle S{\text{ under }}f}$
${\displaystyle f({\mathcal {B}})=\{f(B)~:~B\in {\mathcal {B}}\}}$		Image of ${\displaystyle {\mathcal {B}}{\text{ under }}f}$[14]
${\displaystyle f(S)=\{f(s)~:~s\in S\cap \operatorname {domain} f\}}$	${\displaystyle S}$ is an arbitrary set.	Image of ${\displaystyle S{\text{ under }}f}$
${\displaystyle \operatorname {image} f=f(\operatorname {domain} f)}$		Image of ${\displaystyle f}$

Topology notation

Denote the set of all topologies on a set ${\displaystyle X{\text{ by }}\operatorname {Top} (X).}$ Suppose ${\displaystyle \tau \in \operatorname {Top} (X).}$

Notation and Definition	Assumptions	Name
${\displaystyle \tau (S)=\{O\in \tau ~:~S\subseteq O\}}$	${\displaystyle S\subseteq X}$	Set or prefilter[note 4] of open neighborhoods of ${\displaystyle S{\text{ in }}(X,\tau )}$
${\displaystyle \tau (x)=\{O\in \tau ~:~x\in O\}}$	${\displaystyle x\in X}$	Set or prefilter of open neighborhoods of ${\displaystyle x{\text{ in }}(X,\tau )}$
${\displaystyle {\mathcal {N}}_{\tau }(S)={\mathcal {N}}(S):=\tau (S)^{\uparrow X}}$	${\displaystyle S\subseteq X}$	Set or filter[note 4] of neighborhoods of ${\displaystyle S{\text{ in }}(X,\tau )}$
${\displaystyle {\mathcal {N}}_{\tau }(x)={\mathcal {N}}(x):=\tau (x)^{\uparrow X}}$	${\displaystyle x\in X}$	Set or filter of neighborhoods of ${\displaystyle x{\text{ in }}(X,\tau )}$

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

Nets and their tails

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 ${\displaystyle X.}$

Notation and Definition	Assumptions	Name
${\displaystyle I_{\geq i}=\{j\in I~:~i\leq j\}}$	${\displaystyle i\in I{\text{ and }}(I,\leq )}$ is a directed set	Tail or section of ${\displaystyle I}$ starting at ${\displaystyle i}$
${\displaystyle x_{\geq i}=\left\{x_{j}~:~i\leq j{\text{ and }}j\in I\right\}}$	${\displaystyle i\in I{\text{ and }}x_{\bullet }=\left(x_{i}\right)_{i\in I}}$ is a net	Tail or section of ${\displaystyle x_{\bullet }}$ starting at ${\displaystyle i}$
${\displaystyle \operatorname {Tails} \left(x_{\bullet }\right)=\left\{x_{\geq i}~:~i\in I\right\}}$	${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}$ is a net	Set or prefilter of tails/sections of ${\displaystyle x_{\bullet }.}$ Also called the eventuality filter base generated by (the tails of) ${\displaystyle x_{\bullet }.}$ If ${\displaystyle x_{\bullet }}$ is a sequence then ${\displaystyle \operatorname {Tails} \left(x_{\bullet }\right)}$ is called the sequential filter base instead.[16]
${\displaystyle \operatorname {TailsFilter} \left(x_{\bullet }\right)=\operatorname {Tails} \left(x_{\bullet }\right)^{\uparrow X}}$	${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}$ is a net	(Eventuality) filter of/generated by (tails of) ${\displaystyle x_{\bullet }}$[16]
${\displaystyle f\left(I_{\geq i}\right)=\{f(j)~:~i\leq j{\text{ and }}j\in I\}}$	${\displaystyle i\in I{\text{ and }}f~:~(I,\leq )\to X}$ is a net	Tail or section of ${\displaystyle f}$ starting at ${\displaystyle i}$[16]

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 ${\displaystyle x_{>i}=\left\{x_{j}\in I~:~i>j{\text{ and }}j\in I\right\},}$ 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,^{[note 5]} − which can be rewritten as ${\displaystyle A\cap B\supseteq C}$ (resp. as ${\displaystyle A\cup B\subseteq C}$).
   • If a family ${\displaystyle {\mathcal {B}}}$ has a greatest element with respect to ${\displaystyle \,\supseteq \,}$ (for example, if ${\displaystyle \varnothing \in {\mathcal {B}}}$) then it is necessarily directed downward.
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 ${\displaystyle R,S\subseteq X,}$ ${\displaystyle R\cap S\in {\mathcal {B}}\;{\text{ if and only if }}\;R,S\in {\mathcal {B}}.}$^[10]
   • 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 family ${\displaystyle X\setminus {\mathcal {B}}}$ should not be confused with ${\displaystyle \wp (X)\setminus {\mathcal {B}}=\{S\subseteq X~:~S\notin {\mathcal {B}}\}}$ because these two sets are not equal in general; for instance, ${\displaystyle X\setminus {\mathcal {B}}=\wp (X){\text{ if and only if }}{\mathcal {B}}=\wp (X).}$ The dual of the dual is the original family, meaning ${\displaystyle X\setminus (X\setminus {\mathcal {B}})={\mathcal {B}}.}$ The set ${\displaystyle X}$ belongs to the dual of ${\displaystyle {\mathcal {B}}}$ if and only if ${\displaystyle \varnothing \in {\mathcal {B}}.}$^[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.^[20] 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, which required non–degeneracy.
   • A dual filter on ${\displaystyle X}$ is a family ${\displaystyle {\mathcal {B}}}$ whose dual ${\displaystyle X\setminus {\mathcal {B}}}$ is a filter on ${\displaystyle X.}$ Equivalently, it is an ideal on ${\displaystyle X}$ that does not contain ${\displaystyle X}$ as an element.
4. Prefilter or filter base^[8][21] if ${\displaystyle {\mathcal {B}}\neq \varnothing }$ is proper and directed downward. Equivalently, ${\displaystyle {\mathcal {B}}}$ is 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 (with respect to ${\displaystyle \leq }$) 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]
   • 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 (with respect to ${\displaystyle \leq }$) to the π–system generated by it and both of these families generate the same filter on ${\displaystyle X.}$
6. Filter subbase^[8][22] 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 (i.e. ${\displaystyle \varnothing }$ is not an element).
   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.
   • Assuming ${\displaystyle {\mathcal {B}}}$ is a filter subbase, the filter generated by ${\displaystyle {\mathcal {B}}}$ is the unique smallest (relative to ${\displaystyle \subseteq }$) filter ${\displaystyle {\mathcal {F}}_{\mathcal {B}}{\text{ on }}X}$ containing ${\displaystyle {\mathcal {B}}.}$ It 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 the upward closure of ${\displaystyle \pi ({\mathcal {B}}),}$ meaning ${\displaystyle \pi ({\mathcal {B}})^{\uparrow X}={\mathcal {F}}_{\mathcal {B}}.}$^[9]
   • 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 ${\displaystyle {\mathcal {B}}}$ may 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, as shown in an example below, if a ${\displaystyle \subseteq }$ –smallest prefilter does exist then contrary to usual expectations, it is not necessarily equal to "the prefilter generated by ${\displaystyle {\mathcal {B}}.}$" So unfortunately, "the prefilter generated by" a prefilter ${\displaystyle {\mathcal {B}}}$ might not be ${\displaystyle {\mathcal {B}},}$ 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][23] 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,"^[24] 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 ${\displaystyle X.}$^[25][11] 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 ${\displaystyle X}$^[10] (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^[26] if ${\displaystyle {\mathcal {B}}=\operatorname {Tails} \left(x_{\bullet }\right)}$ for some sequence ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }{\text{ in }}X.}$
• ${\displaystyle {\mathcal {B}}}$ is an elementary filter or a sequential filter on ${\displaystyle X}$^[27] 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.^[28] Every principal filter on a countable set is sequential as is every cofinite filter on a countably infinite set.^[10] The intersection of finitely many sequential filters is again sequential.^[10]
• 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 ${\displaystyle X.}$^[11][25] 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 ${\displaystyle \bigwedge _{{\mathcal {F}}\in \mathbb {F} }{\mathcal {F}}.}$ Said differently, ${\displaystyle \ker \mathbb {F} =\bigcap _{{\mathcal {F}}\in \mathbb {F} }{\mathcal {F}}\in \operatorname {Filters} (X).}$ 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 ${\displaystyle \mathbb {F} .}$^[11]
  • 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 and one of the finest (with respect to ${\displaystyle \,\leq }$) prefilters coarser (with respect to ${\displaystyle \,\leq }$) than both ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {F}};}$ that is, if ${\displaystyle {\mathcal {S}}}$ is a prefilter such that ${\displaystyle {\mathcal {S}}\leq {\mathcal {B}}{\text{ and }}{\mathcal {S}}\leq {\mathcal {F}}}$ then ${\displaystyle {\mathcal {B}}\,(\cup )\,{\mathcal {F}}.}$^[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 (with respect to ${\displaystyle \leq }$) of ${\displaystyle \mathbb {S} .}$^[9]
• Let ${\displaystyle \varnothing \neq \mathbb {F} \subseteq \operatorname {DualIdeals} (X)}$ and let ${\displaystyle \cup \mathbb {F} =\bigcup _{{\mathcal {F}}\in \mathbb {F} }{\mathcal {F}}.}$ The supremum or least upper bound of ${\displaystyle \mathbb {F} {\text{ in }}\operatorname {DualIdeals} (X),}$ denoted by ${\displaystyle \bigvee _{{\mathcal {F}}\in \mathbb {F} }{\mathcal {F}},}$ 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 ${\displaystyle \bigvee _{{\mathcal {F}}\in \mathbb {F} }{\mathcal {F}}=\pi \left(\cup \mathbb {F} \right)^{\uparrow X},}$ 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 ${\displaystyle \bigvee _{{\mathcal {F}}\in \mathbb {F} }{\mathcal {F}}=\pi \left(\cup \mathbb {F} \right)^{\uparrow X}}$ 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 ${\displaystyle \cup \mathbb {F} =\bigcup _{{\mathcal {F}}\in \mathbb {F} }{\mathcal {F}}.}$ The supremum or least upper bound of ${\displaystyle \mathbb {F} {\text{ in }}\operatorname {Filters} (X),}$ denoted by ${\displaystyle \bigvee _{{\mathcal {F}}\in \mathbb {F} }{\mathcal {F}}}$ 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 ${\displaystyle \bigvee _{{\mathcal {F}}\in \mathbb {F} }{\mathcal {F}}=\pi \left(\cup \mathbb {F} \right)^{\uparrow X}}$^[11] (as defined above) and ${\displaystyle \bigvee _{{\mathcal {F}}\in \mathbb {F} }{\mathcal {F}}}$ 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.^[11] 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 ${\displaystyle \wp (X)}$).^[10]
  • 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 ${\displaystyle X}$ (with respect to ${\displaystyle \,\leq }$) 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 ${\displaystyle {\mathcal {B}}\vee {\mathcal {F}}.}$^[10]
• Let ${\displaystyle I{\text{ and }}X}$ be non−empty sets and for every ${\displaystyle i\in I}$ let ${\displaystyle {\mathcal {D}}_{i}}$ be a dual ideal on ${\displaystyle X.}$ If ${\displaystyle {\mathcal {I}}}$ is any dual ideal on ${\displaystyle I}$ then ${\displaystyle \bigcup _{\Xi \in {\mathcal {I}}}\;\;\bigcap _{i\in \Xi }\;{\mathcal {D}}_{i}}$ is a dual ideal on ${\displaystyle X}$ called Kowalsky's dual ideal or Kowalsky's filter.^[17]

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 ${\displaystyle {\overline {\mathcal {B}}}.}$^[29] 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 ${\displaystyle X=\mathbb {R} ^{n}}$ (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 prefilter that is also a proper subset of ${\displaystyle {\mathcal {B}}.}$ The prefilters ${\displaystyle {\mathcal {B}}_{\operatorname {LebFinite} }{\text{ and }}{\mathcal {B}}}$ generate the same filter on ${\displaystyle X.}$
• This example illustrates a class of a filter subbases ${\displaystyle {\mathcal {S}}_{R}}$ where all sets in both ${\displaystyle {\mathcal {S}}_{R}}$ and its generated π–system can be described as sets of the form ${\displaystyle B_{r,s},}$ so that in particular, no more than two variables (specifically, ${\displaystyle r{\text{ and }}s}$) are needed to describe the generated π–system. However, this is not typical and in general, this should not be expected of a filter subbase ${\displaystyle {\mathcal {S}}}$ that is not a π–system. More often, an intersection ${\displaystyle S_{1}\cap \cdots \cap S_{n}}$ of ${\displaystyle n}$ sets from ${\displaystyle {\mathcal {S}}}$ will usually require a description involving ${\displaystyle n}$ variables that cannot be reduced down to only two (consider, for instance, if ${\displaystyle {\mathcal {S}}=\{(0,r)\cup (r,\infty )~:~r>0\}}$). For all ${\displaystyle r,s\in \mathbb {R} ,}$ let ${\displaystyle B_{r,s}=(r,0)\cup (s,\infty ),}$ where ${\displaystyle B_{r,s}=B_{\min(r,s),s}}$ so no generality is lost by adding the assumption ${\displaystyle r\leq s.}$ For all real ${\displaystyle r\leq s{\text{ and }}u\leq v,}$ if ${\displaystyle s\geq 0{\text{ or }}v\geq 0}$ then ${\displaystyle B_{-r,s}\cap B_{-u,v}=B_{-\min(r,u),\max(s,v)}.}$^{[note 6]} For every ${\displaystyle R\subseteq \mathbb {R} ,}$ let ${\displaystyle {\mathcal {S}}_{R}=\left\{B_{-r,r}~:~r\in R\right\}}$ and let ${\displaystyle {\mathcal {B}}_{R}=\left\{B_{-r,s}~:~r\leq s{\text{ with }}r,s\in R\right\}.}$^{[note 7]} Let ${\displaystyle X=\mathbb {R} }$ and suppose ${\displaystyle \varnothing \neq R\subseteq (0,\infty )}$ is not a singleton set. Then ${\displaystyle {\mathcal {S}}_{R}}$ is a filter subbase but not a prefilter and ${\displaystyle {\mathcal {B}}_{R}}$ is the π–system it generates, so that ${\displaystyle {\mathcal {B}}_{R}^{\uparrow X}}$ is the unique smallest filter in ${\displaystyle X=\mathbb {R} }$ containing ${\displaystyle {\mathcal {S}}_{R}.}$ However, ${\displaystyle {\mathcal {S}}_{R}^{\uparrow X}}$ is not a filter on ${\displaystyle X}$ (nor is it a prefilter because it is not directed downward, although it is a filter subbase) and ${\displaystyle {\mathcal {S}}_{R}^{\uparrow X}}$ is a proper subset of the filter ${\displaystyle {\mathcal {B}}_{R}^{\uparrow X}.}$ If ${\displaystyle R,S\subseteq (0,\infty )}$ are non−empty intervals then the filter subbases ${\displaystyle {\mathcal {S}}_{R}{\text{ and }}{\mathcal {S}}_{S}}$ generate the same filter on ${\displaystyle X}$ if and only if ${\displaystyle R=S.}$ If ${\displaystyle {\mathcal {C}}}$ is a family such that ${\displaystyle {\mathcal {S}}_{(0,\infty )}\subseteq {\mathcal {C}}\subseteq {\mathcal {B}}_{(0,\infty )}}$ then ${\displaystyle {\mathcal {C}}}$ is a prefilter if and only if for all real ${\displaystyle 0<r\leq s}$ there exist real ${\displaystyle 0<u\leq v}$ such that ${\displaystyle u\leq r\leq s\leq v{\text{ and }}B_{-u,v}\in {\mathcal {C}}.}$ If ${\displaystyle {\mathcal {C}}}$ is such a prefilter then for any ${\displaystyle C\in {\mathcal {C}}\setminus {\mathcal {S}}_{(0,\infty )},}$ the family ${\displaystyle {\mathcal {C}}\setminus \{C\}}$ is also a prefilter satisfying ${\displaystyle {\mathcal {S}}_{(0,\infty )}\subseteq {\mathcal {C}}\setminus \{C\}\subseteq {\mathcal {B}}_{(0,\infty )}.}$ This shows that there cannot exist a minimal (with respect to ${\displaystyle \subseteq }$) prefilter that both contains ${\displaystyle {\mathcal {S}}_{(0,\infty )}}$ and is a subset of the π–system generated by ${\displaystyle {\mathcal {S}}_{(0,\infty )}.}$ This remains true even if the requirement that the prefilter be a subset of ${\displaystyle {\mathcal {B}}_{(0,\infty )}}$ is removed.

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:

8. Ultra^[8][30] 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 ${\displaystyle S\subseteq \bigcup _{B\in {\mathcal {B}}}B}$ there exists some set ${\displaystyle B\in {\mathcal {B}}}$ such that ${\displaystyle B\cap S{\text{ equals }}B{\text{ or }}\varnothing .}$
      • This characterization of "${\displaystyle {\mathcal {B}}}$ is ultra" does not depend on the set ${\displaystyle X,}$ so mentioning the set ${\displaystyle X}$ is optional when using the term "ultra."
   3. For every set ${\displaystyle S}$ (not necessarily even a subset of ${\displaystyle X}$) there exists some set ${\displaystyle B\in {\mathcal {B}}}$ such that ${\displaystyle B\cap S{\text{ equals }}B{\text{ or }}\varnothing .}$
      • If ${\displaystyle {\mathcal {B}}}$ satisfies this condition then so does every superset ${\displaystyle {\mathcal {F}}\supseteq {\mathcal {B}}.}$ For example, if ${\displaystyle T}$ is any singleton set then ${\displaystyle \{T\}}$ is ultra and consequently, any non–degenerate superset of ${\displaystyle \{T\}}$ (such as its upward closure) is also ultra.
9. Ultra prefilter^[8][30] 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 (with respect to ${\displaystyle \leq }$) 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]
10. Ultrafilter on ${\displaystyle X}$^[8][30] 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 ${\displaystyle X\setminus {\mathcal {B}}.}$
       • The sets ${\displaystyle {\mathcal {B}}{\text{ and }}X\setminus {\mathcal {B}}}$ are disjoint whenever ${\displaystyle {\mathcal {B}}}$ is a prefilter.
    4. ${\displaystyle \wp (X)\setminus {\mathcal {B}}=\{S\in \wp (X):S\not \in {\mathcal {B}}\}}$ is an ideal.^[17]
    5. For any ${\displaystyle R,S\subseteq X,}$ if ${\displaystyle R\cup S=X}$ then ${\displaystyle R\in {\mathcal {B}}{\text{ or }}S\in {\mathcal {B}}.}$
    6. 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.
    7. For any ${\displaystyle R,S\subseteq X,}$ if ${\displaystyle R\cup S\in {\mathcal {B}}{\text{ and }}R\cap S=\varnothing }$ then either ${\displaystyle R\in {\mathcal {B}}{\text{ or }}S\in {\mathcal {B}}.}$
    8. ${\displaystyle {\mathcal {B}}}$ 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 ${\displaystyle \geq }$ is for filters the analog of "is a subnet of," (specifically, "subnet" should mean "AA–subnet," which is defined below) an ultrafilter can be interpreted as being analogous to some sort of "maximally deep net." This idea is actually made rigorous by ultranets.

