Filters in topology

The power set lattice of the set $X:=\{1,2,3,4\},$ with the upper set $\{1,4{\}}^{\uparrow X}$ colored dark green. It is a Template:Em, and even a Template:Em. It is not an Template:Em, as it can be extended to the larger proper filter $\{1{\}}^{\uparrow X}$ by including also the light green elements. Because $\{1{\}}^{\uparrow X}$ cannot be extended any further, it is an ultrafilter.

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

Filters have generalizations called Template:Em (also known as Template:Em) and Template:Em, 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 Template:Em. This establishes a relationship between filters and prefilters that may often be exploited to allow one to use whichever of these two notions is more technically convenient. There is a certain preorder on families of sets (subordination), denoted by $\le ,$ that helps to determine exactly when and how one notion (filter, prefilter, etc.) can or cannot be used in place of another. This preorder's importance is amplified by the fact that it also defines the notion of filter convergence, where by definition, a filter (or prefilter) $ℬ$ Template:Em to a point if and only if $𝒩\le ℬ,$ where $𝒩$ 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 $𝒮\ge ℬ,$ which denotes $ℬ\le 𝒮$ and is expressed by saying that $𝒮$ Template:Em $ℬ,$ also establishes a relationship in which $𝒮$ is to $ℬ$ as a subsequence is to a sequence (that is, the relation $\ge ,$ which is called Template:Em, is for filters the analog of "is a subsequence of").

Filters were introduced by Henri Cartan in 1937Template:Sfn and subsequently used by Bourbaki in their book Template:Em 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 sequence and net convergence, filter convergence is defined Template:Em in terms of subsets of the topological space $X$ and so it provides a notion of convergence that is completely intrinsic to the topological space; indeed, the category of topological spaces can be equivalently defined entirely in terms of filters. Every net induces a canonical filter and dually, every filter induces a canonical net, where this induced net (resp. induced filter) converges to a point if and only if the same is true of the original filter (resp. net). This characterization also holds for many other definitions such as cluster points. These relationships make it possible to switch between filters and nets, and they often also allow one to choose whichever of these two notions (filter or net) is more convenient for the problem at hand. However, assuming that "subnet" is defined using either of its most popular definitions (which are those given by Willard and by Kelley), then in general, this relationship does Template:Em extend to subordinate filters and subnets because as detailed below, there exist subordinate filters whose filter/subordinate-filter relationship cannot be described in terms of the corresponding net/subnet relationship; this issue can however be resolved by using a less commonly encountered definition of "subnet", which is that of an AA-subnet.

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

Motivation

Archetypical example of a filter

The archetypical example of a filter is the [[Neighbourhood filter|Template:Em]] $𝒩(x)$ at a point $x$ in a topological space $(X,\tau ),$ which is the family of sets consisting of all neighborhoods of $x.$ By definition, a neighborhood of some given point $x$ is any subset $B\subseteq X$ whose topological interior contains this point; that is, such that $x\in {\mathrm{Int}}_{X}B.$ Importantly, neighborhoods are Template:Em required to be open sets; those are called Template:Em. Listed below are those fundamental properties of neighborhood filters that ultimately became the definition of a "filter." A Template:Em is a set $ℬ$ of subsets of $X$ that satisfies all of the following conditions:

1. Template:Em:   $X\in ℬ$  –  just as $X\in 𝒩(x),$ since $X$ is always a neighborhood of $x$ (and of anything else that it contains);
2. Template:Em:   $\varnothing \in ̸ℬ$  –  just as no neighborhood of $x$ is empty;
3. Template:Em:   If $B,C\in ℬ\text{ then }B\cap C\in ℬ$  –  just as the intersection of any two neighborhoods of $x$ is again a neighborhood of $x$;
4. Template:Em:   If $B\in ℬ\text{ and }B\subseteq S\subseteq X$ then $S\in ℬ$  –  just as any subset of $X$ that includes a neighborhood of $x$ will necessarily Template:Em a neighborhood of $x$ (this follows from ${\mathrm{Int}}_{X}B\subseteq {\mathrm{Int}}_{X}S$ and the definition of "a neighborhood of $x$").

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

A Template:Em is by definition a map $\mathbb{N}\to X$ from the natural numbers into the space $X.$ 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 Template:Em 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 Template:Em fail to characterize topological properties.

Nets directly generalize the notion of a sequence since nets are, by definition, maps $I\to X$ from an arbitrary directed set $(I,\le )$ into the space $X.$ A sequence is just a net whose domain is $I=\mathbb{N}$ 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 Template:Em the values of a sequence. To see how this is done, consider a sequence $x_{\bullet }={\left(x_i\right)}_{i=1}^{\mathrm{\infty }}\text{ in }X,$ which is by definition just a function $x_{\bullet }:\mathbb{N}\to X$ whose value at $i\in \mathbb{N}$ is denoted by $x_i$ rather than by the usual parentheses notation $x_{\bullet }(i)$ that is commonly used for arbitrary functions. Knowing only the image (sometimes called "the range") $\mathrm{Im}{x}_{\bullet }:=\{x_i:i\in \mathbb{N}\}=\{x_1,x_2,\dots \}$ of the sequence is not enough to characterize its convergence; multiple sets are needed. It turns out that the needed sets are the following, which are called the Template:Em of the sequence $x_{\bullet }$: $$\begin{array}{cccccccccccccc}{x}_{\ge 1}=& \{& & x_1,& & x_2,& & x_3,& & x_4,& & \dots & & \}\\ x_{\ge 2}=& \{& & x_2,& & x_3,& & x_4,& & x_5,& & \dots & & \}\\ x_{\ge 3}=& \{& & x_3,& & x_4,& & x_5,& & x_6,& & \dots & & \}\\ & & & & & & & ⋮& & & & & & \\ x_{\ge n}=& \{& & x_n,& & x_{n+1},& & x_{n+2},& & x_{n+3},& & \dots & & \}\\ & & & & & & & ⋮& & & & & & \\ \end{array}$$

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 $U$ (of this point), there is some integer $n$ such that $U$ contains all of the points $x_n,x_{n+1},\dots .$ This can be reworded as:

Template:In5every neighborhood $U$ must contain some set of the form $\{x_n,x_{n+1},\dots \}$ as a subset.

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

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

Nets versus filters − advantages and disadvantages

Filters and nets each have their own advantages and drawbacks and there's no reason to use one notion exclusively over the other. Depending on what is being proved, a proof may be made significantly easier by using one of these notions instead of the other.Template:Sfn 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,Template:Sfn combinatorics, dynamics, order theory, generalized convergence spaces, Cauchy spaces, and in the definition and use of hyperreal numbers.

Like sequences, nets are Template:Em and so they have the Template:Em. 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 $X$ and a filter on a dense subspace $S\subseteq X.$Template:Sfn

In contrast to nets, filters (and prefilters) are families of Template:Em and so they have the Template:Em. For example, if $f$ is surjective then the Template:Em $f^{-1}(ℬ):=\{f^{-1}(B):B\in ℬ\}$ under $f^{-1}$ of an arbitrary filter or prefilter $ℬ$ is both easily defined and guaranteed to be a prefilter on $f$'s domain, whereas it is less clear how to pullback (unambiguously/without choice) an arbitrary sequence (or net) $y_{\bullet }$ so as to obtain a sequence or net in the domain (unless $f$ is also injective and consequently a bijection, which is a stringent requirement). Similarly, the intersection of any collection of filters is once again a filter whereas it is not clear what this could mean for sequences or nets. Because filters are composed of subsets of the very topological space $X$ 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 Template:Em 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 $X.$ In fact, the class of nets in a given set $X$ is too large to even be a set (it is a proper class); this is because nets in $X$ can have domains of Template:Em cardinality. In contrast, the collection of all filters (and of all prefilters) on $X$ is a set whose cardinality is no larger than that of $\mathrm{\wp }(\mathrm{\wp }(X)).$ Similar to a topology on $X,$ a filter on $X$ is "intrinsic to $X$" in the sense that both structures consist Template:Em of subsets of $X$ and neither definition requires any set that cannot be constructed from $X$ (such as $\mathbb{N}$ or other directed sets, which sequences and nets require).

Preliminaries, notation, and basic notions

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

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

Warning about competing definitions and notation

There are unfortunately several terms in the theory of filters that are defined differently by different authors. These include some of the most important terms such as "filter." While different definitions of the same term usually have significant overlap, due to the very technical nature of filters (and point–set topology), these differences in definitions nevertheless often have important consequences. When reading mathematical literature, it is recommended that readers check how the terminology related to filters is defined by the author. For this reason, this article will clearly state all definitions as they are used. Unfortunately, not all notation related to filters is well established and some notation varies greatly across the literature (for example, the notation for the set of all prefilters on a set) so in such cases this article uses whatever notation is most self describing or easily remembered.

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

Sets operations

The Template:Em or Template:Em in $X$Template:SfnTemplate:Sfn of a family of sets $ℬ\subseteq \mathrm{\wp }(X)$ is

and similarly the Template:Em of $ℬ$ is ${ℬ}^{\downarrow }:=\{S\subseteq B:B\in ℬ\}=\bigcup _{B\in ℬ}\mathrm{\wp }(B).$

Notation and Definition	Name
$\ker ℬ=\bigcap _{B\in ℬ}B$	Template:Em of $ℬ$Template:Sfn
$S\setminus ℬ:=\{S\setminus B:B\in ℬ\}=\{S\}(\setminus )ℬ$	Template:Em where $S$ is a set.Template:Sfn
$ℬ{|}_S:=\{B\cap S:B\in ℬ\}=ℬ(\cap )\{S\}$	Template:EmTemplate:Sfn or Template:Em where $S$ is a set; sometimes denoted by $ℬ\cap S$
$ℬ(\cap )𝒞=\{B\cap C:B\in ℬ\text{ and }C\in 𝒞\}$Template:Sfn	Template:Em (Template:Em) Template:Em ($ℬ\cap 𝒞$ will denote the usual intersection)
$ℬ(\cup )𝒞=\{B\cup C:B\in ℬ\text{ and }C\in 𝒞\}$Template:Sfn	Template:Em (Template:Em) Template:Em ($ℬ\cup 𝒞$ will denote the usual union)
$ℬ(\setminus )𝒞=\{B\setminus C:B\in ℬ\text{ and }C\in 𝒞\}$	Template:Em (Template:Em) Template:Em ($ℬ\setminus 𝒞$ will denote the usual set subtraction)
$\mathrm{\wp }(X)=\{S:S\subseteq X\}$	Template:Em of a set $X$Template:Sfn

Throughout, $f$ is a map.

Notation and Definition	Name
$f^{-1}(ℬ)=\{f^{-1}(B):B\in ℬ\}$Template:Sfn	Template:Em of $ℬ\text{ under }{f}^{-1},$ or the Template:Em of $ℬ$ under $f$
$f(ℬ)=\{f(B):B\in ℬ\}$Template:Sfn	Template:Em of $ℬ$ under $f$
$\mathrm{image}f=f(\mathrm{domain}f)$	Template:Em (or range) of $f$

Topology notation

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

Notation and Definition	Name
$\tau (S)=\{O\in \tau :S\subseteq O\}$	Template:Em or Template:Em Template:Em of $S\text{ in }(X,\tau )$
$\tau (x)=\{O\in \tau :x\in O\}$	Template:Em or Template:Em Template:Em of $x\text{ in }(X,\tau )$
${𝒩}_{\tau }(S)=𝒩(S):=\tau (S{)}^{\uparrow X}$	Template:Em or Template:Em Template:Em of $S\text{ in }(X,\tau )$
${𝒩}_{\tau }(x)=𝒩(x):=\tau (x{)}^{\uparrow X}$	Template:Em or Template:Em of $x\text{ in }(X,\tau )$

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

Template:Anchor Nets and their tails

A Template:Em is a set $I$ together with a preorder, which will be denoted by $\le$ (unless explicitly indicated otherwise), that makes $(I,\le )$ into an (Template:Em) Template:Em;Template:Sfn this means that for all $i,j\in I,$ there exists some $k\in I$ such that $i\le k\text{ and }j\le k.$ For any indices $i\text{ and }j,$ the notation $j\ge i$ is defined to mean $i\le j$ while $i<j$ is defined to mean that $i\le j$ holds but it is Template:Em true that $j\le i$ (if $\le$ is antisymmetric then this is equivalent to $i\le j\text{ and }i\ne j$).

