Filters in topology

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

The powerset lattice of the set

X ≝ {1,2,3,4

}, with the upper set

{1,4} ↑ X

colored dark green. It is a filter, and even a principal filter. It is not an ultrafilter, as it can be extended to the larger nontrivial filter

{1} ↑ X

, by including also the light green elements. Since

{1} ↑ X

cannot be extended any further, it is an ultrafilter.

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

Filters were introduced by Henri Cartan in 1937[1][2] and subsequently used by Bourbaki in their book Topologie Générale as an alternative to the similar notion of a net developed in 1922 by E. H. Moore and H. L. Smith. Filters can also be used to characterize the notions of sequence and net convergence. But unlike[note 1] sequence and net convergence, filter convergence is defined entirely in terms of subsets of the topological space $X$ and so it provides a notion of convergence that is completely intrinsic to the topological space. Every net induces a canonical filter and dually, every filter induces a canonical net, where this induced net (resp. induced filter) converges to a point if and only if the same is true of the original filter (resp. net). This characterization also holds for many other definitions such as cluster points. These relationships make it possible to switch between filters and nets, and they often also allow one to choose whichever of these two notions (filter or net) is more convenient for the problem at hand. However, in general this relationship does not extend to subordinate filters and subnets because as detailed below, there exist subordinate filters whose filter/subordinate–filter relationship cannot be described in terms of the corresponding net/subnet relationship (here it is assumed that "subnet" is defined using any of its most popular definitions, which are given in this article).

Motivation

Archetypical example of a filter

The archetypical example of a filter is the neighborhood filter ${\mathcal {N}}(x)$ at a point $x$ in a topological space $(X,\tau ),$ which by definition is the family of sets consisting of all neighborhoods of $x.$ By definition, a neighborhood of some given point (or subset) is any subset $N\subseteq X$ whose topological interior contains this point (or subset); importantly, neighborhoods are not required to be open sets (those are called open neighborhoods). The fundamental properties shared by neighborhood filters, which are listed below, ultimately became the definition of a "filter." A filter on $X$ is a set ${\mathcal {B}}$ of subsets of $X$ that satisfies all of the following conditions:

Not empty: $X\in {\mathcal {B}}$ – just as $X\in {\mathcal {N}}(x),$ since $X$ is always an (open) neighborhood of $x$ (and of anything else that it contains);
Does not contain the empty set: $\varnothing \not \in {\mathcal {B}}$ – just as no neighborhood of $x$ is empty;
Closed under finite intersections: If $B,C\in {\mathcal {B}}$ then $B\cap C\in {\mathcal {B}}$ – just as the intersection of any two neighborhoods of $x$ is again a neighborhood of $x$ ;
Upward closed: If $B\in {\mathcal {B}}$ and $B\subseteq S\subseteq X$ then $S\in {\mathcal {B}}$ – just as any subset of $X$ that contains a neighborhood of $x$ will necessarily be a neighborhood of $x$ (because $\operatorname {Int} _{X}B\subseteq \operatorname {Int} _{X}S$ and by definition of "neighborhood of $x$ ").

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

A sequence in $X$ is by definition a map $\mathbb {N} \to X$ from the natural numbers, which are an example of a directed set, 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 not be used to describe even basic topological properties like closure or continuity. This failure of sequences was the motivation for defining notions such as nets and filters, which never fail to characterize topological properties.

Nets directly generalize the notion of a sequence since nets are, by definition, maps $I\to X$ from an arbitrary directed set $(I,\leq )$ 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 only values in the range of a sequence. To see how this is done, consider a sequence $x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }$ in $X,$ which is by definition just a map $x_{\bullet }:\mathbb {N} \to X$ whose value at $i\in \mathbb {N}$ is denoted by $x_{i}$ rather than the more common parentheses notation $x_{\bullet }(i).$ Knowing only the range of the sequence is not enough to describe its convergence; multiple sets are needed. It turns out that the needed sets are the following,[note 2] which are called the tails of the sequence $x_{\bullet }$ :

\left\{x_{1},x_{2},x_{3},x_{4},\ldots \right\}

\left\{x_{2},x_{3},x_{4},x_{5},\ldots \right\}

\left\{x_{3},x_{4},x_{5},x_{6},\ldots \right\}

~~~~~~~~~~~~~~~~\vdots

\left\{x_{n},x_{n+1},x_{n+2},x_{n+3},\ldots \right\}

~~~~~~~~~~~~~~~~\vdots

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 $x_{n},x_{n+1},\ldots .$ This can be reworded as:

every neighborhood

U

must contain some set of the form

\{x_{n},x_{n+1},\ldots \}

as a subset.

It is the above characterization that can be used with the above family of tails to determine convergence (or non–convergence) of the sequence $x_{\bullet }:\mathbb {N} \to X.$ With these sets in hand, the map $x_{\bullet }:\mathbb {N} \to X$ 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 maps to families of sets.

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

Nets vs. filters − advantages and disadvantages

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

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

In contrast to nets, filters (and prefilters) are families of sets and so they have the advantages of sets. For example, if $f$ is surjective then the preimage or pullback $f -1 (ℬ) ≝ {f -1 (B) : B \in ℬ }$ of an arbitrary filter or prefilter ${\mathcal {B}}$ is both easily defined and guaranteed to be a prefilter, whereas it is less clear how to define the pullback of an arbitrary sequence (or net) $x •$ so that it is once again a sequence or net (unless $f$ is also injective and consequently a bijection, which is a stringent requirement). 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 the all sets in a filter is sometimes useful in Functional Analysis for instance. Theorems about images or preimages of sets under functions (e.g. continuity's definitions in terms of images or preimages of sets) may also be applied to filters. Special types of filters called ultrafilters have many useful properties that can significantly help in proving results. One downside of nets is their dependence on the directed sets that constitute their domains, which in general may be entirely unrelated to the space $X.$ 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 any cardinality. In contrast, the collection of all filters (and all prefilters) on $X$ is a set. Unlike nets and sequences, the notions of a "filter on $X$ " and of a "topology on $X$ " are both "intrinsic to $X$ " in the sense that both consist entirely of the subsets of $X$ and do not require any set that cannot be constructed from $X$ (such as $ℕ$ 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 $℘(X)$ will denote the powerset of $X.$ A subset of a powerset is called a family of sets (or simply, a family) where it is over $X$ if it is a subset of $℘(X)$ . Families of sets will be denoted by upper case calligraphy letters such as ${\mathcal {B}},$ ${\mathcal {C}},$ and ${\mathcal {F}}.$ Whenever these assumptions are needed, then it should be assumed that $X$ is non–empty and that ${\mathcal {B}},$ ${\mathcal {F}},$ 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 that are used in this article. Unfortunately, not all notation related to filters is well established and some notation varies greatly across the literature (e.g. 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 prefilter is well developed and has a plethora of definitions and notations, many of which are now unceremoniously listed to prevent this article from becoming prolix and to allow for the easy look up of notation and definitions. Their important properties are described later.