Any non–degenerate family that has a singleton set as an element is ultra, in which case it will then be an ultra prefilter if and only if it also has the finite intersection property. The trivial filter ${\displaystyle \{X\}{\text{ on }}X}$ is ultra if and only if ${\displaystyle X}$ is a singleton set.

The ultrafilter lemma

The following important theorem is due to Alfred Tarski (1930).^[31]

A consequence of the ultrafilter lemma is that every filter is equal to the intersection of all ultrafilters containing it.^{[11][proof 1]} 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 for any point ${\displaystyle x,x\not \in \ker {\mathcal {B}}{\text{ if and only if }}X\setminus \{x\}\in {\mathcal {B}}^{\uparrow X}.}$

Properties of kernels

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 ${\displaystyle f(\ker {\mathcal {B}})\subseteq \ker f({\mathcal {B}}){\text{ and }}f^{-1}(\ker {\mathcal {B}})=\ker f^{-1}({\mathcal {B}}).}$ If ${\displaystyle {\mathcal {B}}\leq {\mathcal {C}}{\text{ then }}\ker {\mathcal {C}}\subseteq \ker {\mathcal {B}}}$ while if ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}}$ are equivalent then ${\displaystyle \ker {\mathcal {B}}=\ker {\mathcal {C}}.}$ If ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}}$ are principal then they are equivalent if and only if ${\displaystyle \ker {\mathcal {B}}=\ker {\mathcal {C}}.}$

Classifying families by their kernels[]

A family ${\displaystyle {\mathcal {B}}}$ of sets is/is an:

2. 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.
3. 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.
4. Principal^[7] if ${\displaystyle \ker {\mathcal {B}}\in {\mathcal {B}}.}$
   • A proper principal family of sets is necessarily a prefilter.
5. Discrete or Principal at ${\displaystyle x\in X}$^[25] 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 ${\displaystyle {\mathcal {F}}=\{x\}^{\uparrow X}.}$
6. Countably deep if whenever ${\displaystyle {\mathcal {C}}\subseteq {\mathcal {F}}}$ is a countable subset then ${\displaystyle \ker {\mathcal {C}}\in {\mathcal {B}}.}$^[10]

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 ${\displaystyle \mathbb {R} .}$

For every filter ${\displaystyle {\mathcal {F}}{\text{ on }}X}$ there exists a unique pair of dual ideals ${\displaystyle {\mathcal {F}}^{*}{\text{ and }}{\mathcal {F}}^{\bullet }{\text{ on }}X}$ such that ${\displaystyle {\mathcal {F}}^{*}}$ is free, ${\displaystyle {\mathcal {F}}^{\bullet }}$ is principal, and ${\displaystyle {\mathcal {F}}^{*}\wedge {\mathcal {F}}^{\bullet }={\mathcal {F}},}$ and ${\displaystyle {\mathcal {F}}^{*}{\text{ and }}{\mathcal {F}}^{\bullet }}$ do not mesh (that is, ${\displaystyle {\mathcal {F}}^{*}\vee {\mathcal {F}}^{\bullet }=\wp (X)}$). The dual ideal ${\displaystyle {\mathcal {F}}^{*}}$ is called the free part of ${\displaystyle {\mathcal {F}}}$ while ${\displaystyle {\mathcal {F}}^{\bullet }}$ is called the principal part^[10] where at least one of these dual ideals is filter. If ${\displaystyle {\mathcal {F}}}$ is principal then ${\displaystyle {\mathcal {F}}^{\bullet }:={\mathcal {F}}{\text{ and }}{\mathcal {F}}^{*}:=\wp (X);}$ otherwise, ${\displaystyle {\mathcal {F}}^{\bullet }:=\{\ker {\mathcal {F}}\}^{\uparrow X}}$ and ${\displaystyle {\mathcal {F}}^{*}:={\mathcal {F}}\vee \{X\setminus \left(\ker {\mathcal {F}}\right)\}^{\uparrow X}}$ is a free (non–degenerate) filter.^[10]

Finite prefilters and finite sets

If a filter subbase ${\displaystyle {\mathcal {B}}}$ is finite then it is fixed (that is, not free); this is because ${\displaystyle \ker {\mathcal {B}}=\bigcap _{B\in {\mathcal {B}}}B}$ is a finite intersection and the filter subbase ${\displaystyle {\mathcal {B}}}$ has the finite intersection property. A finite prefilter is necessarily principal, although it does not have to be closed under finite intersections.

If ${\displaystyle X}$ is finite then all of the conclusions above hold for any ${\displaystyle {\mathcal {B}}\subseteq \wp (X).}$ In particular, on a finite set ${\displaystyle X,}$ there are no free filter subbases (or prefilters), all prefilters are principal, and all filters on ${\displaystyle X}$ are principal filters generated by their (non–empty) kernels.

The trivial filter ${\displaystyle \{X\}}$ is always a finite filter on ${\displaystyle X}$ and if ${\displaystyle X}$ is infinite then it is the only finite filter because a non–trivial finite filter on a set ${\displaystyle X}$ is possible if and only if ${\displaystyle X}$ is finite. However, on any infinite set there are non–trivial filter subbases and prefilters that are finite (although they cannot be filters). If ${\displaystyle X}$ is a singleton set then the trivial filter ${\displaystyle \{X\}}$ is the only proper subset of ${\displaystyle \wp (X).}$ This set ${\displaystyle \{X\}}$ is a principal ultra prefilter and any superset ${\displaystyle {\mathcal {F}}\supseteq {\mathcal {B}}}$ (where ${\displaystyle {\mathcal {F}}\subseteq \wp (Y){\text{ and }}X\subseteq Y}$) with the finite intersection property will also be a principal ultra prefilter (even if ${\displaystyle Y}$ is infinite).

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.

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",^[24] 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 be 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

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) ${\displaystyle {\mathcal {C}},}$^{[11][12][13][9][10]} 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 ${\displaystyle F\subseteq C.}$
   • 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 ${\displaystyle C}$ is larger than some set in ${\displaystyle {\mathcal {F}}.}$ The equivalence of (a) and (b) follows immediately.
   • From this characterization, it follows that if ${\displaystyle \left({\mathcal {C}}_{i}\right)_{i\in I}}$ are families of sets, then ${\displaystyle \bigcup _{i\in I}{\mathcal {C}}_{i}\leq {\mathcal {F}}{\text{ if and only if }}{\mathcal {C}}_{i}\leq {\mathcal {F}}{\text{ for all }}i\in I.}$
3. ${\displaystyle {\mathcal {C}}\leq {\mathcal {F}}^{\uparrow X},}$ which is equivalent to ${\displaystyle {\mathcal {C}}\subseteq {\mathcal {F}}^{\uparrow X}}$;
4. ${\displaystyle {\mathcal {C}}^{\uparrow X}\leq {\mathcal {F}}}$;
5. ${\displaystyle {\mathcal {C}}^{\uparrow X}\leq {\mathcal {F}}^{\uparrow X},}$ which is equivalent to ${\displaystyle {\mathcal {C}}^{\uparrow X}\subseteq {\mathcal {F}}^{\uparrow X}}$;

and if in addition ${\displaystyle {\mathcal {F}}}$ is upward closed, which means that ${\displaystyle {\mathcal {F}}={\mathcal {F}}^{\uparrow X},}$ then this list can be extended to include:

6. ${\displaystyle {\mathcal {C}}\subseteq {\mathcal {F}}.}$^[6]
   • So in this case, this definition of "${\displaystyle {\mathcal {F}}}$ is finer than ${\displaystyle {\mathcal {C}}}$" would be identical to the topological definition of "finer" had ${\displaystyle {\mathcal {C}}{\text{ and }}{\mathcal {F}}}$ been topologies on ${\displaystyle X.}$

If an upward closed family ${\displaystyle {\mathcal {F}}}$ is finer than ${\displaystyle {\mathcal {C}}}$ (that is, ${\displaystyle {\mathcal {C}}\leq {\mathcal {F}}}$) but ${\displaystyle {\mathcal {C}}\neq {\mathcal {F}}}$ then ${\displaystyle {\mathcal {F}}}$ is said to be strictly finer than ${\displaystyle {\mathcal {C}}}$ and ${\displaystyle {\mathcal {C}}}$ is strictly coarser than ${\displaystyle {\mathcal {F}}.}$

Two families are comparable if one of these sets is finer than the other.^[11]

For example, if ${\displaystyle {\mathcal {B}}}$ is any family then ${\displaystyle \varnothing \leq {\mathcal {B}}\leq {\mathcal {B}}\leq \{\varnothing \}}$ always holds and furthermore, ${\displaystyle \{\varnothing \}\leq {\mathcal {B}}{\text{ if and only if }}\varnothing \in {\mathcal {B}}.}$

Assume that ${\displaystyle {\mathcal {C}}{\text{ and }}{\mathcal {F}}}$ are families of sets that satisfy ${\displaystyle {\mathcal {B}}\leq {\mathcal {F}}{\text{ and }}{\mathcal {C}}\leq {\mathcal {F}}.}$ Then ${\displaystyle \ker {\mathcal {F}}\subseteq \ker {\mathcal {C}},}$ and ${\displaystyle {\mathcal {C}}\neq \varnothing {\text{ implies }}{\mathcal {F}}\neq \varnothing ,}$ and also ${\displaystyle \varnothing \in {\mathcal {C}}{\text{ implies }}\varnothing \in {\mathcal {F}}.}$ If in addition to ${\displaystyle {\mathcal {C}}\leq {\mathcal {F}},{\mathcal {F}}}$ is a filter subbase and ${\displaystyle {\mathcal {C}}\neq \varnothing ,}$ then ${\displaystyle {\mathcal {C}}}$ is a filter subbase^[9] and also ${\displaystyle {\mathcal {C}}{\text{ and }}{\mathcal {F}}}$ mesh.^{[19][proof 2]} More generally, if both ${\displaystyle \varnothing \neq {\mathcal {B}}\leq {\mathcal {F}}{\text{ and }}\varnothing \neq {\mathcal {C}}\leq {\mathcal {F}}}$ and if the intersection of any two elements of ${\displaystyle {\mathcal {F}}}$ is non–empty, then ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}}$ mesh.^{[proof 2]} Every filter subbase is coarser than both the π–system that it generates and the filter that it generates.^[9]

If ${\displaystyle {\mathcal {C}}{\text{ and }}{\mathcal {F}}}$ are families such that ${\displaystyle {\mathcal {C}}\leq {\mathcal {F}},}$ the family ${\displaystyle {\mathcal {C}}}$ is ultra, and ${\displaystyle \varnothing \not \in {\mathcal {F}},}$ then ${\displaystyle {\mathcal {F}}}$ is necessarily ultra. It follows that any family that is equivalent to an ultra family will necessarily be ultra. In particular, if ${\displaystyle {\mathcal {C}}}$ is a prefilter then either both ${\displaystyle {\mathcal {C}}}$ and the filter ${\displaystyle {\mathcal {C}}^{\uparrow X}}$ it generates are ultra or neither one is ultra. If a filter subbase is ultra then it is necessarily a prefilter, in which case the filter that it generates will also be ultra. A filter subbase ${\displaystyle {\mathcal {B}}}$ that is not a prefilter cannot be ultra; but it is nevertheless still possible for the prefilter and filter generated by ${\displaystyle {\mathcal {B}}}$ to be ultra. If ${\displaystyle S\subseteq X{\text{ and }}{\mathcal {B}}\subseteq \wp (X)}$ is upward closed in ${\displaystyle X}$ then ${\displaystyle S\not \in {\mathcal {B}}{\text{ if and only if }}(X\setminus S)\#{\mathcal {B}}.}$^[10]

Relational properties of subordination

The relation ${\displaystyle \,\leq \,}$ is reflexive and transitive, which makes it into a preorder on ${\displaystyle \wp (\wp (X)).}$^[32]

Symmetry: For any ${\displaystyle {\mathcal {B}}\subseteq \wp (X),{\mathcal {B}}\leq \{X\}{\text{ if and only if }}\{X\}={\mathcal {B}}.}$ So the set ${\displaystyle X}$ has more than one point if and only if the relation ${\displaystyle \,\leq \,{\text{ on }}\operatorname {Filters} (X)}$ is not symmetric.

Antisymmetry: If ${\displaystyle {\mathcal {B}}\subseteq {\mathcal {C}}{\text{ then }}{\mathcal {B}}\leq {\mathcal {C}}}$ but while the converse does not hold in general, it does hold if ${\displaystyle {\mathcal {C}}}$ is upward closed (such as if ${\displaystyle {\mathcal {C}}}$ is a filter). Two filters are equivalent if and only if they are equal, which makes the restriction of ${\displaystyle \,\leq \,}$ to ${\displaystyle \operatorname {Filters} (X)}$ antisymmetric. But in general, ${\displaystyle \,\leq \,}$ is not antisymmetric on ${\displaystyle \operatorname {Prefilters} (X)}$ nor on ${\displaystyle \wp (\wp (X))}$; that is, ${\displaystyle {\mathcal {C}}\leq {\mathcal {B}}{\text{ and }}{\mathcal {B}}\leq {\mathcal {C}}}$ does not necessarily imply ${\displaystyle {\mathcal {B}}={\mathcal {C}}}$; not even if both ${\displaystyle {\mathcal {C}}{\text{ and }}{\mathcal {B}}}$ are prefilters.^[13] For instance, if ${\displaystyle {\mathcal {B}}}$ is a prefilter but not a filter then ${\displaystyle {\mathcal {B}}\leq {\mathcal {B}}^{\uparrow X}{\text{ and }}{\mathcal {B}}^{\uparrow X}\leq {\mathcal {B}}{\text{ but }}{\mathcal {B}}\neq {\mathcal {B}}^{\uparrow X}.}$

Equivalent families of sets[]

The preorder ${\displaystyle \,\leq \,}$ induces its canonical equivalence relation on ${\displaystyle \wp (\wp (X)),}$ where for all ${\displaystyle {\mathcal {B}},{\mathcal {C}}\in \wp (\wp (X)),}$ ${\displaystyle {\mathcal {B}}}$ is equivalent to ${\displaystyle {\mathcal {C}}}$ if any of the following equivalent conditions hold:^[9][6]

1. ${\displaystyle {\mathcal {C}}\leq {\mathcal {B}}{\text{ and }}{\mathcal {B}}\leq {\mathcal {C}}.}$
2. The upward closures of ${\displaystyle {\mathcal {C}}{\text{ and }}{\mathcal {B}}}$ are equal.

Two upward closed (in ${\displaystyle X}$) subsets of ${\displaystyle \wp (X)}$ are equivalent if and only if they are equal.^[9] If ${\displaystyle {\mathcal {B}}\subseteq \wp (X)}$ then necessarily ${\displaystyle \varnothing \leq {\mathcal {B}}\leq \wp (X)}$ and ${\displaystyle {\mathcal {B}}}$ is equivalent to ${\displaystyle {\mathcal {B}}^{\uparrow X}.}$ Every equivalence class other than ${\displaystyle \{\varnothing \}}$ contains a unique representative (that is, element of the equivalence class) that is upward closed in ${\displaystyle X.}$^[9]

Properties preserved between equivalent families

Let ${\displaystyle {\mathcal {B}},{\mathcal {C}}\in \wp (\wp (X))}$ be arbitrary and let ${\displaystyle {\mathcal {F}}}$ be any family of sets. If ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}}$ are equivalent (which implies that ${\displaystyle \ker {\mathcal {B}}=\ker {\mathcal {C}}}$) then for each of the statements/properties listed below, either it is true of both ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}}$ or else it is false of both ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}}$:^[32]

1. Not empty
2. Proper (that is, ${\displaystyle \varnothing }$ is not an element)
   • Moreover, any two degenerate families are necessarily equivalent.
3. Filter subbase
4. Prefilter
   • In which case ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}}$ generate the same filter on ${\displaystyle X}$ (that is, their upward closures in ${\displaystyle X}$ are equal).
5. Free
6. Principal
7. Ultra
8. Is equal to the trivial filter ${\displaystyle \{X\}}$
   • In words, this means that the only subset of ${\displaystyle \wp (X)}$ that is equivalent to the trivial filter is the trivial filter. In general, this conclusion of equality does not extend to non−trivial filters (one exception is when both families are filters).
9. Meshes with ${\displaystyle {\mathcal {F}}}$
10. Is finer than ${\displaystyle {\mathcal {F}}}$
11. Is coarser than ${\displaystyle {\mathcal {F}}}$
12. Is equivalent to ${\displaystyle {\mathcal {F}}}$

Missing from the above list is the word "filter" because this property is not preserved by equivalence. However, if ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}}$ are filters on ${\displaystyle X,}$ then they are equivalent if and only if they are equal; this characterization does not extend to prefilters.

Equivalence of prefilters and filter subbases

If ${\displaystyle {\mathcal {B}}}$ is a prefilter on ${\displaystyle X}$ then the following families are always equivalent to each other:

1. ${\displaystyle {\mathcal {B}}}$;
2. the π–system generated by ${\displaystyle {\mathcal {B}}}$;
3. the filter on ${\displaystyle X}$ generated by ${\displaystyle {\mathcal {B}}}$;

and moreover, these three families all generate the same filter on ${\displaystyle X}$ (that is, the upward closures in ${\displaystyle X}$ of these families are equal).

In particular, every prefilter is equivalent to the filter that it generates. By transitivity, two prefilters are equivalent if and only if they generate the same filter.^{[9][proof 3]} Every prefilter is equivalent to exactly one filter on ${\displaystyle X,}$ which is the filter that it generates (that is, the prefilter's upward closure). Said differently, every equivalence class of prefilters contains exactly one representative that is a filter. In this way, filters can be considered as just being distinguished elements of these equivalence classes of prefilters.^[9]

A filter subbase that is not also a prefilter cannot be equivalent to the prefilter (or filter) that it generates. In contrast, every prefilter is equivalent to the filter that it generates. This is why prefilters can, by and large, be used interchangeably with the filters that they generate while filter subbases cannot. Every filter is both a π–system and a ring of sets.