A Template:EmTemplate:Sfn is a map from a non-empty directed set into $X.$ The notation $x_{\bullet }={\left(x_i\right)}_{i\in I}$ will be used to denote a net with domain $I.$

Notation and Definition	Name
$I_{\ge i}=\{j\in I:j\ge i\}$	Template:Em or Template:Em where $(I,\le )$ is a directed set.
$x_{\ge i}=\{x_j:j\ge i\text{ and }j\in I\}$	Template:Em or Template:Em
$\mathrm{Tails}\left(x_{\bullet }\right)=\{x_{\ge i}:i\in I\}$	Template:Em or Template:Em/Template:Em of $x_{\bullet }.$ Also called the Template:Em generated by (the tails of) $x_{\bullet }={\left(x_i\right)}_{i\in I}.$ If $x_{\bullet }$ is a sequence then $\mathrm{Tails}\left(x_{\bullet }\right)$ is also called the Template:Em.Template:Sfn
$\mathrm{TailsFilter}\left(x_{\bullet }\right)=\mathrm{Tails}{\left(x_{\bullet }\right)}^{\uparrow X}$	Template:Anchor(Template:Em) Template:Em of/generated by (tails of) $x_{\bullet }$Template:Sfn
$f\left(I_{\ge i}\right)=\{f(j):j\ge i\text{ and }j\in I\}$	Template:Em or Template:EmTemplate:Sfn where $(I,\le )$ is a directed set.

Warning about using strict comparison

If $x_{\bullet }={\left(x_i\right)}_{i\in I}$ is a net and $i\in I$ then it is possible for the set $x_{>i}=\{x_j:j>i\text{ and }j\in I\},$ which is called Template:Em, to be empty (for example, this happens if $i$ is an upper bound of the directed set $I$). In this case, the family $\{x_{>i}:i\in I\}$ would contain the empty set, which would prevent it from being a prefilter (defined later). This is the (important) reason for defining $\mathrm{Tails}\left(x_{\bullet }\right)$ as $\{x_{\ge i}:i\in I\}$ rather than $\{x_{>i}:i\in I\}$ or even $\{x_{>i}:i\in I\}\cup \{x_{\ge i}:i\in I\}$ and it is for this reason that in general, when dealing with the prefilter of tails of a net, the strict inequality $<$ may not be used interchangeably with the inequality $\le .$

Filters and prefilters

The following is a list of properties that a family $ℬ$ 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 $ℬ\subseteq \mathrm{\wp }(X).$

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

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

Basic examples

Named examples

• The singleton set $ℬ=\{X\}$ is called the Template:Em or Template:EmTemplate:SfnTemplate:Sfn It is the unique Template:Em filter on $X$ because it is a subset of every filter on $X$; however, it need not be a subset of every prefilter on $X.$
• The dual ideal $\mathrm{\wp }(X)$ is also called Template:EmTemplate:Sfn (despite not actually being a filter). It is the only dual ideal on $X$ that is not a filter on $X.$
• If $(X,\tau )$ is a topological space and $x\in X,$ then the neighborhood filter $𝒩(x)$ at $x$ is a filter on $X.$ By definition, a family $ℬ\subseteq \mathrm{\wp }(X)$ is called a Template:Em (resp. a Template:Em) at $x\text{ for }(X,\tau )$ if and only if $ℬ$ is a prefilter (resp. $ℬ$ is a filter subbase) and the filter on $X$ that $ℬ$ generates is equal to the neighborhood filter $𝒩(x).$ The subfamily $\tau (x)\subseteq 𝒩(x)$ of open neighborhoods is a filter base for $𝒩(x).$ Both prefilters $𝒩(x)\text{ and }\tau (x)$ also form a bases for topologies on $X,$ with the topology generated $\tau (x)$ being coarser than $\tau .$ This example immediately generalizes from neighborhoods of points to neighborhoods of non-empty subsets $S\subseteq X.$
• $ℬ$ is an Template:Em if $ℬ=\mathrm{Tails}\left(x_{\bullet }\right)$ for some sequence of points $x_{\bullet }={\left(x_i\right)}_{i=1}^{\mathrm{\infty }}.$
• $ℬ$ is an Template:Em or a Template:Em on $X$Template:Sfn if $ℬ$ is a filter on $X$ generated by some elementary prefilter. The filter of tails generated by a sequence that is not eventually constant is necessarily Template:Em an ultrafilter.Template:Sfn Every principal filter on a countable set is sequential as is every cofinite filter on a countably infinite set.Template:Sfn The intersection of finitely many sequential filters is again sequential.Template:Sfn
• The set $ℱ$ of all cofinite subsets of $X$ (meaning those sets whose complement in $X$ is finite) is proper if and only if $ℱ$ is infinite (or equivalently, $X$ is infinite), in which case $ℱ$ is a filter on $X$ known as the [[Fréchet filter|Template:Em or the Template:Em]] on $X.$Template:SfnTemplate:Sfn If $X$ is finite then $ℱ$ is equal to the dual ideal $\mathrm{\wp }(X),$ which is not a filter. If $X$ is infinite then the family $\{X\setminus \{x\}:x\in X\}$ of complements of singleton sets is a filter subbase that generates the Fréchet filter on $X.$ As with any family of sets over $X$ that contains $\{X\setminus \{x\}:x\in X\},$ the kernel of the Fréchet filter on $X$ is the empty set: $\ker ℱ=\varnothing .$
• The intersection of all elements in any non-empty family $𝔽\subseteq \mathrm{Filters}(X)$ is itself a filter on $X$ called the Template:Em or Template:Em of $𝔽\text{ in }\mathrm{Filters}(X),$ which is why it may be denoted by $\underset{ℱ\in 𝔽}{\bigwedge }ℱ.$ Said differently, $\ker 𝔽=\bigcap _{ℱ\in 𝔽}ℱ\in \mathrm{Filters}(X).$ Because every filter on $X$ has $\{X\}$ as a subset, this intersection is never empty. By definition, the infimum is the finest/largest (relative to $\subseteq \text{ and }\le$) filter contained as a subset of each member of $𝔽.$Template:Sfn
  • If $ℬ\text{ and }ℱ$ are filters then their infimum in $\mathrm{Filters}(X)$ is the filter $ℬ(\cup )ℱ.$Template:Sfn If $ℬ\text{ and }ℱ$ are prefilters then $ℬ(\cup )ℱ$ is a prefilter that is coarser than both $ℬ\text{ and }ℱ$ (that is, $ℬ(\cup )ℱ\le ℬ\text{ and }ℬ(\cup )ℱ\le ℱ$); indeed, it is one of the finest such prefilters, meaning that if $𝒮$ is a prefilter such that $𝒮\le ℬ\text{ and }𝒮\le ℱ$ then necessarily $𝒮\le ℬ(\cup )ℱ.$Template:Sfn More generally, if $ℬ\text{ and }ℱ$ are non−empty families and if $𝕊:=\{𝒮\subseteq \mathrm{\wp }(X):𝒮\le ℬ\text{ and }𝒮\le ℱ\}$ then $ℬ(\cup )ℱ\in 𝕊$ and $ℬ(\cup )ℱ$ is a greatest element of $(𝕊,\le ).$Template:Sfn
• Let $\varnothing \ne 𝔽\subseteq \mathrm{DualIdeals}(X)$ and let $\cup 𝔽=\bigcup _{ℱ\in 𝔽}ℱ.$ The Template:Em or Template:Em of $𝔽\text{ in }\mathrm{DualIdeals}(X),$ denoted by $\underset{ℱ\in 𝔽}{\bigvee }ℱ,$ is the smallest (relative to $\subseteq$) dual ideal on $X$ containing every element of $𝔽$ as a subset; that is, it is the smallest (relative to $\subseteq$) dual ideal on $X$ containing $\cup 𝔽$ as a subset. This dual ideal is $\underset{ℱ\in 𝔽}{\bigvee }ℱ=\pi {(\cup 𝔽)}^{\uparrow X},$ where $\pi (\cup 𝔽):=\{F_1\cap \cdots \cap F_n:n\in \mathbb{N}\text{ and every }{F}_i\text{ belongs to some }ℱ\in 𝔽\}$ is the Template:Pi-system generated by $\cup 𝔽.$ As with any non-empty family of sets, $\cup 𝔽$ is contained in Template:Em filter on $X$ if and only if it is a filter subbase, or equivalently, if and only if $\underset{ℱ\in 𝔽}{\bigvee }ℱ=\pi {(\cup 𝔽)}^{\uparrow X}$ is a filter on $X,$ in which case this family is the smallest (relative to $\subseteq$) filter on $X$ containing every element of $𝔽$ as a subset and necessarily $𝔽\subseteq \mathrm{Filters}(X).$
• Let $\varnothing \ne 𝔽\subseteq \mathrm{Filters}(X)$ and let $\cup 𝔽=\bigcup _{ℱ\in 𝔽}ℱ.$ The Template:Em or Template:Em of $𝔽\text{ in }\mathrm{Filters}(X),$ denoted by $\underset{ℱ\in 𝔽}{\bigvee }ℱ$ if it exists, is by definition the smallest (relative to $\subseteq$) filter on $X$ containing every element of $𝔽$ as a subset. If it exists then necessarily $\underset{ℱ\in 𝔽}{\bigvee }ℱ=\pi {(\cup 𝔽)}^{\uparrow X}$Template:Sfn (as defined above) and $\underset{ℱ\in 𝔽}{\bigvee }ℱ$ will also be equal to the intersection of all filters on $X$ containing $\cup 𝔽.$ This supremum of $𝔽\text{ in }\mathrm{Filters}(X)$ exists if and only if the dual ideal $\pi {(\cup 𝔽)}^{\uparrow X}$ is a filter on $X.$ The least upper bound of a family of filters $𝔽$ may fail to be a filter.Template:Sfn Indeed, if $X$ contains at least two distinct elements then there exist filters $ℬ\text{ and }𝒞\text{ on }X$ for which there does Template:Em exist a filter $ℱ\text{ on }X$ that contains both $ℬ\text{ and }𝒞.$ If $\cup 𝔽$ is not a filter subbase then the supremum of $𝔽\text{ in }\mathrm{Filters}(X)$ does not exist and the same is true of its supremum in $\mathrm{Prefilters}(X)$ but their supremum in the set of all dual ideals on $X$ will exist (it being the degenerate filter $\mathrm{\wp }(X)$).Template:Sfn
  • If $ℬ\text{ and }ℱ$ are prefilters (resp. filters on $X$) then $ℬ(\cap )ℱ$ is a prefilter (resp. a filter) if and only if it is non-degenerate (or said differently, if and only if $ℬ\text{ and }ℱ$ mesh), in which case it is Template:Em coarsest prefilters (resp. Template:Em coarsest filter) on $X$ that is finer (with respect to $\le$) than both $ℬ\text{ and }ℱ;$ this means that if $𝒮$ is any prefilter (resp. any filter) such that $ℬ\le 𝒮\text{ and }ℱ\le 𝒮$ then necessarily $ℬ(\cap )ℱ\le 𝒮,$Template:Sfn in which case it is denoted by $ℬ\vee ℱ.$Template:Sfn

Other examples

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

Ultrafilters

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

The ultrafilter lemma

The following important theorem is due to Alfred Tarski (1930).Template:Sfn

A consequence of the ultrafilter lemma is that every filter is equal to the intersection of all ultrafilters containing it.Template:Sfn 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 Template:Em 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.