Sets operations

The upward closure or isotonization in

X

[6] of a family of subsets

{\mathcal {B}}\subseteq \wp (X)

is

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

and similarly the downward closure of

{\mathcal {B}}

is

{\mathcal {B}}^{\downarrow }:=\left\{S\subseteq B~:~B\in {\mathcal {B}}\,\right\}=\bigcup _{B\in {\mathcal {B}}}\wp (B).


Notation and Definition	Assumptions	Name
$\operatorname {ker} {\mathcal {B}}=\bigcap _{B\in {\mathcal {B}}}B$		Kernel of ${\mathcal {B}}$ [7]
$℘(X) = {S : S \subseteq X$ }		Power set of a set $X$ [7]
$ℬ \| S = {B \cap S : B \in ℬ } = ℬ (\cap) {S}$	$S$ is a set	Trace of ${\mathcal {B}}$ on $S$ [8] or the restriction of ${\mathcal {B}}$ to $S$
$ℬ (\cap) 𝒞 = {B \cap C : B \in ℬ and C \in 𝒞 }$ [9]		Elementwise (set) intersection ( ${\mathcal {B}}\cap {\mathcal {C}}$ will denote the usual intersection)
$ℬ (\cup) 𝒞 = {B \cup C : B \in ℬ and C \in 𝒞 }$ [9]		Elementwise (set) union ( ${\mathcal {B}}\cup {\mathcal {C}}$ will denote the usual union)
$ℬ (∖) 𝒞 = {B ∖ C : B \in ℬ and C \in 𝒞 }$		Elementwise (set) subtraction
$S ∖ ℬ = {S ∖ B : B \in ℬ } = {S} (∖) ℬ$	$S$ is a set	Dual of ${\mathcal {B}}$ in $S$ or set subtraction[8]
$S ↑ X = {S} ↑ X$	$S$ is a set	Upward closure or Isotonization[7]

The preorder

\leq

is defined on families of sets, say

{\mathcal {C}}

and

{\mathcal {F}},

by declaring that

{\mathcal {C}}\leq {\mathcal {F}}

if and only if

for every

C\in {\mathcal {C}}

there is some

F\in {\mathcal {F}}

such that

F \subseteq C

in which case it is said that ${\mathcal {C}}$ is coarser than ${\mathcal {F}},$ ${\mathcal {F}}$ is finer than (or subordinate to) ${\mathcal {C}},$ [10][11][12] and $ℱ ⊢ 𝒞$ may be written.

Two families

{\mathcal {B}}

and

{\mathcal {C}}

of sets mesh[8] if

B \cap C \neq \emptyset

for all

B\in {\mathcal {B}}

and

C\in {\mathcal {C}}

.


Notation	Definition	Name
$f^{-1}\left({\mathcal {B}}\right)=\left\{f^{-1}(B)~:~B\in {\mathcal {B}}\right\}$	$f$ is a map	Preimage of ${\mathcal {B}}$ under $f$ [13]
$f^{-1}(S)=\left\{x\in \operatorname {domain} f~:~f(x)\in S\right\}$	$f$ is a map and $S$ is a set	Preimage a $S$ under $f$
$f\left({\mathcal {B}}\right)=\{f(B)~:~B\in {\mathcal {B}}\}$	$f$ is a map	Image of ${\mathcal {B}}$ under $f$ [13]
$f(S)=\left\{f(s)~:~s\in S\cap \operatorname {domain} f\right\}$	$f$ is a map and $S$ is a set	Image a $S$ under $f$

Topology notation

The set of all topologies $X$ will be denoted by $Top(X)$ . Suppose $\tau$ is a topology on $X.$


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

Nets and their tails

A directed set is a set

I

together with a preorder, which will be denoted by

\leq

(unless explicitly indicated otherwise), that makes

(I,\leq )

into an (upward) directed set;[14] this means that for all

i,j\in I,

there exists some

k\in I

such that

i\leq k

and

j\leq k.

For any indices

i

and

j,

the notation

j\geq i

is defined to mean

i\leq j

while

i<j

is defined to mean that

i\leq j

holds but it is not true that

j\leq i

(if

\leq

is antisymmetric then this is equivalent to

i\leq j

and

i\neq j

).

A net in $X$ [14] is a map from a non–empty directed set into

X.