Examples of determining equivalence/non–equivalence

Examples: Let ${\displaystyle X=\mathbb {R} }$ and let ${\displaystyle E}$ be the set ${\displaystyle \mathbb {Z} }$ of integers (or the set ${\displaystyle \mathbb {N} }$). Define the sets

All three sets are filter subbases but none are filters on ${\displaystyle X}$ and only ${\displaystyle {\mathcal {B}}}$ is prefilter (in fact, ${\displaystyle {\mathcal {B}}}$ is even free and closed under finite intersections). The set ${\displaystyle {\mathcal {C}}_{\operatorname {closed} }}$ is fixed while ${\displaystyle {\mathcal {C}}_{\operatorname {open} }}$ is free (unless ${\displaystyle E=\mathbb {N} }$). They satisfy ${\displaystyle {\mathcal {C}}_{\operatorname {closed} }\leq {\mathcal {C}}_{\operatorname {open} }\leq {\mathcal {B}},}$ but no two of these families are equivalent; moreover, no two of the filters generated by these three filter subbases are equivalent/equal. This conclusion can be reached by showing that the π–systems that they generate are not equivalent. Unlike with ${\displaystyle {\mathcal {C}}_{\operatorname {open} },}$ every set in the π–system generated by ${\displaystyle {\mathcal {C}}_{\operatorname {closed} }}$ contains ${\displaystyle \mathbb {Z} }$ as a subset,^{[note 8]} which is what prevents their generated π–systems (and hence their generated filters) from being equivalent. If ${\displaystyle E}$ was instead ${\displaystyle \mathbb {Q} {\text{ or }}\mathbb {R} }$ then all three families would be free and although the sets ${\displaystyle {\mathcal {C}}_{\operatorname {closed} }{\text{ and }}{\mathcal {C}}_{\operatorname {open} }}$ would remain not equivalent to each other, their generated π–systems would be equivalent and consequently, they would generate the same filter on ${\displaystyle X}$; however, this common filter would still be strictly coarser than the filter generated by ${\displaystyle {\mathcal {B}}.}$

Set theoretic properties and constructions relevant to topology[]

Trace and meshing[]

If ${\displaystyle {\mathcal {B}}}$ is a prefilter (resp. filter) on ${\displaystyle X{\text{ and }}S\subseteq X}$ then the trace of ${\displaystyle {\mathcal {B}}{\text{ on }}S,}$ which is the family ${\displaystyle {\mathcal {B}}{\big \vert }_{S}:={\mathcal {B}}(\cap )\{S\},}$ is a prefilter (resp. a filter) if and only if ${\displaystyle {\mathcal {B}}{\text{ and }}S}$ mesh (that is, ${\displaystyle \varnothing \not \in {\mathcal {B}}(\cap )\{S\}}$^[11]), in which case the trace of ${\displaystyle {\mathcal {B}}{\text{ on }}S}$ is said to be induced by ${\displaystyle S}$. If ${\displaystyle {\mathcal {B}}}$ is ultra and if ${\displaystyle {\mathcal {B}}{\text{ and }}S}$ mesh then the trace ${\displaystyle {\mathcal {B}}{\big \vert }_{S}}$ is ultra. If ${\displaystyle {\mathcal {B}}}$ is an ultrafilter on ${\displaystyle X}$ then the trace of ${\displaystyle {\mathcal {B}}{\text{ on }}S}$ is a filter on ${\displaystyle S}$ if and only if ${\displaystyle S\in {\mathcal {B}}.}$

For example, suppose that ${\displaystyle {\mathcal {B}}}$ is a filter on ${\displaystyle X{\text{ and }}S\subseteq X}$ is such that ${\displaystyle S\neq X{\text{ and }}X\setminus S\not \in {\mathcal {B}}.}$ Then ${\displaystyle {\mathcal {B}}{\text{ and }}S}$ mesh and ${\displaystyle {\mathcal {B}}\cup \{S\}}$ generates a filter on ${\displaystyle X}$ that is strictly finer than ${\displaystyle {\mathcal {B}}.}$^[11]

When prefilters mesh

Given non–empty families ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}},}$ the family

satisfies ${\displaystyle {\mathcal {C}}\leq {\mathcal {B}}(\cap ){\mathcal {C}}}$ and ${\displaystyle {\mathcal {B}}\leq {\mathcal {B}}(\cap ){\mathcal {C}}.}$ If ${\displaystyle {\mathcal {B}}(\cap ){\mathcal {C}}}$ is proper (resp. a prefilter, a filter subbase) then this is also true of both ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}.}$ In order to make any meaningful deductions about ${\displaystyle {\mathcal {B}}(\cap ){\mathcal {C}}}$ from ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}},{\mathcal {B}}(\cap ){\mathcal {C}}}$ needs to be proper (that is, ${\displaystyle \varnothing \not \in {\mathcal {B}}(\cap ){\mathcal {C}},}$ which is the motivation for the definition of "mesh". In this case, ${\displaystyle {\mathcal {B}}(\cap ){\mathcal {C}}}$ is a prefilter (resp. filter subbase) if and only if this is true of both ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}.}$ Said differently, if ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}}$ are prefilters then they mesh if and only if ${\displaystyle {\mathcal {B}}(\cap ){\mathcal {C}}}$ is a prefilter. Generalizing gives a well known characterization of "mesh" entirely in terms of subordination (that is, ${\displaystyle \,\leq \,}$):

     Two prefilters (resp. filter subbases) ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}}$ mesh if and only if there exists a prefilter (resp. filter subbase) ${\displaystyle {\mathcal {B}}(\cap ){\mathcal {C}}}$ such that ${\displaystyle {\mathcal {C}}\leq {\mathcal {B}}(\cap ){\mathcal {C}}}$ and ${\displaystyle {\mathcal {B}}\leq {\mathcal {B}}(\cap ){\mathcal {C}}.}$

If the least upper bound of two filters ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}}$ exists in ${\displaystyle \operatorname {Filters} (X)}$ then this least upper bound is equal to ${\displaystyle {\mathcal {B}}(\cap ){\mathcal {C}}.}$^[28]

Images and preimages under functions[]

Throughout, ${\displaystyle f:X\to Y{\text{ and }}g:Y\to Z}$ will be maps between non–empty sets.

Images of prefilters

Let ${\displaystyle {\mathcal {B}}\subseteq \wp (Y).}$ Many of the properties that ${\displaystyle {\mathcal {B}}}$ may have are preserved under images of maps; notable exceptions include being upward closed, being closed under finite intersections, and being a filter, which are not necessarily preserved.

Explicitly, if one of the following properties is true of ${\displaystyle {\mathcal {B}}{\text{ on }}Y,}$ then it will necessarily also be true of ${\displaystyle g({\mathcal {B}}){\text{ on }}g(Y)}$ (although possibly not on the codomain ${\displaystyle Z}$ unless ${\displaystyle g}$ is surjective):^{[11][14][33][34][35][31]}

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

Moreover, if ${\displaystyle {\mathcal {B}}\subseteq \wp (Y)}$ is a prefilter then so are both ${\displaystyle g({\mathcal {B}}){\text{ and }}g^{-1}(g({\mathcal {B}})).}$^[11] The image under a map ${\displaystyle f:X\to Y}$ of an ultra set ${\displaystyle {\mathcal {B}}\subseteq \wp (X)}$ is again ultra and if ${\displaystyle {\mathcal {B}}}$ is an ultra prefilter then so is ${\displaystyle f({\mathcal {B}}).}$

If ${\displaystyle {\mathcal {B}}}$ is a filter then ${\displaystyle g({\mathcal {B}})}$ is a filter on the range ${\displaystyle g(Y),}$ but it is a filter on the codomain ${\displaystyle Z}$ if and only if ${\displaystyle g}$ is surjective.^[33] Otherwise it is just a prefilter on ${\displaystyle Z}$ and its upward closure must be taken in ${\displaystyle Z}$ to obtain a filter. The upward closure of ${\displaystyle g({\mathcal {B}}){\text{ in }}Z}$ is

where if ${\displaystyle {\mathcal {B}}}$ is upward closed in ${\displaystyle Y}$ (that is, a filter) then this simplifies to:

If ${\displaystyle X\subseteq Y}$ then taking ${\displaystyle g}$ to be the inclusion map ${\displaystyle X\to Y}$ shows that any prefilter (resp. ultra prefilter, filter subbase) on ${\displaystyle X}$ is also a prefilter (resp. ultra prefilter, filter subbase) on ${\displaystyle Y.}$^[11]

Preimages of prefilters

Let ${\displaystyle {\mathcal {B}}\subseteq \wp (Y).}$ Under the assumption that ${\displaystyle f:X\to Y}$ is surjective:

     ${\displaystyle f^{-1}({\mathcal {B}})}$ is a prefilter (resp. filter subbase, π–system, closed under finite unions, proper) if and only if this is true of ${\displaystyle {\mathcal {B}}.}$

However, if ${\displaystyle {\mathcal {B}}}$ is an ultrafilter on ${\displaystyle Y}$ then even if ${\displaystyle f}$ is surjective (which would make ${\displaystyle f^{-1}({\mathcal {B}})}$ a prefilter), it is nevertheless still possible for the prefilter ${\displaystyle f^{-1}({\mathcal {B}})}$ to be neither ultra nor a filter on ${\displaystyle X}$ ^[34] (see this^{[note 9]} footnote for an example).

If ${\displaystyle f:X\to Y}$ is not surjective then denote the trace of ${\displaystyle {\mathcal {B}}{\text{ on }}f(X)}$ by ${\displaystyle {\mathcal {B}}{\big \vert }_{f(X)},}$ where in this case particular case the trace satisfies:

and consequently also:

This last equality and the fact that the trace ${\displaystyle {\mathcal {B}}{\big \vert }_{f(X)}}$ is a family of sets over ${\displaystyle f(X)}$ means that to draw conclusions about ${\displaystyle f^{-1}({\mathcal {B}}),}$ the trace ${\displaystyle {\mathcal {B}}{\big \vert }_{f(X)}}$ can be used in place of ${\displaystyle {\mathcal {B}}}$ and the surjection ${\displaystyle f:X\to f(X)}$ can be used in place of ${\displaystyle f:X\to Y.}$ For example:^[14][11][35]

     ${\displaystyle f^{-1}({\mathcal {B}})}$ is a prefilter (resp. filter subbase, π–system, proper) if and only if this is true of ${\displaystyle {\mathcal {B}}{\big \vert }_{f(X)}.}$

In this way, the case where ${\displaystyle f}$ is not (necessarily) surjective can be reduced down to the case of a surjective function (which is a case that was described at the start of this subsection).

Even if ${\displaystyle {\mathcal {B}}}$ is an ultrafilter on ${\displaystyle Y,}$ if ${\displaystyle f}$ is not surjective then it is nevertheless possible that ${\displaystyle \varnothing \in {\mathcal {B}}{\big \vert }_{f(X)},}$ which would make ${\displaystyle f^{-1}({\mathcal {B}})}$ degenerate as well. The next characterization shows that degeneracy is the only obstacle. If ${\displaystyle {\mathcal {B}}}$ is a prefilter then the following are equivalent:^[14][11][35]

1. ${\displaystyle f^{-1}({\mathcal {B}})}$ is a prefilter;
2. ${\displaystyle {\mathcal {B}}{\big \vert }_{f(X)}}$ is a prefilter;
3. ${\displaystyle \varnothing \not \in {\mathcal {B}}{\big \vert }_{f(X)}}$;
4. ${\displaystyle {\mathcal {B}}}$ meshes with ${\displaystyle f(X)}$

and moreover, if ${\displaystyle f^{-1}({\mathcal {B}})}$ is a prefilter then so is ${\displaystyle f\left(f^{-1}({\mathcal {B}})\right).}$^[14][11]

If ${\displaystyle S\subseteq Y}$ and if ${\displaystyle \operatorname {In} :S\to Y}$ denotes the inclusion map then the trace of ${\displaystyle {\mathcal {B}}{\text{ on }}S}$ is equal to the preimage ${\displaystyle \operatorname {In} ^{-1}({\mathcal {B}}).}$^[11] This observation allows the results in this subsection to be applied to investigating the trace on a set.

Bijections, injections, and surjections

All properties involving filters are preserved under bijections. This means that if ${\displaystyle {\mathcal {B}}\subseteq \wp (Y){\text{ and }}g:Y\to Z}$ is a bijection, then ${\displaystyle {\mathcal {B}}}$ is a prefilter (resp. ultra, ultra prefilter, filter on ${\displaystyle X,}$ ultrafilter on ${\displaystyle X,}$ filter subbase, π–system, ideal on ${\displaystyle X,}$ etc.) if and only if the same is true of ${\displaystyle g({\mathcal {B}}){\text{ on }}Z.}$^[34]

A map ${\displaystyle g:Y\to Z}$ is injective if and only if for all prefilters ${\displaystyle {\mathcal {B}}{\text{ on }}X,{\mathcal {B}}}$ is equivalent to ${\displaystyle g^{-1}(g({\mathcal {B}})).}$^[28] The image of an ultra family of sets under an injection is again ultra.

The map ${\displaystyle f:X\to Y}$ is a surjection if and only if whenever ${\displaystyle {\mathcal {B}}}$ is a prefilter on ${\displaystyle Y}$ then the same is true of ${\displaystyle f^{-1}({\mathcal {B}}){\text{ on }}X}$ (this result does not require the ultrafilter lemma).

Subordination is preserved by images and preimages[]

The relation ${\displaystyle \,\leq \,}$ is preserved under both images and preimages of families of sets.^[11] This means that for any families ${\displaystyle {\mathcal {C}}{\text{ and }}{\mathcal {F}},}$^[35]

Moreover, the following relations always hold for any family of sets ${\displaystyle {\mathcal {C}}}$:^[35]

where equality will hold if ${\displaystyle f}$ is surjective.^[35] Furthermore,

If ${\displaystyle {\mathcal {B}}\subseteq \wp (X){\text{ and }}{\mathcal {C}}\subseteq \wp (Y)}$ then^[10]

and ${\displaystyle g^{-1}(g({\mathcal {C}}))\leq {\mathcal {C}}}$^[35] where equality will hold if ${\displaystyle g}$ is injective.^[35]

Products of prefilters[]

Suppose ${\displaystyle X_{\bullet }=\left(X_{i}\right)_{i\in I}}$ is a family of one or more non–empty sets, whose product will be denoted by ${\displaystyle \prod X_{\bullet }:=\prod _{i\in I}X_{i},}$ and for every index ${\displaystyle i\in I,}$ let

denote the canonical projection. Let ${\displaystyle {\mathcal {B}}_{\bullet }:=\left({\mathcal {B}}_{i}\right)_{i\in I}}$ be non−empty families, also indexed by ${\displaystyle I,}$ such that ${\displaystyle {\mathcal {B}}_{i}\subseteq \wp \left(X_{i}\right)}$ for each ${\displaystyle i\in I.}$ The product of the families ${\displaystyle {\mathcal {B}}_{\bullet }}$^[11] is defined identically to how the basic open subsets of the product topology are defined (had all of these ${\displaystyle {\mathcal {B}}_{i}}$ been topologies). That is, both the notations denote the family of all subsets ${\displaystyle \prod _{i\in I}S_{i}\subseteq \prod _{}X_{\bullet }}$ such that ${\displaystyle S_{i}=X_{i}}$ for all but finitely many ${\displaystyle i\in I}$ and where ${\displaystyle S_{i}\in {\mathcal {B}}_{i}}$ for any one of these finitely many exceptions (that is, for any ${\displaystyle i}$ such that ${\displaystyle S_{i}\neq X_{i},}$ necessarily ${\displaystyle S_{i}\in {\mathcal {B}}_{i}}$). When every ${\displaystyle {\mathcal {B}}_{i}}$ is a filter subbase then the family ${\displaystyle \bigcup _{i\in I}\Pr {}_{X_{i}}^{-1}\left({\mathcal {B}}_{i}\right)}$ is a filter subbase for the filter on ${\displaystyle \prod X_{\bullet }}$ generated by ${\displaystyle {\mathcal {B}}_{\bullet }.}$^[11] If ${\displaystyle \prod {\mathcal {B}}_{\bullet }}$ is a filter subbase then the filter on ${\displaystyle \prod X_{\bullet }}$ that it generates is called the filter generated by ${\displaystyle {\mathcal {B}}_{\bullet }}$.^[11] If every ${\displaystyle {\mathcal {B}}_{i}}$ is a prefilter on ${\displaystyle X_{i}}$ then ${\displaystyle \prod {\mathcal {B}}_{\bullet }}$ will be a prefilter on ${\displaystyle \prod X_{\bullet }}$ and moreover, this prefilter is equal to the coarsest prefilter ${\displaystyle {\mathcal {F}}{\text{ on }}\prod X_{\bullet }}$ such that ${\displaystyle \Pr {}_{X_{i}}({\mathcal {F}})={\mathcal {B}}_{i}}$ for every ${\displaystyle i\in I.}$^[11] However, ${\displaystyle \prod {\mathcal {B}}_{\bullet }}$ may fail to be a filter on ${\displaystyle \prod X_{\bullet }}$ even if every ${\displaystyle {\mathcal {B}}_{i}}$ is a filter on ${\displaystyle X_{i}.}$^[11]

Set subtraction and some examples[]

Set subtracting away a subset of the kernel

If ${\displaystyle {\mathcal {B}}}$ is a prefilter on ${\displaystyle X,S\subseteq \ker {\mathcal {B}},{\text{ and }}S\not \in {\mathcal {B}}}$ then ${\displaystyle \{B\setminus S~:~B\in {\mathcal {B}}\}}$ is a prefilter, where this latter set is a filter if and only if ${\displaystyle {\mathcal {B}}}$ is a filter and ${\displaystyle S=\varnothing .}$ In particular, if ${\displaystyle {\mathcal {B}}}$ is a neighborhood basis at a point ${\displaystyle x}$ in a topological space ${\displaystyle X}$ having at least 2 points, then ${\displaystyle \{B\setminus \{x\}~:~B\in {\mathcal {B}}\}}$ is a prefilter on ${\displaystyle X.}$ This construction is used to define ${\displaystyle \lim _{\stackrel {x\to x_{0}}{x\neq x_{0}}}f(x)\to y}$ in terms of prefilter convergence.

Using duality between ideals and dual ideals

There is a dual relation ${\displaystyle {\mathcal {B}}\vartriangleleft {\mathcal {C}}}$ or ${\displaystyle {\mathcal {C}}\vartriangleright {\mathcal {B}},}$ which is defined to mean that every ${\displaystyle B\in {\mathcal {B}}}$ is contained in some ${\displaystyle C\in {\mathcal {C}}.}$ Explicitly, this means that for every ${\displaystyle B\in {\mathcal {B}}}$ , there is some ${\displaystyle C\in {\mathcal {C}}}$ such that ${\displaystyle B\subseteq C.}$ This relation is dual to ${\displaystyle \,\leq \,}$ in sense that ${\displaystyle {\mathcal {B}}\vartriangleleft {\mathcal {C}}}$ if and only if ${\displaystyle (X\setminus {\mathcal {B}})\leq (X\setminus {\mathcal {C}}).}$^[6] The relation ${\displaystyle {\mathcal {B}}\vartriangleleft {\mathcal {C}}}$ is closely related to the downward closure of a family in a manner similar to how ${\displaystyle \,\leq \,}$ is related to the upward closure family.

For an example that uses this duality, suppose ${\displaystyle f:X\to Y}$ is a map and ${\displaystyle \Xi \subseteq \wp (Y).}$ Define

which contains the empty set if and only if ${\displaystyle \Xi }$ does. It is possible for ${\displaystyle \Xi }$ to be an ultrafilter and for ${\displaystyle \Xi _{f}}$ to be empty or not closed under finite intersections (see footnote for example).^{[note 10]} Although ${\displaystyle \Xi _{f}}$ does not preserve properties of filters very well, if ${\displaystyle \Xi }$ is downward closed (resp. closed under finite unions, an ideal) then this will also be true for ${\displaystyle \Xi _{f}.}$ Using the duality between ideals and dual ideals allows for a construction of the following filter.