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

(1) $ℬ,$ (2) the Template:Pi-system generated by $ℬ,$ and (3) the filter generated by $ℬ.$

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

Classifying families by their kernels

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

Family of examples: For any non-empty $C\subseteq ℝ,$ the family ${ℬ}_C=\{ℝ\setminus (r+C):r\in ℝ\}$ is free but it is a filter subbase if and only if no finite union of the form $(r_1+C)\cup \cdots \cup (r_n+C)$ covers $ℝ,$ in which case the filter that it generates will also be free. In particular, ${ℬ}_C$ is a filter subbase if $C$ is countable (for example, $C=\mathbb{Q},\mathbb{Z},$ the primes), a meager set in $ℝ,$ a set of finite measure, or a bounded subset of $ℝ.$ If $C$ is a singleton set then ${ℬ}_C$ is a subbase for the Fréchet filter on $ℝ.$

Characterizing fixed ultra prefilters

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

Every filter on $X$ that is principal at a single point is an ultrafilter, and if in addition $X$ is finite, then there are no ultrafilters on $X$ other than these.Template:Sfn

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 $\le$ 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",Template:Sfn where "$ℱ\ge 𝒞$" can be interpreted as "$ℱ$ is a subsequence of $𝒞$" (so "subordinate to" is the prefilter equivalent of "subsequence of"). It is also used to define prefilter convergence in a topological space. The definition of $ℬ$ meshes with $𝒞,$ which is closely related to the preorder $\le ,$ is used in topology to define cluster points.

Two families of sets $ℬ\text{ and }𝒞$ Template:EmTemplate:Sfn and are Template:Em, indicated by writing $ℬ\mathrm{\#}𝒞,$ if $B\cap C\ne \varnothing \text{ for all }B\in ℬ\text{ and }C\in 𝒞.$ If $ℬ\text{ and }𝒞$ do not mesh then they are Template:Em. If $S\subseteq X\text{ and }ℬ\subseteq \mathrm{\wp }(X)$ then $ℬ\text{ and }S$ are said to Template:Em if $ℬ\text{ and }\{S\}$ mesh, or equivalently, if the Template:Em of $ℬ\text{ on }S,$ which is the family $$ℬ{|}_S=\{B\cap S:B\in ℬ\},$$ does not contain the empty set, where the trace is also called the Template:Em of $ℬ\text{ to }S.$

Example: If $x_{i_{\bullet }}={\left(x_{i_n}\right)}_{n=1}^{\mathrm{\infty }}$ is a subsequence of $x_{\bullet }={\left(x_i\right)}_{i=1}^{\mathrm{\infty }}$ then $\mathrm{Tails}\left(x_{i_{\bullet }}\right)$ is subordinate to $\mathrm{Tails}\left(x_{\bullet }\right);$ in symbols: $\mathrm{Tails}\left(x_{i_{\bullet }}\right)⊢\mathrm{Tails}\left(x_{\bullet }\right)$ and also $\mathrm{Tails}\left(x_{\bullet }\right)\le \mathrm{Tails}\left(x_{i_{\bullet }}\right).$ Stated in plain English, the prefilter of tails of a subsequence is always subordinate to that of the original sequence. To see this, let $C:=x_{\ge i}\in \mathrm{Tails}\left(x_{\bullet }\right)$ be arbitrary (or equivalently, let $i\in \mathbb{N}$ be arbitrary) and it remains to show that this set contains some $F:=x_{i_{\ge n}}\in \mathrm{Tails}\left(x_{i_{\bullet }}\right).$ For the set $x_{\ge i}=\{x_i,x_{i+1},\dots \}$ to contain $x_{i_{\ge n}}=\{x_{i_n},x_{i_{n+1}},\dots \},$ it is sufficient to have $i\le i_n.$ Since $i_1<i_2<\cdots$ are strictly increasing integers, there exists $n\in \mathbb{N}$ such that $i_n\ge i,$ and so $x_{\ge i}\supseteq x_{i_{\ge n}}$ holds, as desired. Consequently, $\mathrm{TailsFilter}\left(x_{\bullet }\right)\subseteq \mathrm{TailsFilter}\left(x_{i_{\bullet }}\right).$ The left hand side will be a Template:Em subset of the right hand side if (for instance) every point of $x_{\bullet }$ is unique (that is, when $x_{\bullet }:\mathbb{N}\to X$ is injective) and $x_{i_{\bullet }}$ is the even-indexed subsequence $(x_2,x_4,x_6,\dots )$ because under these conditions, every tail $x_{i_{\ge n}}=\{x_{2n},x_{2n+2},x_{2n+4},\dots \}$ (for every $n\in \mathbb{N}$) of the subsequence will belong to the right hand side filter but not to the left hand side filter.

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

A non-empty family that is coarser than a filter subbase must itself be a filter subbase.Template:Sfn Every filter subbase is coarser than both the Template:Pi-system that it generates and the filter that it generates.Template:Sfn

If $𝒞\text{ and }ℱ$ are families such that $𝒞\le ℱ,$ the family $𝒞$ is ultra, and $\varnothing \in ̸ℱ,$ then $ℱ$ is necessarily ultra. It follows that any family that is equivalent to an ultra family will necessarily Template:Em ultra. In particular, if $𝒞$ is a prefilter then either both $𝒞$ and the filter ${𝒞}^{\uparrow X}$ it generates are ultra or neither one is ultra.

The relation $\le$ is reflexive and transitive, which makes it into a preorder on $\mathrm{\wp }(\mathrm{\wp }(X)).$Template:Sfn The relation $\le \text{ on }\mathrm{Filters}(X)$ is antisymmetric but if $X$ has more than one point then it is Template:Em symmetric.

Equivalent families of sets

The preorder $\le$ induces its canonical equivalence relation on $\mathrm{\wp }(\mathrm{\wp }(X)),$ where for all $ℬ,𝒞\in \mathrm{\wp }(\mathrm{\wp }(X)),$ $ℬ$ is Template:Em to $𝒞$ if any of the following equivalent conditions hold:Template:SfnTemplate:Sfn

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

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

Properties preserved between equivalent families

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

1. Not empty
2. Proper (that is, $\varnothing$ is not an element)
   • Moreover, any two degenerate families are necessarily equivalent.
3. Filter subbase
4. Prefilter
   • In which case $ℬ\text{ and }𝒞$ generate the same filter on $X$ (that is, their upward closures in $X$ are equal).
5. Free
6. Principal
7. Ultra
8. Is equal to the trivial filter $\{X\}$
   • In words, this means that the only subset of $\mathrm{\wp }(X)$ that is equivalent to the trivial filter Template:Em 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 $ℱ$
10. Is finer than $ℱ$
11. Is coarser than $ℱ$
12. Is equivalent to $ℱ$

Missing from the above list is the word "filter" because this property is Template:Em preserved by equivalence. However, if $ℬ\text{ and }𝒞$ are filters on $X,$ then they are equivalent if and only if they are equal; this characterization does Template:Em extend to prefilters.

Equivalence of prefilters and filter subbases

If $ℬ$ is a prefilter on $X$ then the following families are always equivalent to each other:

1. $ℬ$;
2. the Template:Pi-system generated by $ℬ$;
3. the filter on $X$ generated by $ℬ$;

and moreover, these three families all generate the same filter on $X$ (that is, the upward closures in $X$ 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.Template:Sfn Every prefilter is equivalent to exactly one filter on $X,$ 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.Template:Sfn

A filter subbase that is Template:Em also a prefilter canTemplate:Em 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.

Set theoretic properties and constructions relevant to topology

Trace and meshing

If $ℬ$ is a prefilter (resp. filter) on $X\text{ and }S\subseteq X$ then the trace of $ℬ\text{ on }S,$ which is the family $ℬ{|}_S:=ℬ(\cap )\{S\},$ is a prefilter (resp. a filter) if and only if $ℬ\text{ and }S$ mesh (that is, $\varnothing \in ̸ℬ(\cap )\{S\}$Template:Sfn), in which case the trace of $ℬ\text{ on }S$ is said to be Template:Em. The trace is always finer than the original family; that is, $ℬ\le ℬ{|}_S.$ If $ℬ$ is ultra and if $ℬ\text{ and }S$ mesh then the trace $ℬ{|}_S$ is ultra. If $ℬ$ is an ultrafilter on $X$ then the trace of $ℬ\text{ on }S$ is a filter on $S$ if and only if $S\in ℬ.$

For example, suppose that $ℬ$ is a filter on $X\text{ and }S\subseteq X$ is such that $S\ne X\text{ and }X\setminus S\in ̸ℬ.$ Then $ℬ\text{ and }S$ mesh and $ℬ\cup \{S\}$ generates a filter on $X$ that is strictly finer than $ℬ.$Template:Sfn

When prefilters mesh

Given non-empty families $ℬ\text{ and }𝒞,$ the family $$ℬ(\cap )𝒞:=\{B\cap C:B\in ℬ\text{ and }C\in 𝒞\}$$ satisfies $𝒞\le ℬ(\cap )𝒞$ and $ℬ\le ℬ(\cap )𝒞.$ If $ℬ(\cap )𝒞$ is proper (resp. a prefilter, a filter subbase) then this is also true of both $ℬ\text{ and }𝒞.$ In order to make any meaningful deductions about $ℬ(\cap )𝒞$ from $ℬ\text{ and }𝒞,ℬ(\cap )𝒞$ needs to be proper (that is, $\varnothing \in ̸ℬ(\cap )𝒞,$ which is the motivation for the definition of "mesh". In this case, $ℬ(\cap )𝒞$ is a prefilter (resp. filter subbase) if and only if this is true of both $ℬ\text{ and }𝒞.$ Said differently, if $ℬ\text{ and }𝒞$ are prefilters then they mesh if and only if $ℬ(\cap )𝒞$ is a prefilter. Generalizing gives a well known characterization of "mesh" entirely in terms of subordination (that is, $\le$):

Template:In5Two prefilters (resp. filter subbases) $ℬ\text{ and }𝒞$ mesh if and only if there exists a prefilter (resp. filter subbase) $ℱ$ such that $𝒞\le ℱ$ and $ℬ\le ℱ.$

If the least upper bound of two filters $ℬ\text{ and }𝒞$ exists in $\mathrm{Filters}(X)$ then this least upper bound is equal to $ℬ(\cap )𝒞.$Template:Sfn