Notation and Definition	Assumptions	Name
$I_{\geq i}=\left\{j\in I~:~i\leq j\right\}$	$i\in I$ and $(I,\leq )$ is a directed set	Tail or section of $I$ starting at $i$
$f\left(I_{\geq i}\right)=\left\{f(j)~:~i\leq j{\text{ and }}j\in I\right\}$	$i\in I$ and $f~:~(I,\leq )\to X$ is a net	Tail or section of $f$ starting at $i$ [15]
$x_{\geq i}=\left\{x_{j}\in I~:~i\leq j{\text{ and }}j\in I\right\}$	$i\in I$ and $x_{\bullet }=\left(x_{i}\right)_{i\in I}$ is a net	Tail or section of $x_{\bullet }$ starting at $i$
$\operatorname {Tails} \left(x_{\bullet }\right)=x_{\geq \bullet }=\left\{x_{\geq i}~:~i\in I\right\}$	$x_{\bullet }=\left(x_{i}\right)_{i\in I}$ is a net	Set or prefilter of tails/sections of $x_{\bullet }.$ Also called the eventuality filter base generated by (the tails of) $x_{\bullet }.$ If $x_{\bullet }$ is a sequence then $\operatorname {Tails} \left(x_{\bullet }\right)$ is called the sequential filter base instead.[15]
$\operatorname {TailsFilter} \left(x_{\bullet }\right)=\operatorname {Tails} \left(x_{\bullet }\right)^{\uparrow X}$	$x_{\bullet }=\left(x_{i}\right)_{i\in I}$ is a net	(Eventuality) filter of/generated by (tails of) $x_{\bullet }.$ [15]

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 $x_{>i}=\left\{x_{j}\in I~:~i>j{\text{ and }}j\in I\right\},$ which is called the tail of $x_{\bullet }$ after $i$ , to be empty (e.g. this happens if $i$ is an upper bound of the directed set $I$ ). In this case, the family $\left\{x_{>i}~:~i\in I\right\}$ would contain the empty set, which would prevent it from being a prefilter (defined later). This is the (important) reason for defining $\operatorname {Tails} \left(x_{\bullet }\right)$ as $\left\{x_{\geq i}~:~i\in I\right\}$ rather than $\left\{x_{>i}~:~i\in I\right\}$ or even $\left\{x_{>i}~:~i\in I\right\}\cup \left\{x_{\geq i}~:~i\in I\right\}$ and it is for this reason that in general, when dealing with the prefilter of tails of a net, the strict inequality $<$ may not be used interchangeably with the inequality $\leq$ .

Filters and prefilters

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

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

Proper or nondegenerate if $\varnothing \not \in {\mathcal {B}}.$ Otherwise, if $\varnothing \in {\mathcal {B}},$ then it is called improper[16] or degenerate.

Directed downward[14] if whenever $A, B \in ℬ$ then there exists some $C\in {\mathcal {B}}$ such that $C\subseteq A\cap B.$
Alternatively, ${\mathcal {B}}$ directed downward (resp. directed upward) if and only if ${\mathcal {B}}$ is (upward) directed with respect to the preorder $\supseteq$ (resp. $\subseteq$ ), where by definition this means that for all $A, B \in ℬ$ , there exists some "greater" $C\in {\mathcal {B}}$ such that $A \supseteq C$ and $B \supseteq C$ (resp. such that $A \subseteq C$ and $B \subseteq C$ ), which can be rewritten as $A\cap B\supseteq C$ (resp. $A\cup B\subseteq C$ ). This explains the word "directed."

If a family ${\mathcal {B}}$ has a greatest element with respect to $\supseteq$ (for example, if $\varnothing \in {\mathcal {B}}$ ) then it is necessarily directed downward.

Closed under finite intersections (resp. unions) if the intersection (resp. union) of any two elements of ${\mathcal {B}}$ is an element of ${\mathcal {B}}.$
If ${\mathcal {B}}$ is closed under finite intersections then ${\mathcal {B}}$ is necessarily directed downward. The converse is generally false.

Upward closed or Isotone in $X$ [6] if ${\mathcal {B}}\subseteq \wp (X)$ and $ℬ = ℬ ↑ X$ , or equivalently, if whenever $B\in {\mathcal {B}}$ and $C$ satisfies $B \subseteq C \subseteq X$ then $C\in {\mathcal {B}}$ . Similarly, ${\mathcal {B}}$ is downward closed if $ℬ = ℬ ↓$ . An upward (respectively, downward) closed set is also called an upper set or upset (resp. a lower set or down set).
The family ${\mathcal {B}}^{\uparrow X},$ which is the upward closure of ${\mathcal {B}}$ in $X,$ is the unique smallest isotone family of sets over $X$ having ${\mathcal {B}}$ as a subset.

Many of the properties of ${\mathcal {B}}$ 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.

Ultrafilters(X) = Filters(X) \cap UltraPrefilters(X) \subseteq Filters(X) \cup UltraPrefilters(X) \subseteq Prefilters(X) \subseteq FilterSubbases(X)

.

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

Ideal[16][17] if ${\mathcal {B}}\neq \varnothing$ is downward closed and closed under finite unions.

Dual ideal on $X$ [18] if ${\mathcal {B}}\neq \varnothing$ is upward closed in $X$ and also closed under finite intersections.
Explanation of the word "dual": A family ${\mathcal {B}}$ is a dual ideal (resp. an ideal) on $X$ if and only if the dual of ${\mathcal {B}}$ in $X$ , which is the family

$X ∖ ℬ = {X ∖ B : B \in ℬ$ },

is an ideal (resp. a dual ideal) on $X.$ The family $X ∖ ℬ$ should not be confused with $℘(X) ∖ ℬ ≝ {S \subseteq X : S \notin ℬ$ }, where in general $X ∖ ℬ \neq ℘(X) ∖ ℬ$ . The dual of the dual is the original family, meaning $X ∖ (X ∖ ℬ) = ℬ$ ; and also $X$ belongs to the dual of ${\mathcal {B}}$ if and only if $\varnothing \in {\mathcal {B}}.$ [16]

Filter on $X$ [18][8] if ${\mathcal {B}}$ is a proper dual ideal on $X.$ That is, a filter on $X$ is a non-empty subset of $℘(X) ∖ { \emptyset }$ that is closed under finite intersections and upward closed in $X.$ Equivalently, it is a prefilter that is upward closed in $X.$ In words, a filter on $X$ is a family of sets over $X$ that (1) is not empty (or equivalently, it contains $X$ ), (2) is closed under finite intersections, (3) is upward closed in $X,$ and (4) does not have the empty set as an element.
Warning: Some authors, particularly algebrists, use "filter" to mean a dual ideal; others, particularly topologists, use "filter" to mean a proper dual ideal.[19] It is recommended that readers always check how "filter" is defined when reading mathematical literature. This article uses Henri Cartan's original definition of filter, which required propriety.