     Suppose ${\displaystyle {\mathcal {B}}}$ is a filter on ${\displaystyle Y}$ and let ${\displaystyle \Xi :=Y\setminus {\mathcal {B}}}$ be its dual in ${\displaystyle Y.}$ If ${\displaystyle X\not \in \Xi _{f}}$ then ${\displaystyle \Xi _{f}}$'s dual ${\displaystyle X\setminus \Xi _{f}}$ will be a filter.

Other topology related examples

Example: The set ${\displaystyle {\mathcal {B}}}$ of all dense open subsets of a topological space is a proper π–system and 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 ${\displaystyle {\mathcal {B}}.}$

Example: The family ${\displaystyle {\mathcal {B}}_{\operatorname {Open} }}$ of all dense open sets of ${\displaystyle X=\mathbb {R} }$ having finite Lebesgue measure is a proper π–system and a free prefilter. The prefilter ${\displaystyle {\mathcal {B}}_{\operatorname {Open} }}$ 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 ${\displaystyle {\mathcal {B}}_{\operatorname {Open} }}$ is dense in ${\displaystyle X}$ (and also comeagre and non–meager) so the set of all countable intersections of elements of ${\displaystyle {\mathcal {B}}_{\operatorname {Open} }}$ is a prefilter and π–system; it is also finer than, and not equivalent to, ${\displaystyle {\mathcal {B}}_{\operatorname {Open} }.}$

Convergence, limits, and cluster points[]

Throughout, ${\displaystyle (X,\tau )}$ is a topological space.

Prefilters vs. filters

With respect to maps and subsets, the property of being a prefilter is in general more well behaved and better preserved than the property of being a filter. For instance, the image of a prefilter under some map is again a prefilter; but the image of a filter under a non–surjective map is never a filter on the codomain, although it will be a prefilter. The situation is the same with preimages under non–injective maps (even if the map is surjective). If ${\displaystyle S\subseteq X}$ is a proper subset then any filter on ${\displaystyle S}$ will not be a filter on ${\displaystyle X,}$ although it will be a prefilter.

One advantage that filters have is that they are distinguished representatives of their equivalence class (relative to ${\displaystyle \,\leq }$), meaning that any equivalence class of prefilters contains a unique filter. This property may be useful when dealing with equivalence classes of prefilters (for instance, they are useful in construct completions using Cauchy filters). The many properties that characterize ultrafilters are also often useful. They are used to, for example, construct the Stone–Čech compactification. The use of ultrafilters generally requires that the ultrafilter lemma be assumed. But in the many fields where the axiom of choice (or the Hahn–Banach theorem) is assumed, the ultrafilter lemma necessarily holds and does not require an addition assumption.

A note on intuition

Suppose that ${\displaystyle {\mathcal {F}}}$ is a non–principal filter on an infinite set ${\displaystyle X.}$ ${\displaystyle {\mathcal {F}}}$ has one "upward" property (that of being closed upward) and one "downward" property (that of being directed downward). Starting with any ${\displaystyle F_{0}\in {\mathcal {F}},}$ there always exists some ${\displaystyle F_{1}\in {\mathcal {F}}}$ that is a proper subset of ${\displaystyle F_{0}}$; this may be continued ad infinitum to get a sequence ${\displaystyle F_{0}\supset F_{1}\supset \cdots }$ of sets in ${\displaystyle {\mathcal {F}}}$ with each ${\displaystyle F_{i+1}}$ being a proper subset of ${\displaystyle F_{i}.}$ The same is not true going "upward", for if ${\displaystyle F_{0}=X\in {\mathcal {F}}}$ then there is no set in ${\displaystyle {\mathcal {F}}}$ that contains ${\displaystyle X}$ as a proper subset. Thus when it comes to limiting behavior (which is a topic central to the field of topology), going "upward" leads to a dead end, while going "downward" is typically fruitful. So to gain understanding and intuition about how filters (and prefilter) relate to concepts in topology, the "downward" property is usually the one to concentrate on. This is also why so many topological properties can be described by using only prefilters, rather than requiring filters (which only differ from prefilters in that they are also upward closed). The "upward" property of filters is less important for topological intuition but it is sometimes useful to have for technical reasons. For example, with respect to ${\displaystyle \,\subseteq ,}$ every filter subbase is contained in a unique smallest filter but there may not exist a unique smallest prefilter containing it.

Limits and convergence[]

The following well known definition will be generalized to prefilters. A point ${\displaystyle x\in X}$ is called a limit point, cluster point, or accumulation point of a subset ${\displaystyle S\subseteq X}$ if every neighborhood of ${\displaystyle x{\text{ in }}(X,\tau )}$ contains a point of ${\displaystyle S}$ different from ${\displaystyle x,}$ or equivalently, if ${\displaystyle x\in \operatorname {cl} _{(X,\tau )}(S\setminus \{x\}).}$ The set of all limit points of ${\displaystyle S}$ is called the derived set of ${\displaystyle S{\text{ in }}(X,\tau ).}$ The closure of a set ${\displaystyle S\subseteq X}$ is equal to the union of ${\displaystyle S}$ together with the set of all limit points of ${\displaystyle S.}$

     A family ${\displaystyle {\mathcal {B}}}$ is said to converge to a point ${\displaystyle x\in X{\text{ in }}(X,\tau ),}$^[8] written ${\displaystyle {\mathcal {B}}\to x{\text{ or }}\lim {\mathcal {B}}\to x{\text{ in }}(X,\tau ),}$^[29] if ${\displaystyle {\mathcal {B}}\geq {\mathcal {N}}(x),}$ in which case ${\displaystyle x}$ is said to be a limit or limit point^[36] of ${\displaystyle {\mathcal {B}}{\text{ in }}(X,\tau ).}$ Denote^[8] the set of all these limit points by ${\displaystyle \lim {}_{(X,\tau )}{\mathcal {B}}{\text{ and }}\lim {\mathcal {B}}.}$

     As usual, ${\displaystyle \lim {\mathcal {B}}=x}$ is defined to mean that ${\displaystyle {\mathcal {B}}\to x{\text{ in }}(X,\tau )}$ and ${\displaystyle x}$ is the only limit point of ${\displaystyle {\mathcal {B}}{\text{ in }}(X,\tau );}$ that is, if also ${\displaystyle {\mathcal {B}}\to z{\text{ in }}(X,\tau ){\text{ then }}z=x.}$^[29] (If the notation "${\displaystyle \lim {\mathcal {B}}=x}$" did not also require that the limit ${\displaystyle x}$ be unique then the equals sign = would no longer be guaranteed to be transitive).

In words, ${\displaystyle {\mathcal {B}}}$ converges to a point if and only if ${\displaystyle {\mathcal {B}}}$ is finer than the neighborhood filter at that point. Explicitly, ${\displaystyle {\mathcal {N}}(x)\leq {\mathcal {B}}}$ means that every neighborhood ${\displaystyle N{\text{ of }}x}$ contains some ${\displaystyle B\in {\mathcal {B}}}$ as a subset (that is, ${\displaystyle B\subseteq N}$); thus the following then holds: ${\displaystyle {\mathcal {N}}\ni N\supseteq B\in {\mathcal {B}}.}$

More generally, given ${\displaystyle S\subseteq X,}$

     if ${\displaystyle {\mathcal {B}}\geq {\mathcal {N}}(S)}$ then ${\displaystyle {\mathcal {B}}}$ is said to converge to ${\displaystyle S{\text{ in }}(X,\tau )}$ and ${\displaystyle S}$ is called a limit of ${\displaystyle {\mathcal {B}},}$ where this is expressed by writing ${\displaystyle {\mathcal {B}}\to S{\text{ in }}(X,\tau ).}$

In the above definitions, it suffices to check that ${\displaystyle {\mathcal {B}}}$ is finer than some (or equivalently, finer than every) neighborhood base in ${\displaystyle (X,\tau )}$ of the point or set (for example, such as ${\displaystyle \tau (x)=\{U\in \tau :x\in U\}}$ or ${\displaystyle \tau (S)=\bigcap _{s\in S}\tau (s)}$). If ${\displaystyle \varnothing \neq S\subseteq X}$ then because ${\displaystyle {\mathcal {N}}(S)=\bigcap _{s\in S}{\mathcal {N}}(s),}$ a family ${\displaystyle {\mathcal {B}}}$ converges to ${\displaystyle S}$ if and only if ${\displaystyle {\mathcal {B}}\to s{\text{ for all }}s\in S.}$ A family ${\displaystyle {\mathcal {B}}}$ converges to ${\displaystyle S:=\varnothing }$ if and only if ${\displaystyle \varnothing \in {\mathcal {B}},}$ which is why when dealing with convergence of prefilters (or filter subbases), it is typically assumed (often without mention) that ${\displaystyle S\neq \varnothing .}$

Given ${\displaystyle x\in X,}$ the following are equivalent for a prefilter ${\displaystyle {\mathcal {B}}:}$

1. ${\displaystyle {\mathcal {B}}}$ converges to ${\displaystyle x.}$
2. ${\displaystyle {\mathcal {B}}}$ converges to the set ${\displaystyle \{x\}.}$
3. ${\displaystyle {\mathcal {B}}^{\uparrow X}}$ converges to ${\displaystyle x.}$
4. There exists a family equivalent to ${\displaystyle {\mathcal {B}}}$ that converges to ${\displaystyle x.}$

If ${\displaystyle {\mathcal {B}}}$ is a prefilter and ${\displaystyle B\in {\mathcal {B}}}$ then ${\displaystyle {\mathcal {B}}}$ converges to a point (or subset) of ${\displaystyle X}$ if and only if this is true of the trace ${\displaystyle {\mathcal {B}}{\big \vert }_{B}.}$^[37] If ${\displaystyle {\mathcal {B}}}$ is a filter subbase that converges to ${\displaystyle x{\text{ or }}S}$ then this is also true of the filter that it generates (and also of any prefilter equivalent to this filter, such as the π-system generated by ${\displaystyle {\mathcal {B}}}$).

Because subordination is transitive, if ${\displaystyle {\mathcal {B}}\leq {\mathcal {C}}{\text{ then }}\lim {}_{(X,\tau )}{\mathcal {B}}\subseteq \lim {}_{(X,\tau )}{\mathcal {C}}}$ and moreover, for every ${\displaystyle x\in X,}$ both ${\displaystyle \{x\}}$ and the maximal/ultrafilter ${\displaystyle \{x\}^{\uparrow X}}$ converge to ${\displaystyle x{\text{ in }}(X,\tau ).}$ Thus every topological space ${\displaystyle (X,\tau )}$ induces a canonical convergence ${\displaystyle \xi \subseteq X\times \operatorname {Filters} (X)}$ defined by ${\displaystyle (x,{\mathcal {B}})\in \xi {\text{ if and only if }}x\in \lim {}_{(X,\tau )}{\mathcal {B}}.}$ At the other extreme, the neighborhood filter ${\displaystyle {\mathcal {N}}(x)}$ is the smallest (that is, coarsest) filter on ${\displaystyle X}$ that converges to ${\displaystyle x{\text{ in }}(X,\tau );}$ that is, any filter converging to ${\displaystyle x}$ must contain ${\displaystyle {\mathcal {N}}(x)}$ as a subset. Said differently, the family of filters that converge to ${\displaystyle x}$ consists exactly of those filter on ${\displaystyle X}$ that contain ${\displaystyle {\mathcal {N}}(x)}$ as a subset. Consequently, the finer the topology on ${\displaystyle X}$ then the fewer prefilters exist that have any limit points in ${\displaystyle X.}$

Cluster points[]

     Say that ${\displaystyle x\in X}$ is a cluster point or an accumulation point of a family ${\displaystyle {\mathcal {B}}}$^[8] if ${\displaystyle {\mathcal {B}}}$ meshes with the neighborhood filter at ${\displaystyle x}$; that is, if ${\displaystyle {\mathcal {B}}\#{\mathcal {N}}(x).}$ The set of all cluster points of ${\displaystyle {\mathcal {B}}}$ is denoted by ${\displaystyle \operatorname {cl} _{X}{\mathcal {B}}{\text{ or }}\operatorname {cl} {\mathcal {B}}.}$

Explicitly, this means that ${\displaystyle B\cap N\neq \varnothing {\text{ for every }}B\in {\mathcal {B}}}$ and every neighborhood ${\displaystyle N}$ of ${\displaystyle x.}$ When ${\displaystyle {\mathcal {B}}}$ is a prefilter then the definition of "${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {N}}}$ mesh" can be characterized entirely in terms of the preorder ${\displaystyle \,\leq \,.}$

More generally, given ${\displaystyle S\subseteq X,}$ say that

     ${\displaystyle {\mathcal {B}}}$ clusters at ${\displaystyle S}$ if ${\displaystyle {\mathcal {B}}}$ meshes with the neighborhood filter of ${\displaystyle S}$; that is, if ${\displaystyle {\mathcal {B}}\#{\mathcal {N}}(S).}$

In the above definitions, it suffices to check that ${\displaystyle {\mathcal {B}}}$ meshes with some (or equivalently, meshes with every) neighborhood base in ${\displaystyle (X,\tau )}$ of ${\displaystyle x{\text{ or }}S.}$ Two equivalent families of sets have the exact same limit points and also the same cluster points. No matter the topology, for every ${\displaystyle x\in X,}$ both ${\displaystyle \{x\}}$ and the principal ultrafilter ${\displaystyle \{x\}^{\uparrow X}}$ cluster at ${\displaystyle x.}$ For any ${\displaystyle S\subseteq X,}$ if ${\displaystyle {\mathcal {B}}}$ clusters at some ${\displaystyle s\in S}$ then ${\displaystyle {\mathcal {B}}}$ clusters at ${\displaystyle S.}$ No family clusters at ${\displaystyle S:=\varnothing }$ and if ${\displaystyle \varnothing \in {\mathcal {B}}{\text{ then }}\varnothing =\operatorname {cl} {\mathcal {B}}.}$

Given ${\displaystyle x\in X,}$ the following are equivalent for a prefilter ${\displaystyle {\mathcal {B}}{\text{ on }}X}$:

1. ${\displaystyle {\mathcal {B}}}$ clusters at ${\displaystyle x.}$
2. ${\displaystyle {\mathcal {B}}}$ clusters at the set ${\displaystyle \{x\}.}$
3. The family ${\displaystyle {\mathcal {B}}^{\uparrow X}}$ generated by ${\displaystyle {\mathcal {B}}}$ clusters at ${\displaystyle x.}$
4. There exists a family equivalent to ${\displaystyle {\mathcal {B}}}$ that clusters at ${\displaystyle x.}$
5. ${\displaystyle x\in \bigcap _{F\in {\mathcal {B}}}\operatorname {cl} _{X}F.}$
6. ${\displaystyle X\setminus N\not \in {\mathcal {B}}^{\uparrow X}}$ for every neighborhood ${\displaystyle N}$ of ${\displaystyle x.}$
   • If ${\displaystyle {\mathcal {B}}}$ is a filter on ${\displaystyle X}$ then ${\displaystyle x\in \operatorname {cl} _{X}{\mathcal {B}}{\text{ if and only if }}X\setminus N\not \in {\mathcal {B}}}$ for every neighborhood ${\displaystyle N{\text{ of }}x.}$
7. There exists a prefilter ${\displaystyle {\mathcal {F}}}$ subordinate to ${\displaystyle {\mathcal {B}}}$ (that is, ${\displaystyle {\mathcal {F}}\geq {\mathcal {B}}}$) such that ${\displaystyle {\mathcal {F}}\to x.}$
   • This is the filter equivalent of "${\displaystyle x}$ is a cluster point of a sequence if and only if there exists a subsequence converging to ${\displaystyle x.}$
   • In particular, if ${\displaystyle x}$ is a cluster point of a prefilter ${\displaystyle {\mathcal {B}}}$ then ${\displaystyle {\mathcal {B}}(\cap ){\mathcal {N}}(x)}$ is a prefilter subordinate to ${\displaystyle {\mathcal {B}}}$ that converges to ${\displaystyle x{\text{ in }}(X,\tau ).}$

If ${\displaystyle {\mathcal {B}}}$ is an ultra prefilter on ${\displaystyle X{\text{ and }}x\in X,}$ then ${\displaystyle x}$ is a cluster point of ${\displaystyle {\mathcal {B}}{\text{ if and only if }}{\mathcal {B}}\to x{\text{ in }}(X,\tau ).}$^[30]

The set ${\displaystyle \operatorname {cl} _{X}{\mathcal {B}}}$ of all cluster points of a prefilter ${\displaystyle {\mathcal {B}}{\text{ in }}(X,\tau )}$ satisfies

which in particular shows that the set ${\displaystyle \operatorname {cl} _{X}{\mathcal {B}}}$ of all cluster points of any prefilter ${\displaystyle {\mathcal {B}}}$ is a closed subset of ${\displaystyle X.}$^[38][8] This also justifies the notation ${\displaystyle \operatorname {cl} _{X}{\mathcal {B}}}$ for the set of cluster points.^[8]

Properties and relationships[]

Just like sequences and nets, it is possible for a prefilter on an topological space of infinite cardinality to not have any cluster points or limit points.^[38]

If ${\displaystyle x}$ is a limit point of ${\displaystyle {\mathcal {B}}}$ then ${\displaystyle x}$ is necessarily a limit point of any family ${\displaystyle {\mathcal {C}}}$ finer than ${\displaystyle {\mathcal {B}}}$ (that is, if ${\displaystyle {\mathcal {N}}(x)\leq {\mathcal {B}}{\text{ and }}{\mathcal {B}}\leq {\mathcal {C}}}$ then ${\displaystyle {\mathcal {N}}(x)\leq {\mathcal {C}}}$).^[38] In contrast, if ${\displaystyle x}$ is a cluster point of ${\displaystyle {\mathcal {B}}}$ then ${\displaystyle x}$ is necessarily a cluster point of any family ${\displaystyle {\mathcal {C}}}$ coarser than ${\displaystyle {\mathcal {B}}}$ (that is, if ${\displaystyle {\mathcal {N}}(x){\text{ and }}{\mathcal {B}}}$ mesh and ${\displaystyle {\mathcal {C}}\leq {\mathcal {B}}}$ then ${\displaystyle {\mathcal {N}}(x){\text{ and }}{\mathcal {C}}}$ mesh).

Equivalent families and subordination

Any two equivalent families ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}}$ can be used interchangeably in the definitions of "limit of" and "cluster at" because their equivalency guarantees that ${\displaystyle {\mathcal {N}}\leq {\mathcal {B}}}$ if and only if ${\displaystyle {\mathcal {N}}\leq {\mathcal {C}},}$ and also that ${\displaystyle {\mathcal {N}}\#{\mathcal {B}}}$ if and only if ${\displaystyle {\mathcal {N}}\#{\mathcal {C}}.}$ In essence, the preorder ${\displaystyle \,\leq \,}$ is incapable of distinguishing between equivalent families. Given two prefilters, whether or not they mesh can be characterized entirely in terms of subordination. Thus the two most fundamental concepts related to (pre)filters to Topology (that is, limit and cluster points) can both be defined entirely in terms of the subordination relation. This is why the preorder ${\displaystyle \,\leq \,}$ is of such great importance in applying (pre)filters to Topology.