Images and preimages under functions

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

Images of prefilters

Let $ℬ\subseteq \mathrm{\wp }(Y).$ Many of the properties that $ℬ$ 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 $ℬ\text{ on }Y,$ then it will necessarily also be true of $g(ℬ)\text{ on }g(Y)$ (although possibly not on the codomain $Z$ unless $g$ is surjective):Template:SfnTemplate:SfnTemplate:SfnTemplate:SfnTemplate:SfnTemplate:Sfn ultra, ultrafilter, filter, prefilter, filter subbase, dual ideal, upward closed, proper/non-degenerate, ideal, closed under finite unions, downward closed, directed upward. Moreover, if $ℬ\subseteq \mathrm{\wp }(Y)$ is a prefilter then so are both $g(ℬ)\text{ and }{g}^{-1}(g(ℬ)).$Template:Sfn The image under a map $f:X\to Y$ of an ultra set $ℬ\subseteq \mathrm{\wp }(X)$ is again ultra and if $ℬ$ is an ultra prefilter then so is $f(ℬ).$

If $ℬ$ is a filter then $g(ℬ)$ is a filter on the range $g(Y),$ but it is a filter on the codomain $Z$ if and only if $g$ is surjective.Template:Sfn Otherwise it is just a prefilter on $Z$ and its upward closure must be taken in $Z$ to obtain a filter. The upward closure of $g(ℬ)\text{ in }Z$ is $$g(ℬ{)}^{\uparrow Z}=\{S\subseteq Z:B\subseteq g^{-1}(S)\text{ for some }B\in ℬ\}$$ where if $ℬ$ is upward closed in $Y$ (that is, a filter) then this simplifies to: $$g(ℬ{)}^{\uparrow Z}=\{S\subseteq Z:g^{-1}(S)\in ℬ\}.$$

If $X\subseteq Y$ then taking $g$ to be the inclusion map $X\to Y$ shows that any prefilter (resp. ultra prefilter, filter subbase) on $X$ is also a prefilter (resp. ultra prefilter, filter subbase) on $Y.$Template:Sfn

Preimages of prefilters

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

Template:In5$f^{-1}(ℬ)$ is a prefilter (resp. filter subbase, Template:Pi-system, closed under finite unions, proper) if and only if this is true of $ℬ.$

However, if $ℬ$ is an ultrafilter on $Y$ then even if $f$ is surjective (which would make $f^{-1}(ℬ)$ a prefilter), it is nevertheless still possible for the prefilter $f^{-1}(ℬ)$ to be neither ultra nor a filter on $X.$Template:HspTemplate:Sfn

If $f:X\to Y$ is not surjective then denote the trace of $ℬ\text{ on }f(X)$ by $ℬ{|}_{f(X)},$ where in this case particular case the trace satisfies: $$ℬ{|}_{f(X)}=f(f^{-1}(ℬ))$$ and consequently also: $$f^{-1}(ℬ)=f^{-1}\left(ℬ{|}_{f(X)}\right).$$

This last equality and the fact that the trace $ℬ{|}_{f(X)}$ is a family of sets over $f(X)$ means that to draw conclusions about $f^{-1}(ℬ),$ the trace $ℬ{|}_{f(X)}$ can be used in place of $ℬ$ and the Template:Em $f:X\to f(X)$ can be used in place of $f:X\to Y.$ For example:Template:SfnTemplate:SfnTemplate:Sfn

Template:In5$f^{-1}(ℬ)$ is a prefilter (resp. filter subbase, Template:Pi-system, proper) if and only if this is true of $ℬ{|}_{f(X)}.$

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

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

1. $f^{-1}(ℬ)$ is a prefilter;
2. $ℬ{|}_{f(X)}$ is a prefilter;
3. $\varnothing \in ̸ℬ{|}_{f(X)}$;
4. $ℬ$ meshes with $f(X)$

and moreover, if $f^{-1}(ℬ)$ is a prefilter then so is $f(f^{-1}(ℬ)).$Template:SfnTemplate:Sfn

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

Subordination is preserved by images and preimages

The relation $\le$ is preserved under both images and preimages of families of sets.Template:Sfn This means that for Template:Em families $𝒞\text{ and }ℱ,$Template:Sfn $$𝒞\le ℱ\text{ implies }g(𝒞)\le g(ℱ)\text{ and }{f}^{-1}(𝒞)\le f^{-1}(ℱ).$$

Moreover, the following relations always hold for Template:Em family of sets $𝒞$:Template:Sfn $$𝒞\le f(f^{-1}(𝒞))$$ where equality will hold if $f$ is surjective.Template:Sfn Furthermore, $$f^{-1}(𝒞)=f^{-1}\left(f(f^{-1}(𝒞))\right)\text{ and }g(𝒞)=g(g^{-1}(g(𝒞))).$$

If $ℬ\subseteq \mathrm{\wp }(X)\text{ and }𝒞\subseteq \mathrm{\wp }(Y)$ thenTemplate:Sfn $$f(ℬ)\le 𝒞\text{ if and only if }ℬ\le f^{-1}(𝒞)$$ and $g^{-1}(g(𝒞))\le 𝒞$Template:Sfn where equality will hold if $g$ is injective.Template:Sfn

Products of prefilters

Suppose $X_{\bullet }={\left(X_i\right)}_{i\in I}$ is a family of one or more non-empty sets, whose product will be denoted by $\prod _{}{X}_{\bullet }:=\prod _{i\in I}{X}_i,$ and for every index $i\in I,$ let $$\mathrm{Pr}{}_{X_i}:\prod X_{\bullet }\to X_i$$ denote the canonical projection. Let ${ℬ}_{\bullet }:={\left({ℬ}_i\right)}_{i\in I}$ be non−empty families, also indexed by $I,$ such that ${ℬ}_i\subseteq \mathrm{\wp }\left(X_i\right)$ for each $i\in I.$ The Template:Em of the families ${ℬ}_{\bullet }$Template:Sfn is defined identically to how the basic open subsets of the product topology are defined (had all of these ${ℬ}_i$ been topologies). That is, both the notations $$\prod _{}{ℬ}_{\bullet }=\prod _{i\in I}{ℬ}_i$$ denote the family of all cylinder subsets $\prod _{i\in I}{S}_i\subseteq \prod X_{\bullet }$ such that $S_i=X_i$ for all but finitely many $i\in I$ and where $S_i\in {ℬ}_i$ for any one of these finitely many exceptions (that is, for any $i$ such that $S_i\ne X_i,$ necessarily $S_i\in {ℬ}_i$). When every ${ℬ}_i$ is a filter subbase then the family $\bigcup _{i\in I}\mathrm{Pr}{}_{X_i}^{-1}\left({ℬ}_i\right)$ is a filter subbase for the filter on $\prod X_{\bullet }$ generated by ${ℬ}_{\bullet }.$Template:Sfn If $\prod {ℬ}_{\bullet }$ is a filter subbase then the filter on $\prod X_{\bullet }$ that it generates is called the Template:Em.Template:Sfn If every ${ℬ}_i$ is a prefilter on $X_i$ then $\prod {ℬ}_{\bullet }$ will be a prefilter on $\prod X_{\bullet }$ and moreover, this prefilter is equal to the coarsest prefilter $ℱ\text{ on }\prod X_{\bullet }$ such that $\mathrm{Pr}{}_{X_i}(ℱ)={ℬ}_i$ for every $i\in I.$Template:Sfn However, $\prod {ℬ}_{\bullet }$ may fail to be a filter on $\prod X_{\bullet }$ even if every ${ℬ}_i$ is a filter on $X_i.$Template:Sfn

Convergence, limits, and cluster points

Throughout, $(X,\tau )$ 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 Template:Em 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 $S\subseteq X$ is a proper subset then any filter on $S$ will not be a filter on $X,$ although it will be a prefilter.

One advantage that filters have is that they are distinguished representatives of their equivalence class (relative to $\le$), 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 the construction of completions of uniform spaces via 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 $ℱ$ is a non-principal filter on an infinite set $X.$ $ℱ$ has one "upward" property (that of being closed upward) and one "downward" property (that of being directed downward). Starting with any $F_0\in ℱ,$ there always exists some $F_1\in ℱ$ that is a Template:Em subset of $F_0$; this may be continued ad infinitum to get a sequence $F_0⊋F_1⊋\cdots$ of sets in $ℱ$ with each $F_{i+1}$ being a Template:Em subset of $F_i.$ The same is Template:Em true going "upward", for if $F_0=X\in ℱ$ then there is no set in $ℱ$ that contains $X$ 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 $\subseteq ,$ every filter subbase is contained in a unique smallest filter but there may not exist a unique smallest prefilter containing it.

Limits and convergence

Template:In5A family $ℬ$ is said to Template:Em to a point $x$ of $X$Template:Sfn if $ℬ\ge 𝒩(x).$ Explicitly, $𝒩(x)\le ℬ$ means that every neighborhood $N\text{ of }x$ contains some $B\in ℬ$ as a subset (that is, $B\subseteq N$); thus the following then holds: $𝒩\ni N\supseteq B\in ℬ.$ In words, a family converges to a point or subset $x$ if and only if it is Template:Em than the neighborhood filter at $x.$ A family $ℬ$ converging to a point $x$ may be indicated by writing $ℬ\to x\text{ or }\lim ℬ\to x\text{ in }X$Template:Sfn and saying that $x$ is a Template:Em of $ℬ\text{ in }X;$ if this limit $x$ is a point (and not a subset), then $x$ is also called a Template:Em.Template:Sfn As usual, $\lim ℬ=x$ is defined to mean that $ℬ\to x$ and $x\in X$ is the Template:Em limit point of $ℬ;$ that is, if also $ℬ\to z\text{ then }z=x.$Template:Sfn (If the notation "$\lim ℬ=x$" did not also require that the limit point $x$ be unique then the equals sign = would no longer be guaranteed to be transitive). The set of all limit points of $ℬ$ is denoted by $\lim {}_Xℬ\text{ or }\lim ℬ.$Template:Sfn

In the above definitions, it suffices to check that $ℬ$ is finer than some (or equivalently, finer than every) neighborhood base in $(X,\tau )$ of the point (for example, such as $\tau (x)=\{U\in \tau :x\in U\}$ or $\tau (S)=\bigcap _{s\in S}\tau (s)$ when $S\ne \varnothing$).

Examples

If $X:={ℝ}^n$ is Euclidean space and $\Vert x\Vert$ denotes the Euclidean norm (which is the distance from the origin, defined as usual), then all of the following families converge to the origin:

1. the prefilter $\{B_r(0):0<r\le 1\}$ of all open balls centered at the origin, where $B_r(z)=\{x:\Vert x-z\Vert <r\}.$
2. the prefilter $\{B_{\le r}(0):0<r\le 1\}$ of all closed balls centered at the origin, where $B_{\le r}(z)=\{x:\Vert x-z\Vert \le r\}.$ This prefilter is equivalent to the one above.
3. the prefilter $\{R\cap B_{\le r}(0):0<r\le 1\}$ where $R=S_1\cup S_{1/2}\cup S_{1/3}\cup \cdots$ is a union of spheres $S_r=\{x:\Vert x\Vert =r\}$ centered at the origin having progressively smaller radii. This family consists of the sets $S_{1/n}\cup S_{1/(n+1)}\cup S_{1/(n+2)}\cup \cdots$ as $n$ ranges over the positive integers.
4. any of the families above but with the radius $r$ ranging over $1,1/2,1/3,1/4,\dots$ (or over any other positive decreasing sequence) instead of over all positive reals.
   • Drawing or imagining any one of these sequences of sets when $X={ℝ}^2$ has dimension $n=2$ suggests that intuitively, these sets "should" converge to the origin (and indeed they do). This is the intuition that the above definition of a "convergent prefilter" make rigorous.