${\mathcal {B}}$ is a filter on $X$ if and only if its dual $X ∖ ℬ$ is an ideal that does not contain $X$ as an element. If ${\mathcal {B}}$ is an ideal on $X$ that satisfies $X \notin ℬ$ then $X ∖ ℬ$ is called its dual filter on $X.$

Prefilter or filter base[8][20] if ${\mathcal {B}}\neq \varnothing$ is proper and directed downward. Equivalently, ${\mathcal {B}}$ is a prefilter if its upward closure ${\mathcal {B}}^{\uparrow X}$ is a filter. It can also be defined as any family that is equivalent (with respect to $\leq$ ) to some filter.[9] A proper family ${\mathcal {B}}\neq \varnothing$ is a prefilter if and only if $ℬ (\cap) ℬ \leq ℬ$ .[9]
If ${\mathcal {B}}$ is a prefilter then its upward closure ${\mathcal {B}}^{\uparrow X}$ is the unique smallest (relative to $\subseteq$ ) filter on $X$ containing ${\mathcal {B}}$ and it is called the filter generated by ${\mathcal {B}}.$ A filter ${\mathcal {F}}$ is said to be generated by a prefilter ${\mathcal {B}}$ if $ℱ = ℬ ↑ X$ , in which ${\mathcal {B}}$ is called a filter base for ${\mathcal {F}}$ .

Unlike a filter, a prefilter is not necessarily closed under finite intersections.

$π$ –system if ${\mathcal {B}}\neq \varnothing$ is closed under finite intersections. Every non–empty family ${\mathcal {B}}$ is contained in a unique smallest $π$ –system called the $π$ –system generated by ${\mathcal {B}},$ which is sometimes denoted by $π (ℬ)$ . It is equal to the intersection of all $π$ –systems containing ${\mathcal {B}}$ and also to the set of all possible finite intersections of sets from ${\mathcal {B}}$ :
$π$ (ℬ ) = ${B 1 \cap \cdot\cdot\cdot \cap B n : n \geq 1 and B 1, ..., B n \in ℬ$ }.

A $π$ –system is a prefilter if and only if it is proper. Every filter is a proper $π$ –system and every proper $π$ –system is a prefilter but the converses do not hold in general.

A prefilter is equivalent (with respect to $\leq$ ) to the $π$ –system generated by it and both of these families generate the same filter on $X.$

Filter subbase[8][21] and centered[9] if ${\mathcal {B}}\neq \varnothing$ and ${\mathcal {B}}$ satisfies any of the following equivalent conditions:

${\mathcal {B}}$ has the finite intersection property, which means that intersection of any finite family of (one or more) sets in ${\mathcal {B}}$ is not empty; explicitly, this means that whenever $n \geq 1$ and $B 1, ..., B n \in ℬ$ then $\varnothing \neq B_{1}\cap \cdots \cap B_{n}.$

The $π$ –system generated by ${\mathcal {B}}$ is proper (i.e. $\varnothing$ is not an element).

The $π$ –system generated by ${\mathcal {B}}$ is a prefilter.

${\mathcal {B}}$ is a subset of some prefilter.

${\mathcal {B}}$ is a subset of some filter.

The filter generated by ${\mathcal {B}}$ is the unique smallest (relative to $\subseteq$ ) filter $ℱ ℬ$ on $X$ containing ${\mathcal {B}}.$ It is equal to the intersection of all filters on $X$ that have ${\mathcal {B}}$ as a subset. The $π$ –system generated by ${\mathcal {B}},$ denoted by $π (ℬ)$ , will be a prefilter and a subset of $ℱ ℬ$ . Moreover, the filter generated by ${\mathcal {B}}$ is the upward closure of $π (ℬ)$ , meaning $π (ℬ) ↑ X = ℱ ℬ$ .[9]

A smallest (relative to $\subseteq$ ) prefilter containing a filter subbase ${\mathcal {B}}$ will exist only under certain circumstances. For example, (1) ${\mathcal {B}}$ is a prefilter, or (2) the filter (or equivalently, the $π$ –system) generated by ${\mathcal {B}}$ is principle, in which case $ℬ \cup { ker ℬ }$ is the unique smallest prefilter containing ${\mathcal {B}}.$ Otherwise, in general, a $\subseteq$ –smallest prefilter containing ${\mathcal {B}}$ may not exist. For this reason, some authors may refer to the $π$ –system generated by ${\mathcal {B}}$ as the prefilter generated by ${\mathcal {B}}$ . However, as shown in an example below, if such a $\subseteq$ –smallest prefilter does exist then it is not necessarily equal to the prefilter (i.e. $π$ –system) generated by $ℬ$ . So unfortunately, "the prefilter generated by" a prefilter ${\mathcal {B}}$ may not be ${\mathcal {B}},$ which is why this article will prefer the accurate and unambiguous terminology of "the $π$ –system generated by ${\mathcal {B}}$ ".

Subfilter of a filter ${\mathcal {F}}$ and that ${\mathcal {F}}$ is a superfilter of ${\mathcal {B}}$ [16][22] if ${\mathcal {B}}$ is a filter and $ℬ \subseteq ℱ$ where for filters, $ℬ \subseteq ℱ$ if and only if ${\mathcal {B}}\leq {\mathcal {F}}.$
Importantly, the expression "is a superfilter of" is for filters the analog of "is a subsequence of". So despite having the prefix "sub" in common, "is a subfilter of" is actually the reverse of "is a subsequence of."

However, ${\mathcal {B}}\leq {\mathcal {F}}$ can also be written $ℱ ⊢ ℬ$ which is described by saying " ${\mathcal {F}}$ is subordinate to ${\mathcal {B}}.$ " With this terminology, "is subordinate to" becomes for filters (and also for prefilters) the analog of "is a subsequence of,"[23] which makes this one situation where using the term "subordinate" and symbol $⊢$ may be helpful.