Limit and cluster point relationships and sufficient conditions

Every limit point of a prefilter ${\displaystyle {\mathcal {B}}}$ is also a cluster point of ${\displaystyle {\mathcal {B}},}$ since if ${\displaystyle x}$ is a limit point of a prefilter ${\displaystyle {\mathcal {B}}}$ then ${\displaystyle {\mathcal {N}}(x){\text{ and }}{\mathcal {B}}}$ mesh,^[19][38] which makes ${\displaystyle x}$ a cluster point of ${\displaystyle {\mathcal {B}}.}$^[8] Every accumulation point of an ultrafilter is also a limit point.

If ${\displaystyle \varnothing \neq {\mathcal {B}}\subseteq \wp (X){\text{ and }}{\mathcal {S}}\geq {\mathcal {B}}}$ is a filter subbase such that ${\displaystyle {\mathcal {S}}\to x{\text{ in }}(X,\tau )}$ then ${\displaystyle x\in \operatorname {cl} _{X}{\mathcal {B}}.}$ In particular, any limit point of a filter subbase subordinate to ${\displaystyle {\mathcal {B}}\neq \varnothing }$ is necessarily also a cluster point of ${\displaystyle {\mathcal {B}}.}$ If ${\displaystyle x}$ is a cluster point of a prefilter ${\displaystyle {\mathcal {B}}}$ then ${\displaystyle {\mathcal {B}}(\cap ){\mathcal {N}}(x)}$ is a prefilter subordinate to ${\displaystyle {\mathcal {B}}}$ that converges to ${\displaystyle x{\text{ in }}(X,\tau ).}$

If ${\displaystyle S\subseteq X}$ and if ${\displaystyle {\mathcal {B}}}$ is a prefilter on ${\displaystyle S}$ then every cluster point of ${\displaystyle {\mathcal {B}}{\text{ in }}X}$ belongs to ${\displaystyle \operatorname {cl} _{X}S}$ and any point in ${\displaystyle \operatorname {cl} _{X}S}$ is a limit point of a filter on ${\displaystyle S.}$^[38]

Primitive sets

A subset ${\displaystyle P\subseteq X}$ is called primitive^[39] if it is the set of limit points of some ultrafilter on ${\displaystyle X.}$ That is, if there exists an ultrafilter ${\displaystyle {\mathcal {B}}{\text{ on }}X}$ such that ${\displaystyle P}$ is equal to ${\displaystyle \operatorname {lim} _{X}{\mathcal {B}},}$ which recall denotes the set of limit points of ${\displaystyle {\mathcal {B}}{\text{ in }}(X,\tau ).}$

Any closed singleton subset of ${\displaystyle X}$ is a primitive subset of ${\displaystyle X.}$^[39] The image of a primitive subset of ${\displaystyle X}$ under a continuous map ${\displaystyle f:X\to Y}$ is contained in a primitive subset of ${\displaystyle Y.}$^[39]

Assume that ${\displaystyle P,Q\subseteq X}$ are two primitive subset of ${\displaystyle X.}$ If ${\displaystyle U}$ is an open subset of ${\displaystyle X}$ such that ${\displaystyle P\cap Q\neq \varnothing ,}$ then ${\displaystyle U\in {\mathcal {B}}}$ for any ultrafilter ${\displaystyle {\mathcal {B}}{\text{ on }}X}$ such that ${\displaystyle P=\operatorname {lim} _{X}{\mathcal {B}}.}$^[39] In addition, if ${\displaystyle P{\text{ and }}Q}$ are distinct then there exists some ${\displaystyle S\subseteq X}$ and some ultrafilters ${\displaystyle {\mathcal {B}}_{P}{\text{ and }}{\mathcal {B}}_{Q}{\text{ on }}X}$ such that ${\displaystyle P=\operatorname {lim} _{X}{\mathcal {B}}_{P},Q=\operatorname {lim} _{X}{\mathcal {B}}_{Q},S\in {\mathcal {B}}_{P},}$ and ${\displaystyle X\setminus S\in {\mathcal {B}}_{Q}.}$^[39]

Other results

If ${\displaystyle X}$ is a complete lattice then:^{[citation needed]}

• The limit inferior of ${\displaystyle B}$ is the infimum of the set of all cluster points of ${\displaystyle B.}$
• The limit superior of ${\displaystyle B}$ is the supremum of the set of all cluster points of ${\displaystyle B.}$
• ${\displaystyle B}$ is a convergent prefilter if and only if its limit inferior and limit superior agree; in this case, the value on which they agree is the limit of the prefilter.

Limits of functions defined as limits of prefilters[]

     If ${\displaystyle f:X\to Y}$ is a map from a set into a topological space ${\displaystyle Y,y\in Y,{\text{ and }}{\mathcal {B}}\subseteq \wp (X)}$ then ${\displaystyle y}$ is a limit point or limit (respectively, a cluster point) of ${\displaystyle f}$ with respect to ${\displaystyle {\mathcal {B}}}$^[38] if ${\displaystyle y}$ is a limit point (resp. a cluster point) of ${\displaystyle f({\mathcal {B}}){\text{ in }}Y,}$ in which case this may be expressed by writing ${\displaystyle f({\mathcal {B}})\to y,{\text{ or }}\lim f({\mathcal {B}})\to y{\text{ in }}>Y.}$ If the limit ${\displaystyle y}$ is unique then the arrow ${\displaystyle \to }$ may be replaced with an equals sign ${\displaystyle =.}$^[29]

Explicitly, ${\displaystyle y}$ is a limit of ${\displaystyle f}$ with respect to ${\displaystyle {\mathcal {B}}}$ if and only if ${\displaystyle {\mathcal {N}}(y)\leq f({\mathcal {B}}).}$

The definition of a convergent net is a special case of the above definition of a limit of a function. Specifically, if ${\displaystyle x\in X{\text{ and }}\chi :(I,\leq )\to X}$ is a net then

where the left hand side states that ${\displaystyle x}$ is a limit of the net ${\displaystyle \chi }$ while the right hand side states that ${\displaystyle x}$ is a limit of the function ${\displaystyle \chi }$ with respect to ${\displaystyle {\mathcal {B}}:=\operatorname {Tails} (I,\leq )}$ (as defined above).

The table below shows how various types of limits encountered in analysis and topology can be defined in terms of the convergence of images (under ${\displaystyle f}$) of particular prefilters on the domain ${\displaystyle X.}$ This shows that prefilters provide a general framework into which many of the various definitions of limits fit.^[37] The limits in the left–most column are defined in their usual way with their obvious definitions.

Throughout, let ${\displaystyle f:X\to Y}$ be a map between topological spaces, ${\displaystyle x_{0}\in X,{\text{ and }}y\in Y.}$ If ${\displaystyle Y}$ is Hausdorff then all arrows "${\displaystyle \to y}$" in the table may be replaced with equal signs "${\displaystyle =y}$" and "${\displaystyle \lim f({\mathcal {B}})\to y}$" may be replaced with "${\displaystyle \lim f({\mathcal {B}})=y}$".^[29]

Type of limit	if and only if	Definition in terms of prefilters[37]	Assumptions
${\displaystyle \lim _{x\to x_{0}}f(x)\to y}$	⇔	${\displaystyle f({\mathcal {B}})\to y{\text{ where }}{\mathcal {B}}\,:=\,{\mathcal {N}}\left(x_{0}\right)}$
${\displaystyle \lim _{\stackrel {x\to x_{0}}{x\neq x_{0}}}f(x)\to y}$	⇔	${\displaystyle f({\mathcal {B}})\to y{\text{ where }}{\mathcal {B}}\,:=\,\left\{N\setminus \left\{x_{0}\right\}:N\in {\mathcal {N}}\left(x_{0}\right)\right\}}$
${\displaystyle \lim _{\stackrel {x\to x_{0}}{x\in S}}f(x)\to y}$ or ${\displaystyle \lim _{x\to x_{0}}f{\big \vert }_{S}(x)\to y}$	⇔	${\displaystyle f({\mathcal {B}})\to y{\text{ where }}{\mathcal {B}}\,:=\,\left\{N\cap S:N\in {\mathcal {N}}\left(x_{0}\right)\right\}}$	${\displaystyle S\subseteq X{\text{ and }}x_{0}\in \operatorname {cl} _{X}S}$
${\displaystyle \lim _{\stackrel {x\to x_{0}}{x\neq x_{0}}}f(x)\to y}$	⇔	${\displaystyle f({\mathcal {B}})\to y{\text{ where }}{\mathcal {B}}\,:=\,\left\{\left(x_{0}-r,x_{0}\right)\cup \left(x_{0},x_{0}+r\right):0<r\in \mathbb {R} \right\}}$	${\displaystyle x_{0}\in X=\mathbb {R} }$
${\displaystyle \lim _{\stackrel {x\to x_{0}}{x<x_{0}}}f(x)\to y}$	⇔	${\displaystyle f({\mathcal {B}})\to y{\text{ where }}{\mathcal {B}}\,:=\,\left\{\left(x,x_{0}\right):x<x_{0}\right\}}$	${\displaystyle x_{0}\in X=\mathbb {R} }$
${\displaystyle \lim _{\stackrel {x\to x_{0}}{x\leq x_{0}}}f(x)\to y}$	⇔	${\displaystyle f({\mathcal {B}})\to y{\text{ where }}{\mathcal {B}}\,:=\,\left\{\left(x,x_{0}\right]:x<x_{0}\right\}}$	${\displaystyle x_{0}\in X=\mathbb {R} }$
${\displaystyle \lim _{\stackrel {x\to x_{0}}{x>x_{0}}}f(x)\to y}$	⇔	${\displaystyle f({\mathcal {B}})\to y{\text{ where }}{\mathcal {B}}\,:=\,\left\{\left(x_{0},x\right):x_{0}<x\right\}}$	${\displaystyle x_{0}\in X=\mathbb {R} }$
${\displaystyle \lim _{\stackrel {x\to x_{0}}{x\geq x_{0}}}f(x)\to y}$	⇔	${\displaystyle f({\mathcal {B}})\to y{\text{ where }}{\mathcal {B}}\,:=\,\left\{\left[x_{0},x\right):x_{0}\leq x\right\}}$	${\displaystyle x_{0}\in X=\mathbb {R} }$
${\displaystyle \lim _{n\to \infty }f(n)\to y}$	⇔	${\displaystyle f({\mathcal {B}})\to y{\text{ where }}{\mathcal {B}}\,:=\,\{\{n,n+1,\ldots \}~:~n\in \mathbb {N} \}\}}$	${\displaystyle X=\mathbb {N} {\text{ so }}f:\mathbb {N} \to Y}$ is a sequence in ${\displaystyle Y}$
${\displaystyle \lim _{x\to \infty }f(x)\to y}$	⇔	${\displaystyle f({\mathcal {B}})\to y{\text{ where }}{\mathcal {B}}\,:=\,\{(x,\infty ):x\in \mathbb {R} \}}$	${\displaystyle X=\mathbb {R} }$
${\displaystyle \lim _{x\to -\infty }f(x)\to y}$	⇔	${\displaystyle f({\mathcal {B}})\to y{\text{ where }}{\mathcal {B}}\,:=\,\{(-\infty ,x):x\in \mathbb {R} \}}$	${\displaystyle X=\mathbb {R} }$
${\displaystyle \lim _{|x|\to \infty }f(x)\to y}$	⇔	${\displaystyle f({\mathcal {B}})\to y{\text{ where }}{\mathcal {B}}\,:=\,\{X\cap [(-\infty ,x)\cup (x,\infty )]:x\in \mathbb {R} \}}$	${\displaystyle X=\mathbb {R} {\text{ or }}X=\mathbb {Z} }$ for a double-ended sequence
${\displaystyle \lim _{\|x\|\to \infty }f(x)\to y}$	⇔	${\displaystyle f({\mathcal {B}})\to y{\text{ where }}{\mathcal {B}}\,:=\,\{\{x\in X:\|x\|>r\}~:~0<r\in \mathbb {R} \}}$	${\displaystyle (X,\|\cdot \|){\text{ is}}}$ a seminormed space; for example, a Banach space ${\displaystyle {\text{like }}X=\mathbb {C} }$

By defining different prefilters, many other notions of limits that can be defined; for example, ${\displaystyle \lim _{\stackrel {|x|\to |x_{0}|}{|x|\neq |x_{0}|}}f(x)\to y.}$

Filters and nets[]

This article will describe the relationships between prefilters and nets in great detail so as to make it easier to understand later why subnets (with their most commonly used definitions) are not generally equivalent with "sub–prefilters".

Nets to prefilters[]

In the definitions below, the first statement is the standard definition of a limit point of a net (resp. a cluster point of a net) and it is gradually reword it until the corresponding filter concept is reached.

A net ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}{\text{ in }}X}$ is said to converge in ${\displaystyle (X,\tau )}$ to a point ${\displaystyle x\in X,}$ written ${\displaystyle x_{\bullet }\to x{\text{ in }}(X,\tau ),}$ and ${\displaystyle x}$ is called a limit or limit point of ${\displaystyle x_{\bullet },}$^[40] if any of the following equivalent conditions hold:

1. Definition: For every ${\displaystyle N\in {\mathcal {N}}_{\tau }(x),}$ there exists some ${\displaystyle i\in I}$ such that if ${\displaystyle i\leq j\in I{\text{ then }}x_{j}\in N.}$
2. For every ${\displaystyle N\in {\mathcal {N}}_{\tau }(x),}$ there exists some ${\displaystyle i\in I}$ such that the tail of ${\displaystyle x_{\bullet }}$ starting at ${\displaystyle i}$ is contained in ${\displaystyle N}$ (that is, such that ${\displaystyle x_{\geq i}\subseteq N}$).
3. For every ${\displaystyle N\in {\mathcal {N}}_{\tau }(x),}$ there exists some ${\displaystyle B\in \operatorname {Tails} \left(x_{\bullet }\right)}$ such that ${\displaystyle B\subseteq N.}$
4. ${\displaystyle {\mathcal {N}}_{\tau }(x)\leq \operatorname {Tails} \left(x_{\bullet }\right).}$
5. ${\displaystyle \operatorname {Tails} \left(x_{\bullet }\right)\to x{\text{ in }}(X,\tau )}$; that is, the prefilter ${\displaystyle \operatorname {Tails} \left(x_{\bullet }\right)}$ converges to ${\displaystyle x{\text{ in }}(X,\tau ).}$

As usual, ${\displaystyle \lim x_{\bullet }=x}$ is defined to mean that ${\displaystyle x_{\bullet }\to x{\text{ in }}(X,\tau )}$ and ${\displaystyle x}$ is the only limit point of ${\displaystyle x_{\bullet }{\text{ in }}(X,\tau );}$ that is, if also ${\displaystyle x_{\bullet }\to z{\text{ in }}(X,\tau ){\text{ then }}z=x.}$^[40]

A point ${\displaystyle x\in X}$ is called a cluster point or an accumulation point of a net ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}{\text{ in }}(X,\tau )}$ if any of the following equivalent conditions hold:

1. Definition: For every ${\displaystyle N\in {\mathcal {N}}_{\tau }(x)}$ and every ${\displaystyle i\in I,}$ there exists some ${\displaystyle i\leq j\in I}$ such that ${\displaystyle x_{j}\in N.}$
2. For every ${\displaystyle N\in {\mathcal {N}}_{\tau }(x)}$ and every ${\displaystyle i\in I,}$ the tail of ${\displaystyle x_{\bullet }}$ starting at ${\displaystyle i}$ intersects ${\displaystyle N.}$
3. For every ${\displaystyle N\in {\mathcal {N}}_{\tau }(x)}$ and every ${\displaystyle B\in \operatorname {Tails} \left(x_{\bullet }\right),B\cap N\neq \varnothing .}$
4. ${\displaystyle {\mathcal {N}}_{\tau }(x){\text{ and }}\operatorname {Tails} \left(x_{\bullet }\right)}$ mesh (by definition of "mesh").
5. ${\displaystyle x}$ is a cluster point of ${\displaystyle \operatorname {Tails} \left(x_{\bullet }\right){\text{ in }}(X,\tau ).}$

If ${\displaystyle f:X\to Y}$ is a map and ${\displaystyle x_{\bullet }}$ is a net in ${\displaystyle X}$ then ${\displaystyle \operatorname {Tails} \left(f\left(x_{\bullet }\right)\right)=f\left(\operatorname {Tails} \left(x_{\bullet }\right)\right).}$^[4]

Prefilters to nets[]

A pointed set is a pair ${\displaystyle (S,s)}$ consisting of a non–empty set ${\displaystyle S}$ and an element ${\displaystyle s\in S.}$ For any family ${\displaystyle {\mathcal {B}},}$ let

Define a canonical preorder ${\displaystyle \,\leq \,}$ on pointed sets by declaring

If ${\displaystyle s_{0},s_{1}\in S{\text{ then }}\left(S,s_{0}\right)\leq \left(S,s_{1}\right){\text{ and }}\left(S,s_{1}\right)\leq \left(S,s_{0}\right)}$ even if ${\displaystyle s_{0}\neq s_{1},}$ so this preorder is not antisymmetric and given any family of sets ${\displaystyle {\mathcal {B}},}$ ${\displaystyle (\operatorname {PointedSets} ({\mathcal {B}}),\leq )}$ is partially ordered if and only if ${\displaystyle {\mathcal {B}}\neq \varnothing }$ consists entirely of singleton sets. If ${\displaystyle \{x\}\in {\mathcal {B}}{\text{ then }}(\{x\},x)}$ is a maximal element of ${\displaystyle \operatorname {PointedSets} ({\mathcal {B}})}$; moreover, all maximal elements are of this form. If ${\displaystyle \left(B,b_{0}\right)\in \operatorname {PointedSets} ({\mathcal {B}}){\text{ then }}\left(B,b_{0}\right)}$ is a greatest element if and only if ${\displaystyle B=\ker {\mathcal {B}},}$ in which case ${\displaystyle \{(B,b)~:~b\in B\}}$ is the set of all greatest elements. However, a greatest element ${\displaystyle (B,b)}$ is a maximal element if and only if ${\displaystyle B=\{b\}=\ker {\mathcal {B}},}$ so there is at most one element that is both maximal and greatest. There is a canonical map ${\displaystyle \operatorname {Point} _{\mathcal {B}}~:~\operatorname {PointedSets} ({\mathcal {B}})\to X}$ defined by ${\displaystyle (B,b)\mapsto b.}$ If ${\displaystyle i_{0}=\left(B_{0},b_{0}\right)\in \operatorname {PointedSets} ({\mathcal {B}})}$ then the tail of the assignment ${\displaystyle \operatorname {Point} _{\mathcal {B}}}$ starting at ${\displaystyle i_{0}}$ is ${\displaystyle \left\{c~:~(C,c)\in \operatorname {PointedSets} ({\mathcal {B}}){\text{ and }}\left(B_{0},b_{0}\right)\leq (C,c)\right\}=B_{0}.}$