Although $\Vert \cdot \Vert$ was assumed to be the Euclidean norm, the example above remains valid for any other norm on ${ℝ}^n.$

The one and only limit point in $X:=ℝ$ of the free prefilter $\{(0,r):r>0\}$ is $0$ since every open ball around the origin contains some open interval of this form. The fixed prefilter $ℬ:=\{[0,1+r):r>0\}$ does not converges in $ℝ$ to any Template:Em and so $\lim ℬ=\varnothing ,$ although $ℬ$ does converge to the Template:Em $\ker ℬ=[0,1]$ since $𝒩([0,1])\le ℬ.$ However, not every fixed prefilter converges to its kernel. For instance, the fixed prefilter $\{[0,1+r)\cup (1+1/r,\mathrm{\infty }):r>0\}$ also has kernel $[0,1]$ but does not converges (in $ℝ$) to it.

The free prefilter $(ℝ,\mathrm{\infty }):=\{(r,\mathrm{\infty }):r\in ℝ\}$ of intervals does not converge (in $ℝ$) to any point. The same is also true of the prefilter $[ℝ,\mathrm{\infty }):=\{[r,\mathrm{\infty }):r\in ℝ\}$ because it is equivalent to $(ℝ,\mathrm{\infty })$ and equivalent families have the same limits. In fact, if $ℬ$ is any prefilter in any topological space $X$ then for every $S\in {ℬ}^{\uparrow X},$ $ℬ\to S.$ More generally, because the only neighborhood of $X$ is itself (that is, $𝒩(X)=\{X\}$), every non-empty family (including every filter subbase) converges to $X.$

For any point $x,$ its neighborhood filter $𝒩(x)\to x$ always converges to $x.$ More generally, any neighborhood basis at $x$ converges to $x.$ A point $x$ is always a limit point of the principle ultra prefilter $\{\{x\}\}$ and of the ultrafilter that it generates. The empty family $ℬ=\varnothing$ does not converge to any point.

Basic properties

If $ℬ$ converges to a point then the same is true of any family finer than $ℬ.$ This has many important consequences. One consequence is that the limit points of a family $ℬ$ are the same as the limit points of its upward closure: $${\lim }_Xℬ={\lim }_X\left({ℬ}^{\uparrow X}\right).$$ In particular, the limit points of a prefilter are the same as the limit points of the filter that it generates. Another consequence is that if a family converges to a point then the same is true of the family's trace/restriction to any given subset of $X.$ If $ℬ$ is a prefilter and $B\in ℬ$ then $ℬ$ converges to a point of $X$ if and only if this is true of the trace $ℬ{|}_B.$Template:Sfn If a filter subbase converges to a point then do the filter and the Template:Pi-system that it generates, although the converse is not guaranteed. For example, the filter subbase $\{(-\mathrm{\infty },0],[0,\mathrm{\infty })\}$ does not converge to $0$ in $X:=ℝ$ although the (principle ultra) filter that it generates does.

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

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

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

Cluster points

A family $ℬ$ is said to Template:Em a point $x$ of $X$ if it meshes with the neighborhood filter of $x;$ that is, if $ℬ\mathrm{\#}𝒩(x).$ Explicitly, this means that $B\cap N\ne \varnothing \text{ for every }B\in ℬ$ and every neighborhood $N$ of $x.$ In particular, a point $x\in X$ is a Template:Em or an Template:Em of a family $ℬ$Template:Sfn if $ℬ$ meshes with the neighborhood filter at $x:ℬ\mathrm{\#}𝒩(x).$ The set of all cluster points of $ℬ$ is denoted by ${\mathrm{cl}}_Xℬ,$ where the subscript may be dropped if not needed.

In the above definitions, it suffices to check that $ℬ$ meshes with some (or equivalently, meshes with every) neighborhood base in $X$ of $x\text{ or }S.$ When $ℬ$ is a prefilter then the definition of "$ℬ\text{ and }𝒩$ mesh" can be characterized entirely in terms of the subordination preorder $\le .$

Two equivalent families of sets have the exact same limit points and also the same cluster points. No matter the topology, for every $x\in X,$ both $\{x\}$ and the principal ultrafilter $\{x{\}}^{\uparrow X}$ cluster at $x.$ If $ℬ$ clusters to a point then the same is true of any family coarser than $ℬ.$ Consequently, the cluster points of a family $ℬ$ are the same as the cluster points of its upward closure: $${\mathrm{cl}}_Xℬ={\mathrm{cl}}_X\left({ℬ}^{\uparrow X}\right).$$ In particular, the cluster points of a prefilter are the same as the cluster points of the filter that it generates.

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

1. $ℬ$ clusters at $x.$
2. The family ${ℬ}^{\uparrow X}$ generated by $ℬ$ clusters at $x.$
3. There exists a family equivalent to $ℬ$ that clusters at $x.$
4. $x\in \bigcap _{F\in ℬ}{\mathrm{cl}}_{X}F.$Template:Sfn
5. $X\setminus N\in ̸{ℬ}^{\uparrow X}$ for every neighborhood $N$ of $x.$
   • If $ℬ$ is a filter on $X$ then $x\in {\mathrm{cl}}_Xℬ\text{ if and only if }X\setminus N\in ̸ℬ$ for every neighborhood $N\text{ of }x.$
6. There exists a prefilter $ℱ$ subordinate to $ℬ$ (that is, $ℱ\ge ℬ$) that converges to $x.$
   • This is the filter equivalent of "$x$ is a cluster point of a sequence if and only if there exists a subsequence converging to $x.$
   • In particular, if $x$ is a cluster point of a prefilter $ℬ$ then $ℬ(\cap )𝒩(x)$ is a prefilter subordinate to $ℬ$ that converges to $x.$

The set ${\mathrm{cl}}_Xℬ$ of all cluster points of a prefilter $ℬ$ satisfies $${\mathrm{cl}}_Xℬ=\bigcap _{B\in ℬ}{\mathrm{cl}}_{X}B.$$ Consequently, the set ${\mathrm{cl}}_Xℬ$ of all cluster points of Template:Em prefilter $ℬ$ is a closed subset of $X.$Template:SfnTemplate:Sfn This also justifies the notation ${\mathrm{cl}}_Xℬ$ for the set of cluster points.Template:Sfn In particular, if $K\subseteq X$ is non-empty (so that $ℬ:=\{K\}$ is a prefilter) then ${\mathrm{cl}}_X\{K\}={\mathrm{cl}}_{X}K$ since both sides are equal to $\bigcap _{B\in ℬ}{\mathrm{cl}}_{X}B.$

Properties and relationships

Just like sequences and nets, it is possible for a prefilter on a topological space of infinite cardinality to not have Template:Em cluster points or limit points.Template:Sfn

If $x$ is a limit point of $ℬ$ then $x$ is necessarily a limit point of any family $𝒞$ Template:Em than $ℬ$ (that is, if $𝒩(x)\le ℬ\text{ and }ℬ\le 𝒞$ then $𝒩(x)\le 𝒞$).Template:Sfn In contrast, if $x$ is a cluster point of $ℬ$ then $x$ is necessarily a cluster point of any family $𝒞$ Template:Em than $ℬ$ (that is, if $𝒩(x)\text{ and }ℬ$ mesh and $𝒞\le ℬ$ then $𝒩(x)\text{ and }𝒞$ mesh).

Equivalent families and subordination

Any two equivalent families $ℬ\text{ and }𝒞$ can be used Template:Em in the definitions of "limit of" and "cluster at" because their equivalency guarantees that $𝒩\le ℬ$ if and only if $𝒩\le 𝒞,$ and also that $𝒩\mathrm{\#}ℬ$ if and only if $𝒩\mathrm{\#}𝒞.$ In essence, the preorder $\le$ 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 Template:Em in terms of the subordination relation. This is why the preorder $\le$ is of such great importance in applying (pre)filters to Topology.

Limit and cluster point relationships and sufficient conditions

Every limit point of a non-degenerate family $ℬ$ is also a cluster point; in symbols: $${\lim }_Xℬ\subseteq {\mathrm{cl}}_Xℬ.$$ This is because if $x$ is a limit point of $ℬ$ then $𝒩(x)\text{ and }ℬ$ mesh,Template:SfnTemplate:Sfn which makes $x$ a cluster point of $ℬ.$Template:Sfn But in general, a cluster point need not be a limit point. For instance, every point in any given non-empty subset $K\subseteq X$ is a cluster point of the principle prefilter $ℬ:=\{K\}$ (no matter what topology is on $X$) but if $X$ is Hausdorff and $K$ has more than one point then this prefilter has no limit points; the same is true of the filter $\{K{\}}^{\uparrow X}$ that this prefilter generates.

However, every cluster point of an Template:Em prefilter is a limit point. Consequently, the limit points of an Template:Em prefilter $ℬ$ are the same as its cluster points: ${\lim }_Xℬ={\mathrm{cl}}_Xℬ;$ that is to say, a given point is a cluster point of an ultra prefilter $ℬ$ if and only if $ℬ$ converges to that point.Template:SfnTemplate:Sfn Although a cluster point of a filter need not be a limit point, there will always exist a finer filter that does converge to it; in particular, if $ℬ$ clusters at $x$ then $ℬ(\cap )𝒩(x)=\{B\cap N:B\in ℬ,N\in 𝒩(x)\}$ is a filter subbase whose generated filter converges to $x.$

If $\varnothing \ne ℬ\subseteq \mathrm{\wp }(X)\text{ and }𝒮\ge ℬ$ is a filter subbase such that $𝒮\to x\text{ in }X$ then $x\in {\mathrm{cl}}_Xℬ.$ In particular, any limit point of a filter subbase subordinate to $ℬ\ne \varnothing$ is necessarily also a cluster point of $ℬ.$ If $x$ is a cluster point of a prefilter $ℬ$ then $ℬ(\cap )𝒩(x)$ is a prefilter subordinate to $ℬ$ that converges to $x\text{ in }X.$

If $S\subseteq X$ and if $ℬ$ is a prefilter on $S$ then every cluster point of $ℬ\text{ in }X$ belongs to ${\mathrm{cl}}_{X}S$ and any point in ${\mathrm{cl}}_{X}S$ is a limit point of a filter on $S.$Template:Sfn

Primitive sets

A subset $P\subseteq X$ is called Template:EmTemplate:Sfn if it is the set of limit points of some ultrafilter (or equivalently, some ultra prefilter). That is, if there exists an ultrafilter $ℬ\text{ on }X$ such that $P$ is equal to ${\lim }_Xℬ,$ which recall denotes the set of limit points of $ℬ\text{ in }X.$ Since limit points are the same as cluster points for ultra prefilters, a subset is primitive if and only if it is equal to the set ${\mathrm{cl}}_Xℬ$ of cluster points of some ultra prefilter $ℬ.$ For example, every closed singleton subset is primitive.Template:Sfn The image of a primitive subset of $X$ under a continuous map $f:X\to Y$ is contained in a primitive subset of $Y.$Template:Sfn

Assume that $P,Q\subseteq X$ are two primitive subset of $X.$ If $U$ is an open subset of $X$ that intersects $P$ then $U\in ℬ$ for any ultrafilter $ℬ\text{ on }X$ such that $P={\lim }_Xℬ.$Template:Sfn In addition, if $P\text{ and }Q$ are distinct then there exists some $S\subseteq X$ and some ultrafilters ${ℬ}_P\text{ and }{ℬ}_Q\text{ on }X$ such that $P={\lim }_X{ℬ}_P,Q={\lim }_X{ℬ}_Q,S\in {ℬ}_P,$ and $X\setminus S\in {ℬ}_Q.$Template:Sfn