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 $X\neq \varnothing$ whenever this assumption is needed.

Basic examples

Named examples

The singleton set ${\mathcal {B}}=\{X\}$ is called the trivial or indiscrete filter on $X$ .[24][10] It is the unique minimal filter on $X$ because it is a subset of every filter on $X$ ; however, it need not be a subset of every prefilter on $X.$
If $(X,\tau )$ is a topological space and $x\in X,$ then the neighborhood filter ${\mathcal {N}}(x)$ at $x$ is a filter on $X.$ By definition, a family ${\mathcal {B}}$ of subsets of $X$ is called a neighborhood basis (resp. a neighborhood subbasis) at $x$ for $(X,\tau )$ if and only if ${\mathcal {B}}$ is a prefilter (resp. ${\mathcal {B}}$ is a filter subbase) and the filter on $X$ that ${\mathcal {B}}$ generates is equal to the neighborhood filter ${\mathcal {N}}(x).$ The subfamily $\tau (x)\subseteq {\mathcal {N}}(x)$ of open neighborhoods is a filter base for ${\mathcal {N}}(x).$ Both prefilters ${\mathcal {N}}(x)$ 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 $S\subseteq X.$
${\mathcal {B}}$ is an elementary prefilter[25] if ${\mathcal {B}}=\operatorname {Tails} \left(x_{\bullet }\right)$ for some sequence $x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }$ in $X.$
${\mathcal {B}}$ is an elementary filter on $X$ [26] if ${\mathcal {B}}$ 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 not an ultrafilter.[27]
The set ${\mathcal {F}}$ of all cofinite subsets of $X$ (meaning those sets whose complement in $X$ is finite) is proper if and only if ${\mathcal {F}}$ is infinite (or equivalently, $X$ is infinite), in which case ${\mathcal {F}}$ is a filter on $X$ known as the Fréchet or cofinite filter on $X.$ [10][24] If $X$ is finite then ${\mathcal {F}}$ is equal to the dual ideal $℘(X)$ , which is not a filter. If $X$ is infinite then the family $\left\{X\setminus \{x\}~:~x\in X\right\}$ 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 $\left\{X\setminus \{x\}~:~x\in X\right\},$ the kernel of the Fréchet filter on $X$ is the empty set: $\operatorname {ker} {\mathcal {F}}=\varnothing$ .
The intersection of any non–empty set $\mathbb {F}$ $\text{[math]}$ of filters on $X$ $\text{[math]}$ is itself a filter on $X$ $\text{[math]}$ called the infimum or greatest lower bound of $\mathbb {F}$ $\text{[math]}$ in $\operatorname {Filters} (X)$ $\text{[math]}$ . Since every filter on $X$ $\text{[math]}$ has $\{X\}$ $\text{[math]}$ as a subset, this intersection is never empty. By definition, the infimum is the finest/largest (relative to $\subseteq$ $\text{[math]}$ and $\leq$ $\text{[math]}$ ) filter contained as a subset of each member of $\mathbb {F}$ $\text{[math]}$ .[10]
- If ${\mathcal {B}}$ and ${\mathcal {F}}$ are filters then their infimum in $\operatorname {Filters} (X)$ is the filter ${\mathcal {B}}(\cup ){\mathcal {F}}.$ [9] If ${\mathcal {B}}$ and ${\mathcal {F}}$ are prefilters then ${\mathcal {B}}(\cup ){\mathcal {F}}$ is a prefilter and one of the finest (with respect to $\leq$ ) prefilters coarser (with respect to $\leq$ ) than both ${\mathcal {B}}$ and ${\mathcal {F}}$ ; that is, if ${\mathcal {S}}$ is a prefilter such that ${\mathcal {S}}\leq {\mathcal {B}}$ and ${\mathcal {S}}\leq {\mathcal {F}}$ then ${\mathcal {B}}(\cup ){\mathcal {F}}.$ [9] More generally, if ${\mathcal {B}}$ and ${\mathcal {F}}$ are non-empty families and if $𝕊 ≝ { 𝒮 \subseteq ℘(X) : 𝒮 \leq ℬ and 𝒮 \leq ℱ }$ then ${\mathcal {B}}(\cup ){\mathcal {F}}\in \mathbb {S}$ and ${\mathcal {B}}(\cup ){\mathcal {F}}$ is a greatest element (with respect to $\leq$ ) of $𝕊$ .[9]
Let $\mathbb {F}$ $\text{[math]}$ be a set of filters on $X$ $\text{[math]}$ and let ${\mathcal {B}}=\cup _{{\mathcal {F}}\in \mathbb {F} }{\mathcal {F}}.$ $\text{[math]}$ If ${\mathcal {B}}$ $\text{[math]}$ is a filter subbase then the filter on $X$ $\text{[math]}$ generated by ${\mathcal {B}}$ $\text{[math]}$ is the supremum or least upper bound of $\mathbb {F}$ $\text{[math]}$ in $\operatorname {Filters} (X)$ $\text{[math]}$ .[10] By definition, the supremum, if it exists, is the smallest (relative to $\subseteq$ $\text{[math]}$ ) filter containing each member of $\mathbb {F}$ $\text{[math]}$ as a subset. If ${\mathcal {B}}$ $\text{[math]}$ is not a filter subbase, then the supremum of $\mathbb {F}$ $\text{[math]}$ in $\operatorname {Filters} (X)$ $\text{[math]}$ (and also in $\operatorname {Prefilters} (X)$ $\text{[math]}$ ) does not exist.
- If ${\mathcal {B}}$ and ${\mathcal {F}}$ are prefilters (resp. filters on $X$ ) then $ℬ (\cap) ℱ$ is a prefilter (resp. a filter) if and only if it is proper (or said differently, if and only if ${\mathcal {B}}$ and ${\mathcal {F}}$ mesh), in which case it is one of the coarsest (with respect to $\leq$ ) prefilters (resp. the $\leq$ -coarsest filters) that is finer (with respect to $\leq$ ) than both ${\mathcal {B}}$ and ${\mathcal {F}}$ ; that is, if ${\mathcal {S}}$ is a prefilter (resp. filter) such that ${\mathcal {B}}\leq {\mathcal {S}}$ and ${\mathcal {F}}\leq {\mathcal {S}}$ then ${\mathcal {B}}\cap {\mathcal {F}}\leq {\mathcal {S}}.$ [9]
Let $I$ and $X$ be non-empty sets and for every $i\in I$ let ${\mathcal {D}}_{i}$ be a dual ideal on $X.$ If ${\mathcal {I}}$ is any dual ideal on $I$ then $\bigcup _{\Xi \in {\mathcal {I}}}\;\;\bigcap _{i\in \Xi }\;{\mathcal {D}}_{i}$ is a dual ideal on $X$ called Kowalsky's dual ideal or Kowalsky's filter.[16]