Although ${\displaystyle (\operatorname {PointedSets} ({\mathcal {B}}),\leq )}$ is not, in general, a partially ordered set, it is a directed set if (and only if) ${\displaystyle {\mathcal {B}}}$ is a prefilter. So the most immediate choice for the definition of "the net in ${\displaystyle X}$ induced by a prefilter ${\displaystyle {\mathcal {B}}}$" is the assignment ${\displaystyle (B,b)\mapsto b}$ from ${\displaystyle \operatorname {PointedSets} ({\mathcal {B}})}$ into ${\displaystyle X.}$

If ${\displaystyle {\mathcal {B}}}$ is a prefilter on ${\displaystyle X}$ then the net associated with ${\displaystyle {\mathcal {B}}}$ is the map

$${\displaystyle {\begin{alignedat}{4}\operatorname {Net} _{\mathcal {B}}:\;&&(\operatorname {PointedSets} ({\mathcal {B}}),\leq )&&\,\to \;&X\\&&(B,b)&&\,\mapsto \;&b\\\end{alignedat}}}$$

that is, ${\displaystyle \operatorname {Net} _{\mathcal {B}}(B,b):=b.}$

If ${\displaystyle {\mathcal {B}}}$ is a prefilter on ${\displaystyle X{\text{ then }}\operatorname {Net} _{\mathcal {B}}}$ is a net in ${\displaystyle X}$ and the prefilter associated with ${\displaystyle \operatorname {Net} _{\mathcal {B}}}$ is ${\displaystyle {\mathcal {B}}}$; that is:^{[note 11]}

This would not necessarily be true had ${\displaystyle \operatorname {Net} _{\mathcal {B}}}$ been defined on a proper subset of ${\displaystyle \operatorname {PointedSets} ({\mathcal {B}}).}$ For example, suppose ${\displaystyle X}$ has at least two distinct elements, ${\displaystyle {\mathcal {B}}:=\{X\}}$ is the indiscrete filter, and ${\displaystyle x\in X}$ is arbitrary. Had ${\displaystyle \operatorname {Net} _{\mathcal {B}}}$ instead been defined on the singleton set ${\displaystyle D:=\{(X,x)\},}$ where the restriction of ${\displaystyle \operatorname {Net} _{\mathcal {B}}}$ to ${\displaystyle D}$ will temporarily be denote by ${\displaystyle \operatorname {Net} _{D}:D\to X,}$ then the prefilter of tails associated with ${\displaystyle \operatorname {Net} _{D}:D\to X}$ would be the principal prefilter ${\displaystyle \{\,\{x\}\,\}}$ rather than the original filter ${\displaystyle {\mathcal {B}}=\{X\}}$; this means that the equality ${\displaystyle \operatorname {Tails} \left(\operatorname {Net} _{D}\right)={\mathcal {B}}}$ is false, so unlike ${\displaystyle \operatorname {Net} _{\mathcal {B}},}$ the prefilter ${\displaystyle {\mathcal {B}}}$ can not be recovered from ${\displaystyle \operatorname {Net} _{D}.}$ Worse still, while ${\displaystyle {\mathcal {B}}}$ is the unique minimal filter on ${\displaystyle X,}$ the prefilter ${\displaystyle \operatorname {Tails} \left(\operatorname {Net} _{D}\right)=\{\{x\}\}}$ instead generates a maximal filter (that is, an ultrafilter) on ${\displaystyle X.}$

However, if ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}$ is a net in ${\displaystyle X}$ then it is not in general true that ${\displaystyle \operatorname {Net} _{\operatorname {Tails} \left(x_{\bullet }\right)}}$ is equal to ${\displaystyle x_{\bullet }}$ because, for example, the domain of ${\displaystyle x_{\bullet }}$ may be of a completely different cardinality than that of ${\displaystyle \operatorname {Net} _{\operatorname {Tails} \left(x_{\bullet }\right)}}$ (since unlike the domain of ${\displaystyle \operatorname {Net} _{\operatorname {Tails} \left(x_{\bullet }\right)},}$ the domain of an arbitrary net in ${\displaystyle X}$ could have any cardinality).

Ultranets and ultra prefilters

A net ${\displaystyle x_{\bullet }{\text{ in }}X}$ is called an ultranet or universal net in ${\displaystyle X}$ if for every subset ${\displaystyle S\subseteq X,x_{\bullet }}$ is eventually in ${\displaystyle S}$ or it is eventually in ${\displaystyle X\setminus S}$; this happens if and only if ${\displaystyle \operatorname {Tails} \left(x_{\bullet }\right)}$ is an ultra prefilter. A prefilter ${\displaystyle {\mathcal {B}}{\text{ on }}X}$ is an ultra prefilter if and only if ${\displaystyle \operatorname {Net} _{\mathcal {B}}}$ is an ultranet in ${\displaystyle X.}$

Partially ordered net[]

The domain of the canonical net ${\displaystyle \operatorname {Net} _{\mathcal {B}}}$ is in general not partially ordered. However, in 1955 Bruns and Schmidt discovered^[41] a construction that allows for the canonical net to have a domain that is both partially ordered and directed; this was independently rediscovered by Albert Wilansky in 1970.^[4] It begins with the construction of a strict partial order (meaning a transitive and irreflexive relation) ${\displaystyle \,<\,}$ on a subset of ${\displaystyle {\mathcal {B}}\times \mathbb {N} \times X}$ that is similar to the lexicographical order on ${\displaystyle {\mathcal {B}}\times \mathbb {N} }$ of the strict partial orders ${\displaystyle ({\mathcal {B}},\supsetneq ){\text{ and }}(\mathbb {N} ,<).}$ For any ${\displaystyle i=(B,m,b){\text{ and }}j=(C,n,c)}$ in ${\displaystyle {\mathcal {B}}\times \mathbb {N} \times X,}$ declare that ${\displaystyle i<j}$ if and only if

or equivalently, if and only if ${\displaystyle {\text{(1) }}B\supseteq C,{\text{ and (2) if }}B=C{\text{ then }}m<n.}$

The non−strict partial order associated with ${\displaystyle \,<,}$ denoted by ${\displaystyle \,\leq ,}$ is defined by declaring that ${\displaystyle i\leq j\,{\text{ if and only if }}i<j{\text{ or }}i=j.}$ Unwinding these definitions gives the following characterization:

${\displaystyle i\leq j}$ if and only if ${\displaystyle {\text{(1) }}B\supseteq C,{\text{ and (2) if }}B=C{\text{ then }}m\leq n,}$ and also ${\displaystyle {\text{(3) if }}B=C{\text{ and }}m=n{\text{ then }}b=c,}$

which shows that ${\displaystyle \,\leq \,}$ is just the lexicographical order on ${\displaystyle {\mathcal {B}}\times \mathbb {N} \times X}$ induced by ${\displaystyle ({\mathcal {B}},\supseteq ),\,(\mathbb {N} ,\leq ),{\text{ and }}(X,=),}$ where ${\displaystyle X}$ is partially ordered by equality ${\displaystyle \,=.\,}$^{[note 12]} Both ${\displaystyle \,<{\text{ and }}\leq \,}$ are serial and neither possesses a greatest element or a maximal element; this remains true if they are each restricted to the subset of ${\displaystyle {\mathcal {B}}\times \mathbb {N} \times X}$ defined by

where it will henceforth be assumed that they are. Denote the assignment ${\displaystyle i=(B,m,b)\mapsto b}$ from this subset by: If ${\displaystyle i_{0}=\left(B_{0},m_{0},b_{0}\right)\in \operatorname {Poset} _{\mathcal {B}}}$ then just as with ${\displaystyle \operatorname {Net} _{\mathcal {B}}}$ before, the tail of the ${\displaystyle \operatorname {PosetNet} _{\mathcal {B}}}$ starting at ${\displaystyle i_{0}}$ is equal to ${\displaystyle B_{0}.}$ If ${\displaystyle {\mathcal {B}}}$ is a prefilter on ${\displaystyle X}$ then ${\displaystyle \operatorname {PosetNet} _{\mathcal {B}}}$ is a net in ${\displaystyle X}$ whose domain ${\displaystyle \operatorname {Poset} _{\mathcal {B}}}$ is a partially ordered set and moreover, ${\displaystyle \operatorname {Tails} \left(\operatorname {PosetNet} _{\mathcal {B}}\right)={\mathcal {B}}.}$^[4] Because the tails of ${\displaystyle \operatorname {PosetNet} _{\mathcal {B}}{\text{ and }}\operatorname {Net} _{\mathcal {B}}}$ are identical (since both are equal to the prefilter ${\displaystyle {\mathcal {B}}}$), there is typically nothing lost by assuming that the domain of the net associated with a prefilter is both directed and partially ordered.^[4] If the set ${\displaystyle \mathbb {N} }$ is replaced with the positive rational numbers then the strict partial order ${\displaystyle <}$ will also be a dense order.

Subordinate filters and subnets[]

The notion of "${\displaystyle {\mathcal {B}}}$ is subordinate to ${\displaystyle {\mathcal {C}}}$" (written ${\displaystyle {\mathcal {B}}\vdash {\mathcal {C}}}$) is for filters and prefilters what "${\displaystyle x_{n_{\bullet }}=\left(x_{n_{i}}\right)_{i=1}^{\infty }}$ is a subsequence of ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }}$" is for sequences.^[24] For example, if ${\displaystyle \operatorname {Tails} \left(x_{\bullet }\right)=\left\{x_{\geq i}:i\in \mathbb {N} \right\}}$ denotes the set of tails of ${\displaystyle x_{\bullet }}$ and if ${\displaystyle \operatorname {Tails} \left(x_{n_{\bullet }}\right)=\left\{x_{n_{\geq i}}:i\in \mathbb {N} \right\}}$ denotes the set of tails of the subsequence ${\displaystyle x_{n_{\bullet }}}$ (where ${\displaystyle x_{n_{\geq i}}:=\left\{x_{n_{i}}~:~i\in \mathbb {N} \right\}}$) then ${\displaystyle \operatorname {Tails} \left(x_{n_{\bullet }}\right)~\vdash ~\operatorname {Tails} \left(x_{\bullet }\right)}$ (that is, ${\displaystyle \operatorname {Tails} \left(x_{\bullet }\right)\leq \operatorname {Tails} \left(x_{n_{\bullet }}\right)}$) is true but ${\displaystyle \operatorname {Tails} \left(x_{\bullet }\right)~\vdash ~\operatorname {Tails} \left(x_{n_{\bullet }}\right)}$ is in general false. If ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}$ is a net in a topological space ${\displaystyle X}$ and if ${\displaystyle {\mathcal {N}}(x)}$ is the neighborhood filter at a point ${\displaystyle x\in X,}$ then ${\displaystyle x_{\bullet }\to x{\text{ in }}X{\text{ if and only if }}{\mathcal {N}}(x)\leq \operatorname {Tails} \left(x_{\bullet }\right).}$

Subordination analogs of results involving subsequences[]

The following results are the prefilter analogs of statements involving subsequences.^[42] The condition "${\displaystyle {\mathcal {C}}\geq {\mathcal {B}},}$" which is also written ${\displaystyle {\mathcal {C}}\vdash {\mathcal {B}},}$ is the analog of "${\displaystyle {\mathcal {C}}}$ is a subsequence of ${\displaystyle {\mathcal {B}}.}$" So "finer than" and "subordinate to" is the prefilter analog of "subsequence of." Some people prefer saying "subordinate to" instead of "finer than" because it is more reminiscent of "subsequence of."

Non–equivalence of subnets and subordinate filters[]

A subset ${\displaystyle R\subseteq I}$ of a preordered space ${\displaystyle (I,\leq )}$ is frequent or cofinal in ${\displaystyle I}$ if for every ${\displaystyle i\in I}$ there exists some ${\displaystyle r\in R{\text{ such that }}i\leq r.}$ If ${\displaystyle R\subseteq I}$ contains a tail of ${\displaystyle I}$ then ${\displaystyle R}$ is said to be eventual or eventually in ${\displaystyle I}$; explicitly, this means that there exists some ${\displaystyle i\in I{\text{ such that }}I_{\geq i}\subseteq R}$ (that is, ${\displaystyle j\in R{\text{ for all }}j\in I{\text{ satisfying }}i\leq j}$). A subset is eventual if and only if its complement is not frequent (which is termed infrequent).^[43] A map ${\displaystyle h:A\to I}$ between two preordered sets is order–preserving if whenever ${\displaystyle a,b\in A{\text{ satisfy }}a\leq b,{\text{ then }}h(a)\leq h(b).}$

Subnets in the sense of Willard and subnets in the sense of Kelley are the most commonly used definitions of "subnet."^[43] The first definition of a subnet was introduced by John L. Kelley in 1955.^[43] Stephen Willard introduced his own variant of Kelley's definition of subnet in 1970.^[43] AA–subnets were introduced independently by Smiley (1957), Aarnes and Andenaes (1972), and Murdeshwar (1983); AA–subnets were studied in great detail by Aarnes and Andenaes but they are not often used.^[43]

Let ${\displaystyle S=S_{\bullet }~:~(A,\leq )\to X{\text{ and }}N=N_{\bullet }~:~(I,\leq )\to X}$ be nets. Then^[43]

1. ${\displaystyle S_{\bullet }}$ is a Willard–subnet of ${\displaystyle N_{\bullet }}$ or a subnet in the sense of Willard if there exists an order–preserving map ${\displaystyle h:A\to I}$ such that ${\displaystyle S=N\circ h{\text{ and }}h(A)}$ is cofinal in ${\displaystyle I.}$
2. ${\displaystyle S_{\bullet }}$ is a Kelley–subnet of ${\displaystyle N_{\bullet }}$ or a subnet in the sense of Kelley if there exists a map ${\displaystyle h~:~A\to I{\text{ such that }}S=N\circ h}$ and whenever ${\displaystyle E\subseteq I}$ is eventually in ${\displaystyle I}$ then ${\displaystyle h^{-1}(E)}$ is eventually in ${\displaystyle A.}$
3. ${\displaystyle S_{\bullet }}$ is an AA–subnet of ${\displaystyle N_{\bullet }}$ or a subnet in the sense of Aarnes and Andenaes if any of the following equivalent conditions are satisfied:
   1. ${\displaystyle \operatorname {Tails} \left(N_{\bullet }\right)\leq \operatorname {Tails} \left(S_{\bullet }\right).}$
   2. ${\displaystyle \operatorname {TailsFilter} \left(N_{\bullet }\right)\subseteq \operatorname {TailsFilter} \left(S_{\bullet }\right).}$
   3. If ${\displaystyle J}$ is eventually in ${\displaystyle I{\text{ then }}S^{-1}(N(J))}$ is eventually in ${\displaystyle A.}$
   4. For any subset ${\displaystyle R\subseteq X,{\text{ if }}\operatorname {Tails} \left(S_{\bullet }\right){\text{ and }}\{R\}}$ mesh, then so do ${\displaystyle \operatorname {Tails} \left(N_{\bullet }\right){\text{ and }}\{R\}.}$
   5. For any subset ${\displaystyle R\subseteq X,{\text{ if }}\operatorname {Tails} \left(S_{\bullet }\right)\leq \{R\}{\text{ then }}\operatorname {Tails} \left(N_{\bullet }\right)\leq \{R\}.}$

Kelley did not require the map ${\displaystyle h}$ to be order preserving while the definition of an AA–subnet does away entirely with any map between the two nets' domains and instead focuses entirely on ${\displaystyle X}$ − the nets' common codomain. Every Willard–subnet is a Kelley–subnet and both are AA–subnets.^[43] In particular, if ${\displaystyle y_{\bullet }=\left(y_{a}\right)_{a\in A}}$ is a Willard–subnet or a Kelley–subnet of ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}$ then ${\displaystyle \operatorname {Tails} \left(x_{\bullet }\right)\leq \operatorname {Tails} \left(y_{\bullet }\right).}$

AA–subnets have a defining characterization that immediately shows that they are fully interchangeable with sub(ordinate)filters.^[43][44] Explicitly, what is meant is that the following statement is true for AA–subnets:

     If ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {F}}}$ are prefilters then ${\displaystyle {\mathcal {B}}\leq {\mathcal {F}}{\text{ if and only if }}\operatorname {Net} _{\mathcal {F}}}$ is an AA–subset of ${\displaystyle \;\operatorname {Net} _{\mathcal {B}}.}$

If "AA–subnet" is replaced by "Willard–subnet" or "Kelley–subnet" then the above statement becomes false. In particular, the problem is that the following statement is in general false:

     False statement: If ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {F}}}$ are prefilters such that ${\displaystyle {\mathcal {B}}\leq {\mathcal {F}}{\text{ then }}\operatorname {Net} _{\mathcal {F}}}$ is a Kelley–subset of ${\displaystyle \;\operatorname {Net} _{\mathcal {B}}.}$

Since every Willard–subnet is a Kelley–subnet, this statement remains false if the word "Kelley–subnet" is replaced with "Willard–subnet".

• Counter example: For all ${\displaystyle n\in \mathbb {N} ,}$ let ${\displaystyle B_{n}=\{1\}\cup \mathbb {N} _{\geq n}.}$ Let ${\displaystyle {\mathcal {B}}=\{B_{n}~:~n\in \mathbb {N} \},}$ which is a proper π–system, and let ${\displaystyle {\mathcal {F}}=\{\{1\}\}\cup {\mathcal {B}},}$ where both families are prefilters on the natural numbers ${\displaystyle X:=\mathbb {N} =\{1,2,\ldots \}.}$ Because ${\displaystyle {\mathcal {B}}\leq {\mathcal {F}},{\mathcal {F}}}$ is to ${\displaystyle {\mathcal {B}}}$ as a subsequence is to a sequence. So ideally, ${\displaystyle S=\operatorname {Net} _{\mathcal {F}}}$ should be a subnet of ${\displaystyle B=\operatorname {Net} _{\mathcal {B}}.}$ Let ${\displaystyle I:=\operatorname {PointedSets} ({\mathcal {B}})}$ be the domain of ${\displaystyle \operatorname {Net} _{\mathcal {B}},}$ so ${\displaystyle I}$ contains a cofinal subset that is order isomorphic to ${\displaystyle \mathbb {N} }$ and consequently contains neither a maximal nor greatest element. Let ${\displaystyle A:=\operatorname {PointedSets} ({\mathcal {F}})=\{M\}\cup I,{\text{ where }}M:=(1,\{1\})}$ is both a maximal and greatest element of ${\displaystyle A.}$ The directed set ${\displaystyle A}$ also contains a subset that is order isomorphic to ${\displaystyle \mathbb {N} }$ (because it contains ${\displaystyle I,}$ which contains such a subset) but no such subset can be cofinal in ${\displaystyle A}$ because of the maximal element ${\displaystyle M.}$ Consequently, any order–preserving map ${\displaystyle h:A\to I}$ must be eventually constant (with value ${\displaystyle h(M)}$) where ${\displaystyle h(M)}$ is then a greatest element of the range ${\displaystyle \operatorname {image} h.}$ Because of this, there can be no order preserving map ${\displaystyle h:A\to I}$ that satisfies the conditions required for ${\displaystyle \operatorname {Net} _{\mathcal {F}}}$ to be a Willard–subnet of ${\displaystyle \operatorname {Net} _{\mathcal {B}}}$ (because the range of such a map ${\displaystyle h}$ cannot be cofinal in ${\displaystyle I}$). Suppose for the sake of contradiction that there exists a map ${\displaystyle h:A\to I}$ such that ${\displaystyle h^{-1}\left(I_{\geq i}\right)}$ is eventually in ${\displaystyle A}$ for all ${\displaystyle i\in I.}$ Because ${\displaystyle h(M)\in I,}$ there exist ${\displaystyle n,n_{0}\in \mathbb {N} }$ such that ${\displaystyle h(M)=\left(n_{0},B_{n}\right){\text{ with }}n_{0}\in B_{n}.}$ For every ${\displaystyle i\in I,}$ because ${\displaystyle h^{-1}\left(I_{\geq i}\right)}$ is eventually in ${\displaystyle A,}$ it is necessary that ${\displaystyle h(M)\in I_{\geq i}.}$ In particular, if ${\displaystyle i:=\left(n+2,B_{n+2}\right)}$ then ${\displaystyle h(M)\geq i=\left(n+2,B_{n+2}\right),}$ which by definition is equivalent to ${\displaystyle B_{n}\subseteq B_{n+2},}$ which is false. Consequently, ${\displaystyle \operatorname {Net} _{\mathcal {F}}}$ is not a Kelley–subnet of ${\displaystyle \operatorname {Net} _{\mathcal {B}}.}$^[44]

If "subnet" is defined to mean Willard–subnet or Kelley–subnet then nets and filters are not completely interchangeable because there exists a filter–sub(ordinate)filter relationships that cannot be expressed in terms of a net–subnet relationship between the two induced nets. In particular, the problem is that Kelley–subnets and Willard–subnets are not fully interchangeable with subordinate filters. If the notion of "subnet" is not used or if "subnet" is defined to mean AA–subnet, then this ceases to be a problem and so it becomes correct to say that nets and filters are interchangeable. Despite the fact that AA–subnets do not have the problem that Willard and Kelley subnets have, they are not widely used or known about.^[43][44]

Topologies and prefilters[]

Throughout, ${\displaystyle (X,\tau )}$ is a topological space.