Other results

If $X$ is a complete lattice then:Template:Citation needed

• The limit inferior of $B$ is the infimum of the set of all cluster points of $B.$
• The limit superior of $B$ is the supremum of the set of all cluster points of $B.$
• $B$ 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

Suppose $f:X\to Y$ is a map from a set into a topological space $Y,$ $ℬ\subseteq \mathrm{\wp }(X),$ and $y\in Y.$ If $y$ is a limit point (respectively, a cluster point) of $f(ℬ)\text{ in }Y$ then $y$ is called a Template:Em or Template:Em (respectively, a Template:Em) Template:EmTemplate:Sfn Explicitly, $y$ is a limit of $f$ with respect to $ℬ$ if and only if $𝒩(y)\le f(ℬ),$ which can be written as $f(ℬ)\to y\text{ or }\lim f(ℬ)\to y\text{ in }Y$ (by definition of this notation) and stated as $f$ Template:EmTemplate:Sfn If the limit $y$ is unique then the arrow $\to$ may be replaced with an equals sign $=.$Template:Sfn The neighborhood filter $𝒩(y)$ can be replaced with any family equivalent to it and the same is true of $ℬ.$

The definition of a convergent net is a special case of the above definition of a limit of a function. Specifically, if $x\in X\text{ and }\chi :(I,\le )\to X$ is a net then $$\chi \to x\text{ in }X\text{ if and only if }\chi (\mathrm{Tails}(I,\le ))\to x\text{ in }X,$$ where the left hand side states that $x$ is a [[Convergent net|limit Template:Em]] $\chi$ while the right hand side states that $x$ is a limit Template:Em $\chi$ with respect to $ℬ:=\mathrm{Tails}(I,\le )$ (as just 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 $f$) of particular prefilters on the domain $X.$ This shows that prefilters provide a general framework into which many of the various definitions of limits fit.Template:Sfn The limits in the left-most column are defined in their usual way with their obvious definitions.

Throughout, let $f:X\to Y$ be a map between topological spaces, $x_0\in X,\text{ and }y\in Y.$ If $Y$ is Hausdorff then all arrows Template:Nowrap in the table may be replaced with equal signs Template:Nowrap and Template:Nowrap may be replaced with Template:NowrapTemplate:Sfn

Type of limit	Template:Nowarp	Definition in terms of prefiltersTemplate:Sfn	Assumptions
$\underset{x\to x_0}{\lim }f(x)\to y$	⇔	$f(ℬ)\to y\text{ where }ℬ:=𝒩\left(x_0\right)$
$\underset{\stackrel{x\to x_0}{x\ne x_0}}{\lim }f(x)\to y$	⇔	$f(ℬ)\to y\text{ where }ℬ:=\{N\setminus \left\{x_0\right\}:N\in 𝒩\left(x_0\right)\}$
$\underset{\stackrel{x\to x_0}{x\in S}}{\lim }f(x)\to y$ or $\underset{x\to x_0}{\lim }f{|}_S(x)\to y$	⇔	$f(ℬ)\to y\text{ where }ℬ:=𝒩\left(x_0\right){|}_S:=\{N\cap S:N\in 𝒩\left(x_0\right)\}$	$S\subseteq X\text{ and }{x}_0\in {\mathrm{cl}}_{X}S$
$\underset{\stackrel{x\to x_0}{x\ne x_0}}{\lim }f(x)\to y$	⇔	$f(ℬ)\to y\text{ where }ℬ:=\{(x_0-r,x_0)\cup (x_0,x_0+r):0<r\in ℝ\}$	$x_0\in X=ℝ$
$\underset{\stackrel{x\to x_0}{x<x_0}}{\lim }f(x)\to y$	⇔	$f(ℬ)\to y\text{ where }ℬ:=\{(x,x_0):x<x_0\}$	$x_0\in X=ℝ$
$\underset{\stackrel{x\to x_0}{x\le x_0}}{\lim }f(x)\to y$	⇔	$f(ℬ)\to y\text{ where }ℬ:=\{(x,x_0]:x<x_0\}$	$x_0\in X=ℝ$
$\underset{\stackrel{x\to x_0}{x>x_0}}{\lim }f(x)\to y$	⇔	$f(ℬ)\to y\text{ where }ℬ:=\{(x_0,x):x_0<x\}$	$x_0\in X=ℝ$
$\underset{\stackrel{x\to x_0}{x\ge x_0}}{\lim }f(x)\to y$	⇔	$f(ℬ)\to y\text{ where }ℬ:=\{[x_0,x):x_0\le x\}$	$x_0\in X=ℝ$
$\underset{n\to \mathrm{\infty }}{\lim }f(n)\to y$	⇔	$f(ℬ)\to y\text{ where }ℬ:=\{\{n,n+1,\dots \}:n\in \mathbb{N}\}\}$	$X=\mathbb{N}\text{ so }f:\mathbb{N}\to Y$ is a sequence in $Y$
$\underset{x\to \mathrm{\infty }}{\lim }f(x)\to y$	⇔	$f(ℬ)\to y\text{ where }ℬ:=(ℝ,\mathrm{\infty }):=\{(x,\mathrm{\infty }):x\in ℝ\}$	$X=ℝ$
$\underset{x\to -\mathrm{\infty }}{\lim }f(x)\to y$	⇔	$f(ℬ)\to y\text{ where }ℬ:=(-\mathrm{\infty },ℝ):=\{(-\mathrm{\infty },x):x\in ℝ\}$	$X=ℝ$
$\underset{|x|\to \mathrm{\infty }}{\lim }f(x)\to y$	⇔	$f(ℬ)\to y\text{ where }ℬ:=\{X\cap [(-\mathrm{\infty },x)\cup (x,\mathrm{\infty })]:x\in ℝ\}$	$X=ℝ\text{ or }X=\mathbb{Z}$ for a double-ended sequence
$\underset{\Vert x\Vert \to \mathrm{\infty }}{\lim }f(x)\to y$	⇔	$f(ℬ)\to y\text{ where }ℬ:=\{\{x\in X:\Vert x\Vert >r\}:0<r\in ℝ\}$	$(X,\Vert \cdot \Vert )\text{ is}$ a seminormed space; Template:Nowrap $\text{like }X=ℂ$

By defining different prefilters, many other notions of limits can be defined; for example, $\underset{\stackrel{|x|\to |x_0|}{|x|\ne |x_0|}}{\lim }f(x)\to y.$

Divergence to infinity

Divergence of a real-valued function to infinity can be defined/characterized by using the prefilters $$(ℝ,\mathrm{\infty }):=\{(r,\mathrm{\infty }):r\in ℝ\}\text{ and }(-\mathrm{\infty },ℝ):=\{(-\mathrm{\infty },r):r\in ℝ\},$$ where $f\to \mathrm{\infty }$ along $ℬ$ if and only if $(ℝ,\mathrm{\infty })\le f(ℬ)$ and similarly, $f\to -\mathrm{\infty }$ along $ℬ$ if and only if $(-\mathrm{\infty },ℝ)\le f(ℬ).$ The family $(ℝ,\mathrm{\infty })$ can be replaced by any family equivalent to it, such as $[ℝ,\mathrm{\infty }):=\{[r,\mathrm{\infty }):r\in ℝ\}$ for instance (in real analysis, this would correspond to replacing the strict inequality Template:Nowrap in the definition with Template:Nowrap and the same is true of $ℬ$ and $(-\mathrm{\infty },ℝ).$

So for example, if $ℬ:=𝒩\left(x_0\right)$ then $\underset{x\to x_0}{\lim }f(x)\to \mathrm{\infty }$ if and only if $(ℝ,\mathrm{\infty })\le f(ℬ)$ holds. Similarly, $\underset{x\to x_0}{\lim }f(x)\to -\mathrm{\infty }$ if and only if $(-\mathrm{\infty },ℝ)\le f\left(𝒩\left(x_0\right)\right),$ or equivalently, if and only if $(-\mathrm{\infty },ℝ]\le f\left(𝒩\left(x_0\right)\right).$

More generally, if $f$ is valued in $Y={ℝ}^n\text{ or }Y={ℂ}^n$ (or some other seminormed vector space) and if $B_{\ge r}:=\{y\in Y:|y|\ge r\}=Y\setminus B_{<r}$ then $\underset{x\to x_0}{\lim }|f(x)|\to \mathrm{\infty }$ if and only if $B_{\ge ℝ}\le f\left(𝒩\left(x_0\right)\right)$ holds, where $B_{\ge ℝ}:=\{B_{\ge r}:r\in ℝ\}.$

Filters and netsTemplate:Anchor

This section will describe the relationships between prefilters and nets in great detail because of how important these details are applying filters to topology − particularly in switching from utilizing nets to utilizing filters and vice verse.

Nets to prefilters

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

If $f:X\to Y$ is a map and $x_{\bullet }$ is a net in $X$ then $\mathrm{Tails}\left(f\left(x_{\bullet }\right)\right)=f(\mathrm{Tails}\left(x_{\bullet }\right)).$Template:Sfn

Prefilters to nets

A Template:Em is a pair $(S,s)$ consisting of a non-empty set $S$ and an element $s\in S.$ For any family $ℬ,$ let $$\mathrm{PointedSets}(ℬ):=\{(B,b):B\in ℬ\text{ and }b\in B\}.$$

Define a canonical preorder $\le$ on pointed sets by declaring $$(R,r)\le (S,s)\text{ if and only if }R\supseteq S.$$

There is a canonical map ${\mathrm{Point}}_{ℬ}:\mathrm{PointedSets}(ℬ)\to X$ defined by $(B,b)\mapsto b.$ If $i_0=(B_0,b_0)\in \mathrm{PointedSets}(ℬ)$ then the tail of the assignment ${\mathrm{Point}}_{ℬ}$ starting at $i_0$ is $\{c:(C,c)\in \mathrm{PointedSets}(ℬ)\text{ and }(B_0,b_0)\le (C,c)\}=B_0.$

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

If $ℬ$ is a prefilter on $X\text{ then }{\mathrm{Net}}_{ℬ}$ is a net in $X$ and the prefilter associated with ${\mathrm{Net}}_{ℬ}$ is $ℬ$; that is: $$\mathrm{Tails}\left({\mathrm{Net}}_{ℬ}\right)=ℬ.$$ This would not necessarily be true had ${\mathrm{Net}}_{ℬ}$ been defined on a proper subset of $\mathrm{PointedSets}(ℬ).$

If $x_{\bullet }$ is a net in $X$ then it is Template:Em in general true that ${\mathrm{Net}}_{\mathrm{Tails}\left(x_{\bullet }\right)}$ is equal to $x_{\bullet }$ because, for example, the domain of $x_{\bullet }$ may be of a completely different cardinality than that of ${\mathrm{Net}}_{\mathrm{Tails}\left(x_{\bullet }\right)}$ (since unlike the domain of ${\mathrm{Net}}_{\mathrm{Tails}\left(x_{\bullet }\right)},$ the domain of an arbitrary net in $X$ could have Template:Em cardinality).

Partially ordered net

The domain of the canonical net ${\mathrm{Net}}_{ℬ}$ is in general not partially ordered. However, in 1955 Bruns and Schmidt discovered a construction (detailed here: Filter (set theory)#Partially ordered net) 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.Template:Sfn Because the tails of this partially ordered net are identical to the tails of ${\mathrm{Net}}_{ℬ}$ (since both are equal to the prefilter $ℬ$), there is typically nothing lost by assuming that the domain of the net associated with a prefilter is both directed Template:Em partially ordered.Template:Sfn If can further be assumed that the partially ordered domain is also a dense order.