Other examples

Let $X=\{p,1,2,3\}$ and let $ℬ = { {p}, {p, 1, 2 }, {p, 1, 3 }$ }, which makes ${\mathcal {B}}$ a prefilter and a filter subbase that is not closed under finite intersections. Because ${\mathcal {B}}$ is a prefilter, the smallest prefilter containing ${\mathcal {B}}$ is ${\mathcal {B}}.$ The $π$ –system generated by ${\mathcal {B}}$ is ${ {p, 1 } } \cup ℬ$ . In particular, the smallest prefilter containing the filter subbase ${\mathcal {B}}$ is not equal to the set of all finite intersections of sets in ${\mathcal {B}}.$ The filter on $X$ generated by ${\mathcal {B}}$ is $ℬ ↑ X = {S \subseteq X : p \in S} = { {p} \cup T : T \subseteq { 1, 2, 3 }$ }. All three of ${\mathcal {B}},$ the $π$ –system ${\mathcal {B}}$ generates, and ${\mathcal {B}}^{\uparrow X}$ are examples of fixed, principal, ultra prefilters that are principal at the point $p$ ; ${\mathcal {B}}^{\uparrow X}$ is also an ultrafilter on $X.$
A prefilter ${\mathcal {B}}$ on a topological space $X$ is finer than the prefilter ${ cl' X B : B \in ℬ$ }.[28]
The set ${\mathcal {B}}$ of all dense open subsets of a (non–empty) topological space $X$ is a proper $π$ –system and so also a prefilter. If $X=\mathbb {R} ^{n}$ (with $1 \leq n \in ℕ$ ), then the set $ℬ LebFinite$ of all $B\in {\mathcal {B}}$ such that $B$ has finite Lebesgue measure is a proper $π$ –system and prefilter that is also a proper subset of ${\mathcal {B}}.$ The prefilters $ℬ LebFinite$ and ${\mathcal {B}}$ generate the same filter on $X.$
This example illustrates a class of a filter subbases $𝒮 R$ where all sets in both $𝒮 R$ and its generated $π$ -system can be described as sets of the form $B_{r,s},$ so that in particular, no more than two variables (i.e. $r$ and $s$ ) are needed to describe the generated $π$ -system. However, this is not typical and in general, this should not be expected of a filter subbase ${\mathcal {S}}$ that is not a $π$ -system. More often, an intersection $S_{1}\cap \cdots S_{n}$ of $n$ sets from ${\mathcal {S}}$ will usually require a description involving $n$ variables that cannot be reduced down to only two (consider, for instance, if $𝒮 R$ was instead $\left\{(0,r)\cup (r,\infty )~:~r\in R\right\}$ ). For all $r,s\in \mathbb {R} ,$ , let $B_{r,s}=(r,0)\cup (s,\infty ),$ where $B_{r,s}=B_{\min(r,s),s}$ so no generality is lost by adding the assumption $r \leq s$ . For all real $r\leq s$ and $u\leq v,$ if $s\geq 0$ or $v\geq 0$ then $B_{-r,s}\cap B_{-u,v}=B_{-\min(r,u),\max(s,v)}.$ [note 5] For every $R \subseteq ℝ$ , let $𝒮 R = {B - r, r : r \in R}$ and let $ℬ R = {B - r, s : r \leq s with r, s \in R$ }. [note 6] Let $X=\mathbb {R}$ and suppose $\varnothing \neq R\subseteq (0,\infty )$ is not a singleton set. Then $𝒮 R$ is a filter subbase but not a prefilter and $ℬ R$ is the $π$ -system it generates, so that $ℬ R ↑ X$ is the unique smallest filter in $X=\mathbb {R}$ containing $𝒮 R$ . However, $𝒮 R ↑ X$ is not a filter on $X$ (nor is it a prefilter because it is not directed downward, although it is a filter subbase) and $𝒮 R ↑ X$ is a proper subset of the filter $ℬ R ↑ X$ . If $R,S\subseteq (0,\infty )$ are non-empty intervals then the filter subbases $𝒮 R$ and $𝒮 S$ generate the same filter on $X$ if and only if $R=S.$ If ${\mathcal {C}}$ is a family such that $𝒮 (0, \infty) \subseteq 𝒞 \subseteq ℬ (0, \infty)$ then ${\mathcal {C}}$ is a prefilter if and only if for all real $0<r\leq s$ there exist real $0<u\leq v$ such that $u\leq r\leq s\leq v$ and $B - u, v \in 𝒞$ . If ${\mathcal {C}}$ is such a prefilter then for any $C \in 𝒞 ∖ 𝒮 (0, \infty)$ , the family $𝒞 ∖ {C}$ is also a prefilter satisfying $𝒮 (0, \infty) \subseteq 𝒞 ∖ {C} \subseteq ℬ (0, \infty)$ . This shows that there cannot exist a minimal (with respect to $\subseteq$ ) prefilter that both contains $𝒮 (0, \infty)$ and is a subset of the $π$ -system generated by $𝒮 (0, \infty)$ . This remains true even if the requirement that the prefilter be a subset of $ℬ (0, \infty)$ is removed.