Examples of relationships between filters and topologies[]

Bases and prefilters

Let ${\displaystyle {\mathcal {B}}\neq \varnothing }$ be a family of sets that covers ${\displaystyle X}$ and define ${\displaystyle {\mathcal {B}}_{x}=\{B\in {\mathcal {B}}~:~x\in B\}}$ for every ${\displaystyle x\in X.}$ The definition of a base for some topology can be immediately reworded as: ${\displaystyle {\mathcal {B}}}$ is a base for some topology on ${\displaystyle X}$ if and only if ${\displaystyle {\mathcal {B}}_{x}}$ is a filter base for every ${\displaystyle x\in X.}$ If ${\displaystyle \tau }$ is a topology on ${\displaystyle X}$ and ${\displaystyle {\mathcal {B}}\subseteq \tau }$ then the definitions of ${\displaystyle {\mathcal {B}}}$ is a basis (resp. subbase) for ${\displaystyle \tau }$ can be reworded as:

     ${\displaystyle {\mathcal {B}}}$ is a base (resp. subbase) for ${\displaystyle \tau }$ if and only if for every ${\displaystyle x\in X,{\mathcal {B}}_{x}}$ is a filter base (resp. filter subbase) that generates the neighborhood filter of ${\displaystyle (X,\tau )}$ at ${\displaystyle x.}$

Neighborhood filters

The archetypical example of a filter is the set of all neighborhoods of a point in a topological space. Any neighborhood basis of a point in (or of a subset of) a topological space is a prefilter. In fact, the definition of a neighborhood base can be equivalently restated as: "a neighborhood base is any prefilter that is equivalent the neighborhood filter."

Neighborhood bases at points are examples of prefilters that are fixed but may or may not be principal. If ${\displaystyle X=\mathbb {R} }$ has its usual topology and if ${\displaystyle x\in X,}$ then any neighborhood filter base ${\displaystyle {\mathcal {B}}}$ of ${\displaystyle x}$ is fixed by ${\displaystyle x}$ (in fact, it is even true that ${\displaystyle \ker {\mathcal {B}}=\{x\}}$) but ${\displaystyle {\mathcal {B}}}$ is not principal since ${\displaystyle \{x\}\not \in {\mathcal {B}}.}$ In contrast, a topological space has the discrete topology if and only if the neighborhood filter of every point is a principal filter generated by exactly one point. This shows that a non–principal filter on an infinite set is not necessarily free.

The neighborhood filter of every point ${\displaystyle x}$ in topological space ${\displaystyle X}$ is fixed since its kernel contains ${\displaystyle x}$ (and possibly other points if, for instance, ${\displaystyle X}$ is not a T₁ space). This is also true of any neighborhood basis at ${\displaystyle x.}$ For any point ${\displaystyle x}$ in a T₁ space (for example, a Hausdorff space), the kernel of the neighborhood filter of ${\displaystyle x}$ is equal to the singleton set ${\displaystyle \{x\}.}$

However, it is possible for a neighborhood filter at a point to be principal but not discrete (that is, not principal at a single point). A neighborhood basis ${\displaystyle {\mathcal {B}}}$ of a point ${\displaystyle x}$ in a topological space is principal if and only if the kernel of ${\displaystyle {\mathcal {B}}}$ is an open set. If in addition the space is T₁ then ${\displaystyle \ker {\mathcal {B}}=\{x\}}$ so that this basis ${\displaystyle {\mathcal {B}}}$ is principal if and only if ${\displaystyle \{x\}}$ is an open set.

Generating topologies from filters and prefilters

Suppose ${\displaystyle {\mathcal {B}}\subseteq \wp (X)}$ is not empty (and ${\displaystyle X\neq \varnothing }$). If ${\displaystyle {\mathcal {B}}}$ is a filter on ${\displaystyle X}$ then ${\displaystyle \{\varnothing \}\cup {\mathcal {B}}}$ is a topology on ${\displaystyle X}$ but the converse is in general false. This shows that in a sense, filters are almost topologies. Topologies of the form ${\displaystyle \{\varnothing \}\cup {\mathcal {B}}}$ where ${\displaystyle {\mathcal {B}}}$ is an ultrafilter on ${\displaystyle X}$ are an even more specialized subclass of such topologies; they have the property that every proper subset ${\displaystyle \varnothing \neq S\subseteq X}$ is either open or closed, but (unlike the discrete topology) never both. These spaces are, in particular, examples of door spaces.

If ${\displaystyle {\mathcal {B}}}$ is a prefilter (resp. filter subbase, π–system, proper) on ${\displaystyle X}$ then the same is true of both ${\displaystyle \{X\}\cup {\mathcal {B}}}$ and the set ${\displaystyle {\mathcal {B}}_{\cup }}$ of all possible unions of one or more elements of ${\displaystyle {\mathcal {B}}.}$ If ${\displaystyle {\mathcal {B}}}$ is closed under finite intersections then the set ${\displaystyle \tau _{\mathcal {B}}=\{\varnothing ,X\}\cup {\mathcal {B}}_{\cup }}$ is a topology on ${\displaystyle X}$ with both ${\displaystyle \{X\}\cup {\mathcal {B}}_{\cup }{\text{ and }}\{X\}\cup {\mathcal {B}}}$ being bases for it. If the π–system ${\displaystyle {\mathcal {B}}}$ covers ${\displaystyle X}$ then both ${\displaystyle {\mathcal {B}}_{\cup }{\text{ and }}{\mathcal {B}}}$ are also bases for ${\displaystyle \tau _{\mathcal {B}}.}$ If ${\displaystyle \tau }$ is a topology on ${\displaystyle X}$ then ${\displaystyle \tau \setminus \{\varnothing \}}$ is a prefilter (or equivalently, a π–system) if and only if it has the finite intersection property (that is, it is a filter subbase), in which case a subset ${\displaystyle {\mathcal {B}}\subseteq \tau }$ will be a basis for ${\displaystyle \tau }$ if and only if ${\displaystyle {\mathcal {B}}\setminus \{\varnothing \}}$ is equivalent to ${\displaystyle \tau \setminus \{\varnothing \},}$ in which case ${\displaystyle {\mathcal {B}}\setminus \{\varnothing \}}$ will be a prefilter.

Topologies on directed sets and net convergence

Let ${\displaystyle (I,\leq )}$ be a non–empty directed set and let ${\displaystyle \operatorname {Tails} (I)=\left\{I_{\geq i}~:~i\in I\right\},}$ where ${\displaystyle I_{\geq i}=\{j\in I~:~i\leq j\}.}$ Then ${\displaystyle \operatorname {Tails} (I)}$ is a prefilter that covers ${\displaystyle I}$ and if ${\displaystyle I}$ is totally ordered then ${\displaystyle \operatorname {Tails} (I)}$ is also closed under finite intersections. This particular prefilter ${\displaystyle \operatorname {Tails} (I)}$ forms a base for a topology on ${\displaystyle I}$ in which all sets of the form ${\displaystyle I_{>i}=\{j\in I~:~i<j\}}$ are also open. The same is true of the topology ${\displaystyle \tau _{I}:=\{\varnothing \}\cup \operatorname {FilterTails} (I){\text{ on }}I,}$ where ${\displaystyle \operatorname {FilterTails} (I)}$ is the filter on ${\displaystyle I}$ generated by ${\displaystyle \operatorname {Tails} (I).}$ With this topology, convergent nets can be viewed as continuous functions in the following way. Let ${\displaystyle (X,\tau )}$ be a topological space, let ${\displaystyle x\in X,}$ let ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}~:~I\to X}$ be a net in ${\displaystyle X,}$ and let ${\displaystyle \tau (x)\subseteq \tau }$ denote the set of all open neighborhoods of ${\displaystyle x.}$ If the net ${\displaystyle x_{\bullet }}$ converges to ${\displaystyle x{\text{ in }}(X,\tau )}$ then ${\displaystyle x_{\bullet }~:~\left(I,\tau _{I}\right)\to \left(X,\{\varnothing \}\cup \tau (x)\right)}$ is necessarily continuous although in general, the converse is false (for example, consider if ${\displaystyle x_{\bullet }}$ is constant and not equal to ${\displaystyle x}$). But if in addition to continuity, the preimage under ${\displaystyle x_{\bullet }}$ of every ${\displaystyle N\in \tau (x)}$ is not empty, then the net ${\displaystyle x_{\bullet }}$ will necessarily converge to ${\displaystyle x{\text{ in }}(X,\tau ).}$ In this way, the empty set is all that separates net convergence and continuity.

Topological properties and prefilters[]

Throughout ${\displaystyle (X,\tau )}$ will be a topological space with ${\displaystyle X\neq \varnothing .}$

Neighborhoods and topologies

The neighborhood filter of a non–empty subset ${\displaystyle S\subseteq X}$ in a topological space ${\displaystyle X}$ is equal to the intersection of all neighborhood filters of all points in ${\displaystyle S.}$^[28] If ${\displaystyle S\subseteq X}$ then ${\displaystyle S}$ is open in ${\displaystyle X}$ if and only if whenever ${\displaystyle {\mathcal {F}}}$ is a filter on ${\displaystyle X}$ and ${\displaystyle s\in S,}$ then ${\displaystyle {\mathcal {F}}\to s{\text{ in }}(X,\tau ){\text{ implies }}S\in {\mathcal {F}}.}$

Suppose ${\displaystyle \sigma {\text{ and }}\tau }$ are topologies on ${\displaystyle X.}$ Then ${\displaystyle \tau }$ is finer than ${\displaystyle \sigma }$ (that is, ${\displaystyle \sigma \subseteq \tau }$) if and only if whenever ${\displaystyle x\in X{\text{ and }}{\mathcal {B}}}$ is a filter on ${\displaystyle X,}$ if ${\displaystyle {\mathcal {B}}\to x{\text{ in }}(X,\tau )}$ then ${\displaystyle {\mathcal {B}}\to x{\text{ in }}(X,\sigma ).}$^[39] Consequently, ${\displaystyle \sigma =\tau }$ if and only if for every filter ${\displaystyle {\mathcal {B}}{\text{ on }}X}$ and every ${\displaystyle x\in X,{\mathcal {B}}\to x{\text{ in }}(X,\sigma )}$ if and only if ${\displaystyle {\mathcal {B}}\to x{\text{ in }}(X,\tau ).}$^[29] However, it is possible that ${\displaystyle \sigma \neq \tau }$ while also for every filter ${\displaystyle {\mathcal {B}}{\text{ on }}X,{\mathcal {B}}}$ converges to some point of ${\displaystyle X{\text{ in }}(X,\sigma )}$ if and only if ${\displaystyle {\mathcal {B}}}$ converges to some point of ${\displaystyle X{\text{ in }}(X,\tau ).}$^[29]

Closure

If ${\displaystyle x\in X{\text{ and }}S\subseteq X{\text{ with }}S\neq \varnothing }$ then the following are equivalent:

1. ${\displaystyle x\in \operatorname {ck} _{X}S}$
2. ${\displaystyle x}$ is a limit point of the prefilter ${\displaystyle \{S\}}$ (that is, ${\displaystyle \{S\}\to x{\text{ in }}X}$).
3. There exists a prefilter ${\displaystyle {\mathcal {F}}\subseteq \wp (X){\text{ on }}X}$ such that ${\displaystyle S\in {\mathcal {F}}{\text{ and }}{\mathcal {F}}\to x{\text{ in }}X.}$
4. There exists a prefilter ${\displaystyle {\mathcal {F}}\subseteq \wp (S){\text{ on }}S}$ such that ${\displaystyle {\mathcal {F}}\to x{\text{ in }}X.}$^[42]
5. ${\displaystyle x}$ is a cluster point of the prefilter ${\displaystyle \{S\}.}$
6. The prefilter ${\displaystyle \{S\}}$ meshes with the neighborhood filter ${\displaystyle {\mathcal {N}}(x).}$
7. The prefilter ${\displaystyle \{S\}}$ meshes with some (or equivalently, with every) prefilter of ${\displaystyle {\mathcal {N}}(x).}$

The following are equivalent:

1. ${\displaystyle x}$ is a limit points of ${\displaystyle S{\text{ in }}X.}$
2. There exists a prefilter ${\displaystyle {\mathcal {F}}\subseteq \wp (S){\text{ on }}\{S\}\setminus \{x\}}$ such that ${\displaystyle {\mathcal {F}}\to x{\text{ in }}X.}$^[42]

Closed sets

If ${\displaystyle S\subseteq X}$ is not empty then the following are equivalent:

1. ${\displaystyle S}$ is a closed subset of ${\displaystyle X.}$
2. If ${\displaystyle x\in X{\text{ and }}{\mathcal {F}}\subseteq \wp (S)}$ is a prefilter on ${\displaystyle S}$ such that ${\displaystyle {\mathcal {F}}\to x{\text{ in }}X,}$ then ${\displaystyle x\in S.}$
3. If ${\displaystyle x\in X{\text{ and }}{\mathcal {F}}\subseteq \wp (S)}$ is a prefilter on ${\displaystyle S}$ such that ${\displaystyle x}$ is an accumulation points of ${\displaystyle {\mathcal {F}}{\text{ in }}X,}$ then ${\displaystyle x\in S.}$^[42]
4. If ${\displaystyle x\in X}$ is such that the neighborhood filter ${\displaystyle {\mathcal {N}}(x)}$ meshes with ${\displaystyle \{S\}}$ then ${\displaystyle x\in S.}$
   • The proof of this characterization depends the ultrafilter lemma, which depends on the axiom of choice.

Hausdorffness

The following are equivalent:

1. ${\displaystyle X}$ is Hausdorff.
2. Every prefilter on ${\displaystyle X}$ converges to at most one point in ${\displaystyle X.}$^[8]
3. The above statement but with the word "prefilter" replaced by any one of the following: filter, ultra prefilter, ultrafilter.^[8]

Compactness

As discussed in this article, the Ultrafilter Lemma is closely related to many important theorems involving compactness.

The following are equivalent:

1. ${\displaystyle (X,\tau )}$ is a compact space.
2. Every ultrafilter on ${\displaystyle X}$ converges to at least one point in ${\displaystyle X.}$
   • That this condition implies compactness can be proven by using only the ultrafilter lemma. That compactness implies this condition can be proven without the ultrafilter lemma (or even the axiom of choice).
3. The above statement but with the words "prefilter" replaced by any one of the following: filter, ultrafilter.^[45]
4. For every filter ${\displaystyle {\mathcal {C}}{\text{ on }}X}$ there exists a filter ${\displaystyle {\mathcal {F}}{\text{ on }}X}$ such that ${\displaystyle {\mathcal {C}}\leq {\mathcal {F}}}$ and ${\displaystyle {\mathcal {F}}}$ converges to some point of ${\displaystyle X.}$
5. For every prefilter ${\displaystyle {\mathcal {C}}{\text{ on }}X}$ there exists a prefilter ${\displaystyle {\mathcal {F}}{\text{ on }}X}$ such that ${\displaystyle {\mathcal {C}}\leq {\mathcal {F}}}$ and ${\displaystyle {\mathcal {F}}}$ converges to some point of ${\displaystyle X.}$
6. Every maximal (i.e. ultra) prefilter on ${\displaystyle X}$ converges to at least one point in ${\displaystyle X.}$^[8]
7. The above statement but with the words "maximal prefilter" replaced by any one of the following: prefilter, filter, ultra prefilter, ultrafilter.
8. Every prefilter on ${\displaystyle X}$ has at least one cluster point in ${\displaystyle X.}$^[8]
   • That this condition is equivalent to compactness can be proven by using only the ultrafilter lemma.
9. Alexander subbase theorem: There exists a subbase ${\displaystyle {\mathcal {S}}{\text{ for }}\tau }$ such that every cover of ${\displaystyle X}$ by sets in ${\displaystyle {\mathcal {S}}}$ has a finite subcover.
   • That this condition is equivalent to compactness can be proven by using only the ultrafilter lemma.

If ${\displaystyle (X,\tau )}$ is topological space and ${\displaystyle {\mathcal {F}}}$ is the set of all complements of compact subsets of ${\displaystyle (X,\tau ),}$ then ${\displaystyle {\mathcal {F}}}$ is a filter on ${\displaystyle X}$ if and only if ${\displaystyle (X,\tau )}$ is not compact.

Continuity

Let ${\displaystyle f:X\to Y}$ is a map between topological spaces ${\displaystyle (X,\tau ){\text{ and }}(Y,\upsilon ).}$

Given ${\displaystyle x\in X,}$ the following are equivalent:

1. ${\displaystyle f:X\to Y}$ is continuous at ${\displaystyle x.}$
2. Definition: For every neighborhood ${\displaystyle V}$ of ${\displaystyle f(x){\text{ in }}Y}$ there exists some neighborhood ${\displaystyle N}$ of ${\displaystyle x{\text{ in }}X}$ such that ${\displaystyle f(N)\subseteq V.}$
3. ${\displaystyle f({\mathcal {N}}(x))}$ is a filter base for ${\displaystyle {\mathcal {N}}(f(x))}$; that is, the upward closure of ${\displaystyle f({\mathcal {N}}(x)){\text{ in }}Y}$ is equal to ${\displaystyle {\mathcal {N}}(f(x)).}$^[42]
4. ${\displaystyle f({\mathcal {N}}(x))\to f(x){\text{ in }}Y.}$^[42]
5. If ${\displaystyle {\mathcal {B}}}$ is a filter on ${\displaystyle X}$ such that ${\displaystyle {\mathcal {B}}\to x{\text{ in }}X}$ then ${\displaystyle f({\mathcal {B}})\to f(x){\text{ in }}Y.}$
6. The above statement but with the word "filter" replaced by "prefilter".