Subordinate filters and subnets

The notion of "$ℬ$ is subordinate to $𝒞$" (written $ℬ⊢𝒞$) is for filters and prefilters what "$x_{n_{\bullet }}={\left(x_{n_i}\right)}_{i=1}^{\mathrm{\infty }}$ is a subsequence of $x_{\bullet }={\left(x_i\right)}_{i=1}^{\mathrm{\infty }}$" is for sequences.Template:Sfn For example, if $\mathrm{Tails}\left(x_{\bullet }\right)=\{x_{\ge i}:i\in \mathbb{N}\}$ denotes the set of tails of $x_{\bullet }$ and if $\mathrm{Tails}\left(x_{n_{\bullet }}\right)=\{x_{n_{\ge i}}:i\in \mathbb{N}\}$ denotes the set of tails of the subsequence $x_{n_{\bullet }}$ (where $x_{n_{\ge i}}:=\{x_{n_j}:j\ge i\text{ and }j\in \mathbb{N}\}$) then $\mathrm{Tails}\left(x_{n_{\bullet }}\right)⊢\mathrm{Tails}\left(x_{\bullet }\right)$ (which by definition means $\mathrm{Tails}\left(x_{\bullet }\right)\le \mathrm{Tails}\left(x_{n_{\bullet }}\right)$) is true but $\mathrm{Tails}\left(x_{\bullet }\right)⊢\mathrm{Tails}\left(x_{n_{\bullet }}\right)$ is in general false. If $x_{\bullet }={\left(x_i\right)}_{i\in I}$ is a net in a topological space $X$ and if $𝒩(x)$ is the neighborhood filter at a point $x\in X,$ then $x_{\bullet }\to x\text{ if and only if }𝒩(x)\le \mathrm{Tails}\left(x_{\bullet }\right).$

If $f:X\to Y$ is an surjective open map, $x\in X,$ and $𝒞$ is a prefilter on $Y$ that converges to $f(x),$ then there exist a prefilter $ℬ$ on $X$ such that $ℬ\to x$ and $f(ℬ)$ is equivalent to $𝒞$ (that is, $𝒞\le f(ℬ)\le 𝒞$).Template:Sfn

Subordination analogs of results involving subsequences

The following results are the prefilter analogs of statements involving subsequences.Template:Sfn The condition "$𝒞\ge ℬ,$" which is also written $𝒞⊢ℬ,$ is the analog of "$𝒞$ is a subsequence of $ℬ.$" 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

Subnets in the sense of Willard and subnets in the sense of Kelley are the most commonly used definitions of "subnet."Template:Sfn The first definition of a subnet ("Kelley-subnet") was introduced by John L. Kelley in 1955.Template:Sfn Stephen Willard introduced in 1970 his own variant ("Willard-subnet") of Kelley's definition of subnet.Template:Sfn 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.Template:Sfn

A subset $R\subseteq I$ of a preordered space $(I,\le )$ is Template:Em or Template:Em in $I$ if for every $i\in I$ there exists some $r\in R$ such that $i\le r.$ If $R\subseteq I$ contains a tail of $I$ then $R$ is said to be Template:Em in $I$}}; explicitly, this means that there exists some $i\in I$ such that $I_{\ge i}\subseteq R$ (that is, $j\in R$ for all $j\in I$ satisfying $i\le j$). A subset is eventual if and only if its complement is not frequent (which is termed Template:Em).Template:Sfn A map $h:A\to I$ between two preordered sets is Template:Em if whenever $a,b\in A$ satisfy $a\le b,$ then $h(a)\le h(b).$

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

Example: If $I=\mathbb{N}$ and $x_{\bullet }={\left(x_i\right)}_{i\in I}$ is a constant sequence and if $A=\{1\}$ and $s_1:=x_1$ then ${\left(s_a\right)}_{a\in A}$ is an AA-subnet of $x_{\bullet }$ but it is neither a Willard-subnet nor a Kelley-subnet of $x_{\bullet }.$

AA-subnets have a defining characterization that immediately shows that they are fully interchangeable with sub(ordinate)filters.Template:Sfn Explicitly, what is meant is that the following statement is true for AA-subnets:

Template:In5If $ℬ\text{ and }ℱ$ are prefilters then $ℬ\le ℱ$ if and only if ${\mathrm{Net}}_{ℱ}$ is an AA-subnet of ${\mathrm{Net}}_{ℬ}.$

If "AA-subnet" is replaced by "Willard-subnet" or "Kelley-subnet" then the above statement becomes Template:Em. In particular, as this counter-example demonstrates, the problem is that the following statement is in general false:

Template:In5Template:Em statement: If $ℬ\text{ and }ℱ$ are prefilters such that $ℬ\le ℱ\text{ then }{\mathrm{Net}}_{ℱ}$ is a Kelley-subnet of ${\mathrm{Net}}_{ℬ}.$

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

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 Template:Em 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.Template:Sfn

Topologies and prefilters

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

Examples of relationships between filters and topologies

Bases and prefilters

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

Template:In5$ℬ$ is a base (resp. subbase) for $\tau$ if and only if for every $x\in X,{ℬ}_x$ is a filter base (resp. filter subbase) that generates the neighborhood filter of $(X,\tau )$ at $x.$

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 $X=ℝ$ has its usual topology and if $x\in X,$ then any neighborhood filter base $ℬ$ of $x$ is fixed by $x$ (in fact, it is even true that $\ker ℬ=\{x\}$) but $ℬ$ is Template:Em principal since $\{x\}\in ̸ℬ.$ 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 $x$ in topological space $X$ is fixed since its kernel contains $x$ (and possibly other points if, for instance, $X$ is not a T₁ space). This is also true of any neighborhood basis at $x.$ For any point $x$ in a T₁ space (for example, a Hausdorff space), the kernel of the neighborhood filter of $x$ is equal to the singleton set $\{x\}.$

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

Generating topologies from filters and prefilters

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

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

Topological properties and prefilters

Neighborhoods and topologies

The neighborhood filter of a nonempty subset $S\subseteq X$ in a topological space $X$ is equal to the intersection of all neighborhood filters of all points in $S.$Template:Sfn A subset $S\subseteq X$ is open in $X$ if and only if whenever $ℱ$ is a filter on $X$ and $s\in S,$ then $ℱ\to s\text{ in }X\text{ implies }S\in ℱ.$

Suppose $\sigma \text{ and }\tau$ are topologies on $X.$ Then $\tau$ is finer than $\sigma$ (that is, $\sigma \subseteq \tau$) if and only if whenever $x\in X\text{ and }ℬ$ is a filter on $X,$ if $ℬ\to x\text{ in }(X,\tau )$ then $ℬ\to x\text{ in }(X,\sigma ).$Template:Sfn Consequently, $\sigma =\tau$ if and only if for every filter $ℬ\text{ on }X$ and every $x\in X,ℬ\to x\text{ in }(X,\sigma )$ if and only if $ℬ\to x\text{ in }(X,\tau ).$Template:Sfn However, it is possible that $\sigma \ne \tau$ while also for every filter $ℬ\text{ on }X,ℬ$ converges to Template:Em point of $X\text{ in }(X,\sigma )$ if and only if $ℬ$ converges to Template:Em point of $X\text{ in }(X,\tau ).$Template:Sfn

Closure

If $ℬ$ is a prefilter on a subset $S\subseteq X$ then every cluster point of $ℬ\text{ in }X$ belongs to ${\mathrm{cl}}_{X}S.$Template:Sfn

If $x\in X\text{ and }S\subseteq X$ is a non-empty subset, then the following are equivalent:

1. $x\in {\mathrm{cl}}_{X}S$
2. $x$ is a limit point of a prefilter on $S.$ Explicitly: there exists a prefilter $ℱ\subseteq \mathrm{\wp }(S)\text{ on }S$ such that $ℱ\to x\text{ in }X.$Template:Sfn
3. $x$ is a limit point of a filter on $S.$Template:Sfn
4. There exists a prefilter $ℱ\text{ on }X$ such that $S\in ℱ\text{ and }ℱ\to x\text{ in }X.$
5. The prefilter $\{S\}$ meshes with the neighborhood filter $𝒩(x).$ Said differently, $x$ is a cluster point of the prefilter $\{S\}.$
6. The prefilter $\{S\}$ meshes with some (or equivalently, with every) filter base for $𝒩(x)$ (that is, with every neighborhood basis at $x$).

The following are equivalent:

1. $x$ is a limit points of $S\text{ in }X.$
2. There exists a prefilter $ℱ\subseteq \mathrm{\wp }(S)\text{ on }\{S\}\setminus \{x\}$ such that $ℱ\to x\text{ in }X.$Template:Sfn

Closed sets

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

1. $S$ is a closed subset of $X.$
2. If $x\in X\text{ and }ℱ\subseteq \mathrm{\wp }(S)$ is a prefilter on $S$ such that $ℱ\to x\text{ in }X,$ then $x\in S.$
3. If $x\in X\text{ and }ℱ\subseteq \mathrm{\wp }(S)$ is a prefilter on $S$ such that $x$ is an accumulation points of $ℱ\text{ in }X,$ then $x\in S.$Template:Sfn
4. If $x\in X$ is such that the neighborhood filter $𝒩(x)$ meshes with $\{S\}$ then $x\in S.$

Hausdorffness

The following are equivalent:

1. $X$ is a Hausdorff space.
2. Every prefilter on $X$ converges to at most one point in $X.$Template:Sfn
3. The above statement but with the word "prefilter" replaced by any one of the following: filter, ultra prefilter, ultrafilter.Template:Sfn

Compactness

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

The following are equivalent:

1. $(X,\tau )$ is a compact space.
2. Every ultrafilter on $X$ converges to at least one point in $X.$Template:Sfn
   • 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 word "ultrafilter" replaced by "ultra prefilter".Template:Sfn
4. For every filter $𝒞\text{ on }X$ there exists a filter $ℱ\text{ on }X$ such that $𝒞\le ℱ$ and $ℱ$ converges to some point of $X.$
5. The above statement but with each instance of the word "filter" replaced by: prefilter.
6. Every filter on $X$ has at least one cluster point in $X.$Template:Sfn
   • That this condition is equivalent to compactness can be proven by using only the ultrafilter lemma.
7. The above statement but with the word "filter" replaced by "prefilter".Template:Sfn
8. Alexander subbase theorem: There exists a subbase $𝒮\text{ for }\tau$ such that every cover of $X$ by sets in $𝒮$ has a finite subcover.
   • That this condition is equivalent to compactness can be proven by using only the ultrafilter lemma.

If $ℱ$ is the set of all complements of compact subsets of a given topological space $X,$ then $ℱ$ is a filter on $X$ if and only if $X$ is Template:Em compact.

Continuity

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

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

1. $f:X\to Y$ is continuous at $x.$
2. Definition: For every neighborhood $V$ of $f(x)\text{ in }Y$ there exists some neighborhood $N$ of $x\text{ in }X$ such that $f(N)\subseteq V.$
3. $f(𝒩(x))\to f(x)\text{ in }Y.$Template:Sfn
4. If $ℬ$ is a filter on $X$ such that $ℬ\to x\text{ in }X$ then $f(ℬ)\to f(x)\text{ in }Y.$
5. The above statement but with the word "filter" replaced by "prefilter".