Ultrafilters

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

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

Ultra[8][29] if $\varnothing \not \in {\mathcal {B}}$ and any of the following equivalent conditions are satisfied:

For every set $S\subseteq X$ there exists some set $B\in {\mathcal {B}}$ such that $B \subseteq S$ or $B \subseteq X ∖ S$ (or equivalently, such that $B\cap S$ equals $B$ or $\varnothing$ ).

For every set $S \subseteq \cup B \in ℬ B$ there exists some set $B\in {\mathcal {B}}$ such that $B\cap S$ equals $B$ or $\varnothing .$
This characterization of " ${\mathcal {B}}$ is ultra" does not depend on the set $X,$ so mentioning the set $X$ is optional when using the term "ultra."

For every set $S$ (not necessarily even a subset of $X$ ) there exists some set $B\in {\mathcal {B}}$ such that $B\cap S$ equals $B$ or $\varnothing .$
If ${\mathcal {B}}$ satisfies this condition then so does every superset $ℱ \supseteq ℬ$ . In particular, a set ${\mathcal {F}}$ is ultra if and only if $\varnothing \not \in {\mathcal {F}}$ and ${\mathcal {F}}$ contains as a subset some ultra family of sets.

Ultra prefilter[8][29] if it is a prefilter that is also ultra. Equivalently, it is a filter subbase that is ultra.
A filter subbase that is ultra is necessarily a prefilter.[proof 1]

Ultrafilter on $X$ [8][29] if it is a filter on $X$ that is ultra. Equivalently, an ultrafilter on $X$ is a filter ${\mathcal {B}}$ on $X$ that satisfies any of the following equivalent conditions:

${\mathcal {B}}$ is generated by an ultra prefilter;

For any $S\subseteq X,$ $S\in {\mathcal {B}}$ or $X ∖ S \in ℬ$ .[16]

$ℬ \cup (X ∖ ℬ) = ℘(X)$ . This condition can be restated as: $℘(X)$ is partitioned by ${\mathcal {B}}$ and its dual $X ∖ ℬ$ .
The sets ${\mathcal {B}}$ and $X ∖ ℬ$ are disjoint whenever ${\mathcal {B}}$ is a prefilter.

For any $R,S\subseteq X,$ if $R \cup S \in ℬ$ then $R\in {\mathcal {B}}$ or $S\in {\mathcal {B}}$ (a filter with this property is called a prime filter).
This property extends to any finite union of two or more sets.

For any $R,S\subseteq X,$ if $R\cup S=X$ then $R\in {\mathcal {B}}$ or $S\in {\mathcal {B}}.$

For any $R,S\subseteq X,$ if $R \cup S \in ℬ$ and $R\cap S=\varnothing$ then either $R\in {\mathcal {B}}$ or $S\in {\mathcal {B}}.$

${\mathcal {B}}$ is a maximal filter on $X$ ; meaning that if ${\mathcal {F}}$ is a filter on $X$ such that $ℬ \subseteq ℱ$ then $ℬ = ℱ$ .
An ultra prefilter has a similar characterization in terms of maximality with respect to $\leq$ , where in the special case of filters, ${\mathcal {B}}\leq {\mathcal {F}}$ if and only if $ℬ \subseteq ℱ$ .

Because $\geq$ is for filters the analog of "is a subnet of," (specifically, "subnet" should mean "AA-subnet," which is defined below) an ultrafilter can be interpreted as being analogous to some sort of "maximally deep net." This idea is actually made rigorous by ultranets.

The ultrafilter lemma/principle/theorem[10] (Tarski (1930)[30]) — Every filter on a set $X$ is a subset of some ultrafilter on $X.$

A consequence of the ultrafilter lemma is that every filter is equal to the intersection of all ultrafilters containing it.[10]

Examples

Any family that has a singleton set as an element is ultra, in which case it will then be an ultra prefilter if and only if it also has the finite intersection property.
The trivial filter $\{X\}$ on $X$ is ultra if and only if $X$ is a singleton set.
A family ${\mathcal {B}}\subseteq \wp (X)$ of sets is an ultra prefilter if and only if it is equivalent (with respect to $\leq$ ) to some ultrafilter on $X,$ in which case this ultrafilter is necessarily equal to the upward closure ${\mathcal {B}}^{\uparrow X}.$ Consequently, a family ${\mathcal {B}}\subseteq \wp (X)$ of sets is an ultra prefilter if and only if ${\mathcal {B}}^{\uparrow X}$ is an ultrafilter on $X.$

Free, principal, and kernels

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

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

If ${\mathcal {B}}\subseteq \wp (X)$ then for any point $x,$ $x\not \in \operatorname {ker} {\mathcal {B}}$ if and only if $X\setminus \{x\}\in {\mathcal {B}}^{\uparrow X}.$

Properties of kernels

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

(1)

{\mathcal {B}},

(2) the

π

–system generated by

{\mathcal {B}},

and (3) the filter generated by

{\mathcal {B}}.

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

Classifying families of sets by their kernels

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

Free[7] if $ker ℬ = \emptyset$ , or equivalently, if ${X ∖ {x} : x \in X} \subseteq ℬ ↑ X$ ; this can be restated as ${X ∖ {x} : x \in X} \leq ℬ$ .
A filter ${\mathcal {F}}$ on $X$ is free if and only if $X$ is infinite and ${\mathcal {F}}$ contains the Fréchet filter on $X$ as a subset.

Fixed if $ker ℬ \neq \emptyset$ in which case, ${\mathcal {B}}$ is said to be fixed by any point $x \in ker ℬ$ .
Any fixed family is necessarily a filter subbase.

Principal[7] if $ker ℬ \in ℬ$ .
A proper principal family of sets is necessarily a prefilter.

Discrete or Principal at $x\in X$ [24] if ${x} = ker ℬ \in ℬ$ .
The principal filter at $x$ on $X$ is the filter ${x} ↑ X$ . A filter ${\mathcal {F}}$ is principal at $x$ if and only if $ℱ = {x} ↑ X$ .