The following are equivalent:

1. ${\displaystyle f:X\to Y}$ is continuous.
2. If ${\displaystyle x\in X{\text{ and }}{\mathcal {B}}}$ is a prefilter on ${\displaystyle X}$ such that ${\displaystyle {\mathcal {B}}\to x{\text{ in }}X}$ then ${\displaystyle f({\mathcal {B}})\to f(x){\text{ in }}Y.}$
3. If ${\displaystyle x\in X}$ is a limit point of a prefilter ${\displaystyle {\mathcal {B}}{\text{ on }}X}$ then ${\displaystyle f(x)}$ is a limit point of ${\displaystyle f({\mathcal {B}}){\text{ in }}Y.}$
4. Any one of the above two statements but with the word "prefilter" replaced by any one of the following: filter.

If ${\displaystyle {\mathcal {B}}}$ is a prefilter on ${\displaystyle X,x\in X}$ is a cluster point of ${\displaystyle {\mathcal {B}},{\text{ and }}f:X\to Y}$ is continuous, then ${\displaystyle f(x)}$ is a cluster point in ${\displaystyle Y}$ of the prefilter ${\displaystyle f({\mathcal {B}}).}$^[39]

Products

Suppose ${\displaystyle X_{\bullet }:=\left(X_{i}\right)_{i\in I}}$ is a non–empty family of non–empty topological spaces and that is a family of prefilters where each ${\displaystyle {\mathcal {B}}_{i}}$ is a prefilter on ${\displaystyle X_{i}.}$ Then the product ${\displaystyle {\mathcal {B}}_{\bullet }}$ of these prefilters (defined above) is a prefilter on the product space ${\displaystyle \prod X_{\bullet },}$ which as usual, is endowed with the product topology.

If ${\displaystyle x_{\bullet }:=\left(x_{i}\right)_{i\in I}\in \prod X_{\bullet },}$ then ${\displaystyle {\mathcal {B}}_{\bullet }\to x_{\bullet }{\text{ in }}\prod X_{\bullet }}$ if and only if ${\displaystyle {\mathcal {B}}_{i}\to x_{i}{\text{ in }}X_{i}{\text{ for every }}i\in I.}$

Suppose ${\displaystyle X{\text{ and }}Y}$ are topological spaces, ${\displaystyle {\mathcal {B}}}$ is a prefilter on ${\displaystyle X}$ having ${\displaystyle x\in X}$ as a cluster point, and ${\displaystyle {\mathcal {C}}}$ is a prefilter on ${\displaystyle Y}$ having ${\displaystyle y\in Y}$ as a cluster point. Then ${\displaystyle (x,y)}$ is a cluster point of ${\displaystyle {\mathcal {B}}\times {\mathcal {C}}}$ in the product space ${\displaystyle X\times Y.}$^[39] However, if ${\displaystyle X=Y=\mathbb {Q} }$ then there exist sequences ${\displaystyle x_{\bullet }:=\left(x_{i}\right)_{i=1}^{\infty }\subseteq X{\text{ and }}y_{\bullet }:=\left(y_{i}\right)_{i=1}^{\infty }\subseteq Y}$ such that both of these sequences have a cluster point in ${\displaystyle \mathbb {Q} }$ but the sequence ${\displaystyle \left(x_{i},y_{i}\right)_{i=1}^{\infty }\subseteq X\times Y}$ does not have a cluster point in ${\displaystyle X\times Y.}$^[39]

Example application: The ultrafilter lemma along with the axioms of ZF imply Tychonoff's theorem for compact Hausdorff spaces:

Examples of applications of prefilters[]

Uniformities and Cauchy prefilters[]

A uniform space is a set ${\displaystyle X}$ equipped with a filter on ${\displaystyle X\times X}$ that has certain properties. A base or fundamental system of entourages is a prefilter on ${\displaystyle X\times X}$ whose upward closure is a uniform space. A prefilter ${\displaystyle {\mathcal {B}}}$ on a uniform space ${\displaystyle X}$ with uniformity ${\displaystyle {\mathcal {F}}}$ is called a Cauchy prefilter if for every entourage ${\displaystyle N\in {\mathcal {F}},}$ there exists some ${\displaystyle B\in {\mathcal {B}}}$ that is ${\displaystyle N}$–small, which means that ${\displaystyle B\times B\subseteq N.}$ A minimal Cauchy filter is a minimal element (with respect to ${\displaystyle \,\leq \,}$ or equivalently, to ${\displaystyle \,\subseteq }$) of the set of all Cauchy filters on ${\displaystyle X.}$ A uniform space ${\displaystyle (X,{\mathcal {F}})}$ is called complete (resp. sequentially complete) if every Cauchy prefilter (resp. every elementary Cauchy prefilter) on ${\displaystyle X}$ converges to at least one point of ${\displaystyle X.}$

Uniform spaces were the result of attempts to generalize notions such as "uniform continuity" and "uniform convergence" that are present in metric spaces. Every topological vector space, and more generally, every topological group can be made into a uniform space in a canonical way. Every uniformity also generates a canonical induced topology. Filters and prefilters play an important role in the theory of uniform spaces. For example, the completion of a Hausdorff uniform space (even if it is not metrizable) is typically constructed by using minimal Cauchy filters. Nets are less ideal for this construction because their domains are extremely varied (e.g. the class of all Cauchy nets is not a set); sequences cannot be used in the general case because the topology might not be metrizable, first–countable, or even sequential.

Convergence of nets of sets[]

There is often a personal preference of nets over filters or filters over nets. This example shows that the choice between nets and filters is not a dichotomy by combining them together.

If ${\displaystyle S}$ is a subset of a topological space ${\displaystyle (X,\tau )}$ then the set ${\displaystyle \tau (S)}$ of open neighborhoods of ${\displaystyle S{\text{ in }}(X,\tau )}$ is a prefilter if and only if ${\displaystyle S\neq \varnothing .}$ The same is true of the set ${\displaystyle {\mathcal {N}}_{\tau }(S):=\tau (S)^{\uparrow X}}$ of all neighborhoods of ${\displaystyle S{\text{ in }}(X,\tau ).}$ The following definition generalizes the notion of the set of tails of a net of points in ${\displaystyle X}$ to nets of subsets of ${\displaystyle X.}$

A net of sets in ${\displaystyle X}$ is a net into the power set ${\displaystyle \wp (X)}$ of ${\displaystyle X}$; that is, a net of sets in ${\displaystyle X}$ is a function from a non–empty directed set into ${\displaystyle \wp (X).}$ A net ${\displaystyle S_{\bullet }=\left(S_{i}\right)_{i\in I}}$ of sets in ${\displaystyle X}$ is called a net of singleton (resp. non–empty, finite, compact, etc.) sets in ${\displaystyle X}$ if every ${\displaystyle S_{i}}$ has this property. However, a "net in ${\displaystyle X}$" will always refer to a net valued in ${\displaystyle X}$ and never to a net valued in ${\displaystyle \wp (X)}$ although for emphasis or contrast, a net in ${\displaystyle X}$ may also be referred to as a net of points in ${\displaystyle X}$.

(Nets of points ${\displaystyle \leftrightarrow }$ Nets of singleton sets): Every net ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}$ of points in ${\displaystyle X}$ can be uniquely associated with the canonical net of (singleton) sets ${\displaystyle \left(\left\{x_{i}\right\}\right)_{i\in I}}$ that it indices. Conversely, every net of singleton sets in ${\displaystyle X}$ is uniquely associated with a canonical net of points in ${\displaystyle X}$ (defined in the obvious way). Consideration of this bijective correspondence leads naturally to the following definition, which is completely analogous to the previously given definition of the tails of a net (of points) in ${\displaystyle X.}$

Suppose ${\displaystyle S_{\bullet }=\left(S_{i}\right)_{i\in I}}$ is a net of sets in ${\displaystyle X.}$ Define for every index ${\displaystyle i}$ the tail of ${\displaystyle S_{\bullet }}$ starting at ${\displaystyle i}$ to be the set

and define the set or family of tails generated by ${\displaystyle S_{\bullet }}$ to be the family where if ${\displaystyle \varnothing \notin \operatorname {Tails} \left(S_{\bullet }\right)}$ then this set is called the prefilter or filter base of tails generated by ${\displaystyle S_{\bullet }}$ while the upward closure of ${\displaystyle \operatorname {Tails} \left(S_{\bullet }\right){\text{ in }}X}$ is known as the filter of tails or eventuality filter in ${\displaystyle X}$ generated by ${\displaystyle S_{\bullet }.}$

Given any net ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}$ of points in ${\displaystyle X,}$ it is readily seen that ${\displaystyle \operatorname {Tails} \left(x_{\bullet }\right)=\operatorname {Tails} \left(\left\{x_{\bullet }\right\}_{\bullet }\right),}$ where ${\displaystyle \left\{x_{\bullet }\right\}_{\bullet }:=\left(\left\{x_{i}\right\}\right)_{i\in I}}$ is the canonical net of singleton sets associated with ${\displaystyle x_{\bullet }.}$ This makes it apparent that the following definition of "convergence of a net of sets" in ${\displaystyle X}$ is indeed a generalization of the original definition of "convergence of a net of points" in ${\displaystyle X}$ (because ${\displaystyle x_{\bullet }\to R}$ if and only if ${\displaystyle \left(\left\{x_{i}\right\}\right)_{i\in I}\to R}$).

A net of sets ${\displaystyle S_{\bullet }}$ is said to converge in ${\displaystyle (X,\tau )}$ to a subset ${\displaystyle R\subseteq X,}$ written ${\displaystyle S_{\bullet }\to R{\text{ in }}(X,\tau ),}$ if ${\displaystyle \,\operatorname {Tails} \left(S_{\bullet }\right)\to R{\text{ in }}(X,\tau ),}$ which recall was defined to mean that ${\displaystyle {\mathcal {N}}_{\tau }(R)\leq \operatorname {Tails} \left(S_{\bullet }\right).}$ Similarly, ${\displaystyle S_{\bullet }}$ is said to converge in ${\displaystyle (X,\tau )}$ to a point ${\displaystyle x\in X}$ if ${\displaystyle \operatorname {Tails} \left(S_{\bullet }\right)\to x{\text{ in }}(X,\tau )}$ (that is, if and only if ${\displaystyle {\mathcal {N}}_{\tau }(x)\leq \operatorname {Tails} \left(S_{\bullet }\right)}$).

Topologizing the set of prefilters[]

Starting with nothing more than a set ${\displaystyle X,}$ it is possible to topologize the set

of all filter bases on ${\displaystyle X}$ with the Stone topology, which is named after Marshall Harvey Stone.

To reduce confusion, this article will adhere to the following notational conventions:

• Lower case letters for elements ${\displaystyle x\in X.}$
• Upper case letters for subsets ${\displaystyle S\subseteq X.}$
• Upper case calligraphy letters for subsets ${\displaystyle {\mathcal {B}}\subseteq \wp (X)}$ (or equivalently, for elements ${\displaystyle {\mathcal {B}}\in \wp (\wp (X)),}$ such as prefilters).
• Upper case double–struck letters for subsets ${\displaystyle \mathbb {P} \subseteq \wp (\wp (X)).}$

For every ${\displaystyle S\subseteq X,}$ let

where ${\displaystyle \mathbb {O} (X)=\mathbb {P} {\text{ and }}\mathbb {O} (\varnothing )=\varnothing .}$^{[note 13]} These sets will be the basic open subsets of the Stone topology. If ${\displaystyle R\subseteq S\subseteq X}$ then

From this inclusion, it is possible to deduce all of the subset inclusions displayed below with the exception of ${\displaystyle \mathbb {O} (R\cap S)~\supseteq ~\mathbb {O} (R)\cap \mathbb {O} (S).}$^{[note 14]} For all ${\displaystyle R\subseteq S\subseteq X,}$

where in particular, the equality ${\displaystyle \mathbb {O} (R\cap S)=\mathbb {O} (R)\cap \mathbb {O} (S)}$ shows that the family ${\displaystyle \{\mathbb {O} (S)~:~S\subseteq X\}}$ is a ${\displaystyle \pi }$–system that forms a basis for a topology on ${\displaystyle \mathbb {P} }$ called the Stone topology. It is henceforth assumed that ${\displaystyle \mathbb {P} }$ carries this topology and that any subset of ${\displaystyle \mathbb {P} }$ carries the induced subspace topology.

In contrast to most other general constructions of topologies (for example, the product, quotient, subspace topologies, etc.), this topology on ${\displaystyle \mathbb {P} }$ was defined without using anything other than the set ${\displaystyle X;}$ there were no preexisting structures or assumptions on ${\displaystyle X}$ so this topology is completely independent of everything other than ${\displaystyle X}$ (and its subsets).

The following criteria can be used for checking for points of closure and neighborhoods. If ${\displaystyle \mathbb {B} \subseteq \mathbb {P} {\text{ and }}{\mathcal {F}}\in \mathbb {P} }$ then:

• Closure in ${\displaystyle \mathbb {P} }$: ${\displaystyle \ {\mathcal {F}}}$ belongs to the closure of ${\displaystyle \mathbb {B} {\text{ in }}\mathbb {P} }$ if and only if ${\displaystyle {\mathcal {F}}\subseteq \bigcup _{{\mathcal {B}}\in \mathbb {B} }{\mathcal {B}}^{\uparrow X}.}$
• Neighborhoods in ${\displaystyle \mathbb {P} }$: ${\displaystyle \ \mathbb {B} }$ is a neighborhood of ${\displaystyle {\mathcal {F}}{\text{ in }}\mathbb {P} }$ if and only if there exists some ${\displaystyle F\in {\mathcal {F}}}$ such that ${\displaystyle \mathbb {O} (F)=\left\{{\mathcal {B}}\in \mathbb {P} ~:~F\in {\mathcal {B}}^{\uparrow X}\right\}\subseteq \mathbb {B} }$ (that is, such that for all ${\displaystyle {\mathcal {B}}\in \mathbb {P} ,{\text{ if }}F\in {\mathcal {B}}^{\uparrow X}{\text{ then }}{\mathcal {B}}\in \mathbb {B} }$).

It will be henceforth assumed that ${\displaystyle X\neq \varnothing }$ because otherwise ${\displaystyle \mathbb {P} =\varnothing }$ and the topology is ${\displaystyle \{\varnothing \},}$ which is uninteresting.

Subspace of ultrafilters

The set of ultrafilters on ${\displaystyle X}$ (with the subspace topology) is a Stone space, meaning that it is compact, Hausdorff, and totally disconnected. If ${\displaystyle X}$ has the discrete topology then the map ${\displaystyle \beta :X\to \operatorname {UltraFilters} (X),}$ defined by sending ${\displaystyle x\in X}$ to the principal ultrafilter at ${\displaystyle x,}$ is a topological embedding whose image is a dense subset of ${\displaystyle \operatorname {UltraFilters} (X)}$ (see the article Stone–Čech compactification for more details).

Relationships between topologies on ${\displaystyle X}$ and the Stone topology on ${\displaystyle \mathbb {P} }$

Every ${\displaystyle \tau \in \operatorname {Top} (X)}$ induces a canonical map ${\displaystyle {\mathcal {N}}_{\tau }:X\to \operatorname {Filters} (X)}$ defined by ${\displaystyle x\mapsto {\mathcal {N}}_{\tau }(x),}$ which sends ${\displaystyle x\in X}$ to the neighborhood filter of ${\displaystyle x{\text{ in }}(X,\tau ).}$ The map ${\displaystyle {\mathcal {N}}_{\tau }:X\to \operatorname {Filters} (X)}$ is injective if and only if ${\displaystyle \tau {\text{ is }}T_{0}}$ (that is, a Kolmogorov space) and moreover, if ${\displaystyle \tau ,\sigma \in \operatorname {Top} (X)}$ then ${\displaystyle \tau =\sigma {\text{ if and only if }}{\mathcal {N}}_{\tau }={\mathcal {N}}_{\sigma }.}$ Thus every ${\displaystyle \tau \in \operatorname {Top} (X)}$ can be identified with the canonical map ${\displaystyle {\mathcal {N}}_{\tau },}$ which allows ${\displaystyle \operatorname {Top} (X)}$ to be canonically identified as a subset of ${\displaystyle \operatorname {Func} (X;\mathbb {P} )}$ (as a side note, it is now possible to place on ${\displaystyle \operatorname {Func} (X;\mathbb {P} ),}$ and thus also on ${\displaystyle \operatorname {Top} (X),}$ the topology of pointwise convergence on ${\displaystyle X}$ so that it now makes sense to talk about things such as topologies converging pointwise on ${\displaystyle X}$). For every ${\displaystyle \tau \in \operatorname {Top} (X),}$ the surjection ${\displaystyle {\mathcal {N}}_{\tau }:(X,\tau )\to \operatorname {image} {\mathcal {N}}_{\tau }}$ is continuous, closed, and open. In particular, for every ${\displaystyle T_{0}}$ topology ${\displaystyle \tau {\text{ on }}X,}$ the map ${\displaystyle {\mathcal {N}}_{\tau }:(X,\tau )\to \mathbb {P} }$ is a topological embedding.

In addition, if ${\displaystyle {\mathfrak {F}}:X\to \operatorname {Filters} (X)}$ is a map such that ${\displaystyle x\in \ker {\mathfrak {F}}(x):=\bigcap _{F\in {\mathfrak {F}}(x)}F{\text{ for every }}x\in X}$ (which is true of ${\displaystyle {\mathfrak {F}}:={\mathcal {N}}_{\tau },}$ for instance), then for every ${\displaystyle x\in X{\text{ and }}F\in {\mathfrak {F}}(x),}$ the set ${\displaystyle {\mathfrak {F}}(F):=\{{\mathfrak {F}}(f):f\in F\}}$ is a neighborhood of ${\displaystyle {\mathfrak {F}}(x){\text{ in }}\operatorname {image} {\mathfrak {F}}}$ (where ${\displaystyle \operatorname {image} {\mathfrak {F}}}$ has the subspace topology inherited from ${\displaystyle \mathbb {P} }$).

See also[]

• Characterizations of the category of topological spaces
• Convergence space – Generalization of the notion of convergence that is found in general topology
• Filtration (mathematics)
• Filtration (probability theory) – model of information available at a given point of a random process
• Filtration (abstract algebra)
• Fréchet filter
• Generic filter
• Ideal (set theory) – Non-empty family of sets that is closed under finite unions and subsets
• Stone–Čech compactification#Construction using ultrafilters
• The fundamental theorem of ultraproducts

Notes[]

Proofs

Citations[]

References[]

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