The following are equivalent:

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

If $ℬ$ is a prefilter on $X,x\in X$ is a cluster point of $ℬ,\text{ and }f:X\to Y$ is continuous, then $f(x)$ is a cluster point in $Y$ of the prefilter $f(ℬ).$Template:Sfn

A subset $D$ of a topological space $X$ is dense in $X$ if and only if for every $x\in X,$ the trace ${𝒩}_X(x){|}_D$ of the neighborhood filter ${𝒩}_X(x)$ along $D$ does not contain the empty set (in which case it will be a filter on $D$).

Suppose $f:D\to Y$ is a continuous map into a Hausdorff regular space $Y$ and that $D$ is a dense subset of a topological space $X.$ Then $f$ has a continuous extension $F:X\to Y$ if and only if for every $x\in X,$ the prefilter $f({𝒩}_X(x){|}_D)$ converges to some point in $Y.$ Furthermore, this continuous extension will be unique whenever it exists.Template:Sfn

Products

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

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

Suppose $X\text{ and }Y$ are topological spaces, $ℬ$ is a prefilter on $X$ having $x\in X$ as a cluster point, and $𝒞$ is a prefilter on $Y$ having $y\in Y$ as a cluster point. Then $(x,y)$ is a cluster point of $ℬ\times 𝒞$ in the product space $X\times Y.$Template:Sfn However, if $X=Y=\mathbb{Q}$ then there exist sequences ${\left(x_i\right)}_{i=1}^{\mathrm{\infty }}\subseteq X\text{ and }{\left(y_i\right)}_{i=1}^{\mathrm{\infty }}\subseteq Y$ such that both of these sequences have a cluster point in $\mathbb{Q}$ but the sequence ${(x_i,y_i)}_{i=1}^{\mathrm{\infty }}\subseteq X\times Y$ does Template:Em have a cluster point in $X\times Y.$Template:Sfn

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

Template:Collapse top Let $X_{\bullet }:={\left(X_i\right)}_{i\in I}$ be compact [[Hausdorff space|Template:Em]] topological spaces. Assume that the ultrafilter lemma holds (because of the Hausdorff assumption, this proof does Template:Em need the full strength of the axiom of choice; the ultrafilter lemma suffices). Let $X:=\prod X_{\bullet }$ be given the product topology (which makes $X$ a Hausdorff space) and for every $i,$ let $\mathrm{Pr}{}_i:X\to X_i$ denote this product's projections. If $X=\varnothing$ then $X$ is compact and the proof is complete so assume $X\ne \varnothing .$ Despite the fact that $X\ne \varnothing ,$ because the axiom of choice is not assumed, the projection maps $\mathrm{Pr}{}_i:X\to X_i$ are not guaranteed to be surjective.

Let $ℬ$ be an ultrafilter on $X$ and for every $i,$ let ${ℬ}_i$ denote the ultrafilter on $X_i$ generated by the ultra prefilter $\mathrm{Pr}{}_i(ℬ).$ Because $X_i$ is compact and Hausdorff, the ultrafilter ${ℬ}_i$ converges to a unique limit point $x_i\in X_i$ (because of $x_i$'s uniqueness, this definition does not require the axiom of choice). Let $x:={\left(x_i\right)}_{i\in I}$ where $x$ satisfies $\mathrm{Pr}{}_i(x)=x_i$ for every $i.$ The characterization of convergence in the product topology that was given above implies that $ℬ\to x\text{ in }X.$ Thus every ultrafilter on $X$ converges to some point of $X,$ which implies that $X$ is compact (recall that this implication's proof only required the ultrafilter lemma). $◼$ Template:Collapse bottom

Examples of applications of prefilters

Uniformities and Cauchy prefilters

A uniform space is a set $X$ equipped with a filter on $X\times X$ that has certain properties. A Template:Em or Template:Em is a prefilter on $X\times X$ whose upward closure is a uniform space. A prefilter $ℬ$ on a uniform space $X$ with uniformity $ℱ$ is called a Template:Em if for every entourage $N\in ℱ,$ there exists some $B\in ℬ$ that is Template:Em, which means that $B\times B\subseteq N.$ A Template:Em is a minimal element (with respect to $\le$ or equivalently, to $\subseteq$) of the set of all Cauchy filters on $X.$ Examples of minimal Cauchy filters include the neighborhood filter ${𝒩}_X(x)$ of any point $x\in X.$ Every convergent filter on a uniform space is Cauchy. Moreover, every cluster point of a Cauchy filter is a limit point.

A uniform space $(X,ℱ)$ is called Template:Em (resp. Template:Em) if every Cauchy prefilter (resp. every elementary Cauchy prefilter) on $X$ converges to at least one point of $X$ (replacing all instance of the word "prefilter" with "filter" results in equivalent statement). Every compact uniform space is complete because any Cauchy filter has a cluster point (by compactness), which is necessarily also a limit point (since the filter is Cauchy).

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 (for example, 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. The set of all Template:Em on a Hausdorff topological vector space (TVS) $X$ can made into a vector space and topologized in such a way that it becomes a completion of $X$ (with the assignment $x\mapsto {𝒩}_X(x)$ becoming a linear topological embedding that identifies $X$ as a dense vector subspace of this completion).

More generally, a [[Cauchy space|Template:Em]] is a pair $(X,ℭ)$ consisting of a set $X$ together a family $ℭ\subseteq \mathrm{\wp }(\mathrm{\wp }(X))$ of (proper) filters, whose members are declared to be "Template:Em", having all of the following properties:

1. For each $x\in X,$ the discrete ultrafilter at $x$ is an element of $ℭ.$
2. If $F\in ℭ$ is a subset of a proper filter $G,$ then $G\in ℭ.$
3. If $F,G\in ℭ$ and if each member of $F$ intersects each member of $G,$ then $F\cap G\in ℭ.$

The set of all Cauchy filters on a uniform space forms a Cauchy space. Every Cauchy space is also a convergence space. A map $f:X\to Y$ between two Cauchy spaces is called Template:Em if the image of every Cauchy filter in $X$ is a Cauchy filter in $Y.$ Unlike the category of topological spaces, the category of Cauchy spaces and Cauchy continuous maps is Cartesian closed, and contains the category of proximity spaces.

Topologizing the set of prefilters

Starting with nothing more than a set $X,$ it is possible to topologize the set $$\mathbb{P}:=\mathrm{Prefilters}(X)$$ of all filter bases on $X$ with the Template:Em, which is named after Marshall Harvey Stone.

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

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

For every $S\subseteq X,$ let $$𝕆(S):=\{ℬ\in \mathbb{P}:S\in {ℬ}^{\uparrow X}\}$$ where $𝕆(X)=\mathbb{P}\text{ and }𝕆(\varnothing )=\varnothing .$ These sets will be the basic open subsets of the Stone topology. If $R\subseteq S\subseteq X$ then $$\{ℬ\in \mathrm{\wp }(\mathrm{\wp }(X)):R\in {ℬ}^{\uparrow X}\}\subseteq \{ℬ\in \mathrm{\wp }(\mathrm{\wp }(X)):S\in {ℬ}^{\uparrow X}\}.$$

From this inclusion, it is possible to deduce all of the subset inclusions displayed below with the exception of $𝕆(R\cap S)\supseteq 𝕆(R)\cap 𝕆(S).$ For all $R\subseteq S\subseteq X,$ $$𝕆(R\cap S)=𝕆(R)\cap 𝕆(S)\subseteq 𝕆(R)\cup 𝕆(S)\subseteq 𝕆(R\cup S)$$ where in particular, the equality $𝕆(R\cap S)=𝕆(R)\cap 𝕆(S)$ shows that the family $\{𝕆(S):S\subseteq X\}$ is a $\pi$-system that forms a basis for a topology on $\mathbb{P}$ called the Template:Em. It is henceforth assumed that $\mathbb{P}$ carries this topology and that any subset of $\mathbb{P}$ 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 $\mathbb{P}$ was defined withTemplate:Em using anything other than the set $X;$ there were Template:Em preexisting structures or assumptions on $X$ so this topology is completely independent of everything other than $X$ (and its subsets).

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

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

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

Subspace of ultrafilters

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

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

Every $\tau \in \mathrm{Top}(X)$ induces a canonical map ${𝒩}_{\tau }:X\to \mathrm{Filters}(X)$ defined by $x\mapsto {𝒩}_{\tau }(x),$ which sends $x\in X$ to the neighborhood filter of $x\text{ in }(X,\tau ).$ If $\tau ,\sigma \in \mathrm{Top}(X)$ then $\tau =\sigma$ if and only if ${𝒩}_{\tau }={𝒩}_{\sigma }.$ Thus every topology $\tau \in \mathrm{Top}(X)$ can be identified with the canonical map ${𝒩}_{\tau }\in \mathrm{Func}(X;\mathbb{P}),$ which allows $\mathrm{Top}(X)$ to be canonically identified as a subset of $\mathrm{Func}(X;\mathbb{P})$ (as a side note, it is now possible to place on $\mathrm{Func}(X;\mathbb{P}),$ and thus also on $\mathrm{Top}(X),$ the topology of pointwise convergence on $X$ so that it now makes sense to talk about things such as sequences of topologies on $X$ converging pointwise). For every $\tau \in \mathrm{Top}(X),$ the surjection ${𝒩}_{\tau }:(X,\tau )\to \mathrm{image}{𝒩}_{\tau }$ is always continuous, closed, and open, but it is injective if and only if $\tau \text{ is }{T}_0$ (that is, a Kolmogorov space). In particular, for every $T_0$ topology $\tau \text{ on }X,$ the map ${𝒩}_{\tau }:(X,\tau )\to \mathbb{P}$ is a topological embedding (said differently, every Kolmogorov space is a topological subspace of the space of prefilters).

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

See also

• Template:Annotated link
• Template:Annotated link
• Template:Annotated link
• Template:Annotated link
• Template:Annotated link
• Template:Annotated link
• Template:Annotated link
• Template:Annotated link
• Template:Annotated link
• Template:Annotated link

Notes

Proofs

References

• Template:Adams Franzosa Introduction to Topology Pure and Applied
• Template:Arkhangel'skii Ponomarev Fundamentals of General Topology Problems and Exercises
• Template:Berberian Lectures in Functional Analysis and Operator Theory
• Template:Bourbaki General Topology Part I Chapters 1-4
• Template:Bourbaki General Topology Part II Chapters 5-10
• Template:Bourbaki Topological Vector Spaces Part 1 Chapters 1–5
• Template:Cite book
• Template:Cite journal
• Template:Cite journal
• Template:Comfort Negrepontis The Theory of Ultrafilters 1974
• Template:Császár General Topology
• Template:Dixmier General Topology
• Template:Dolecki Mynard Convergence Foundations Of Topology
• Template:Dugundji Topology
• Template:Dunford Schwartz Linear Operators Part 1 General Theory
• Template:Edwards Functional Analysis Theory and Applications
• Template:Howes Modern Analysis and Topology 1995
• Template:Jarchow Locally Convex Spaces
• Template:Cite book
• Template:Joshi Introduction to General Topology
• Template:Kelley General Topology
• Template:Köthe Topological Vector Spaces I
• Template:Cite web (Provides an introductory review of filters in topology and in metric spaces.)
• Template:Narici Beckenstein Topological Vector Spaces
• Template:Robertson Topological Vector Spaces
• Template:Schaefer Wolff Topological Vector Spaces
• Template:Schechter Handbook of Analysis and Its Foundations
• Template:Schubert Topology
• Template:Trèves François Topological vector spaces, distributions and kernels
• Template:Wilansky Modern Methods in Topological Vector Spaces
• Template:Wilansky Topology for Analysis 2008
• Template:Willard General Topology