Family of examples: For any non–empty $C \subseteq ℝ$ , the family $ℬ C = { ℝ ∖ (r + C) : r \in ℝ }$ is free but it is a filter subbase if and only if no finite union of the form $(r 1 + C) \cup \cdot\cdot\cdot \cup (r n + C)$ 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 (e.g. $C = ℚ$ , $ℤ$ , 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 subbasis for the Fréchet filter on $ℝ$ .

Characterizations of fixed ultra prefilters

If a family of sets ${\mathcal {B}}$ is fixed (i.e. $ker ℬ \neq \emptyset$ ) then ${\mathcal {B}}$ is ultra if and only if some element of ${\mathcal {B}}$ is a singleton set, in which case ${\mathcal {B}}$ will necessarily be a prefilter. Every principal prefilter is fixed, so a principal prefilter ${\mathcal {B}}$ is ultra if and only if $\operatorname {ker} {\mathcal {B}}$ 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.[7]

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

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

${\mathcal {F}}$ is fixed, or equivalently, not free, meaning $ker ℱ \neq \emptyset$ .
${\mathcal {F}}$ is principle, meaning $ker ℱ \in ℱ$ .
Some element of ${\mathcal {F}}$ is a finite set.
Some element of ${\mathcal {F}}$ is a singleton set.
${\mathcal {F}}$ is principle at some point of $X,$ which means $ker ℱ = {x} \in ℱ$ for $x\in X.$
${\mathcal {F}}$ does not contain the Fréchet filter on $X.$

Finite prefilters and finite sets

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

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

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

Finer/coarser, subordination, and meshing

Throughout ${\mathcal {B}},$ ${\mathcal {C}},$ and ${\mathcal {F}}$ will be any subsets of $℘(X)$ .

The preorder $\leq$ that is defined below is of fundamental importance for the use of prefilters (and filters) in topology. For instance, this preorder is used to define the prefilter equivalent of "subsequence",[23] where " ${\mathcal {F}}\geq {\mathcal {C}}$ " can be interpreted as " ${\mathcal {F}}$ is a subsequence of ${\mathcal {C}}$ " (so "subordinate to" is the prefilter equivalent of "subsequence of"). It is also be used to define prefilter convergence in a topological space. The definition of ${\mathcal {B}}$ meshes with ${\mathcal {C}},$ which is closely related to the preorder $\leq$ , is used in Topology to define cluster points.

Two families of sets ${\mathcal {B}}$ and ${\mathcal {C}}$ mesh[8] and are compatible, indicated by writing $ℬ # 𝒞$ , if $B\cap C\neq \varnothing$ for all $B\in {\mathcal {B}}$ and $C\in {\mathcal {C}}$ . If ${\mathcal {B}}$ and ${\mathcal {C}}$ do not mesh then they are dissociated. If $S$ is a set (but not necessarily a family of sets) then ${\mathcal {B}}$ and $S$ are said to mesh if ${\mathcal {B}}$ and $\{S\}$ mesh, or equivalently, if the trace $ℬ | S = {B \cap S : B \in ℬ }$ of ${\mathcal {B}}$ on $S$ does not contain the empty set.

Declare that ${\mathcal {C}}\leq {\mathcal {F}},$ ${\mathcal {F}}\geq {\mathcal {C}},$ and $ℱ ⊢ 𝒞$ , stated as ${\mathcal {C}}$ is coarser than ${\mathcal {F}}$ and ${\mathcal {F}}$ is finer than (or subordinate to) ${\mathcal {C}},$ [10][11][12][9] if any of the following equivalent conditions hold:

Definition: Every $C\in {\mathcal {C}}$ contains some $F\in {\mathcal {F}}.$ Explicitly, this means that for every $C\in {\mathcal {C}},$ there is some $F\in {\mathcal {F}}$ such that $F\subseteq C.$
Said more briefly, ${\mathcal {C}}\leq {\mathcal {F}}$ if every set in ${\mathcal {C}}$ is larger than some set in ${\mathcal {F}}.$ Here, a "larger set" means a superset.

$\{C\}\leq {\mathcal {F}}$ for every $C\in {\mathcal {C}}.$
In words, $\{C\}\leq {\mathcal {F}}$ states exactly that $C$ is larger than some set in ${\mathcal {F}}.$ The equivalence of (a) and (b) follows immediately.

From this characterization, it follows that if $\left({\mathcal {C}}_{i}\right)_{i\in I}$ are families of sets, then $\cup i \in I 𝒞 i \leq ℱ$ if and only if $𝒞 i \leq ℱ$ for all $i\in I.$

${\mathcal {C}}\leq {\mathcal {F}}^{\uparrow X},$ which is equivalent to ${\mathcal {C}}\subseteq {\mathcal {F}}^{\uparrow X}$ ;

${\mathcal {C}}^{\uparrow X}\leq {\mathcal {F}}$ ;

${\mathcal {C}}^{\uparrow X}\leq {\mathcal {F}}^{\uparrow X},$ which is equivalent to ${\mathcal {C}}^{\uparrow X}\subseteq {\mathcal {F}}^{\uparrow X}$ ;

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

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

If an upward closed family ${\mathcal {F}}$ is finer than ${\mathcal {C}}$ (i.e. ${\mathcal {C}}\leq {\mathcal {F}}$ ) but ${\mathcal {C}}\neq {\mathcal {F}}$ then ${\mathcal {F}}$ is said to be strictly finer than ${\mathcal {C}}$ and ${\mathcal {C}}$ is strictly coarser than ${\mathcal {F}}.$ Two families ${\mathcal {C}}$ and ${\mathcal {F}}$ are comparable if one of these sets is finer than the other.[10]

Proof

Throughout this proof, "set" will mean "subset of $X$ " unless indicated otherwise. A "larger set" means a superset. This proof is written with the aim of making the proof of each implication as intuitively clear as possible. For this reason, it is written in a more conversational style and it also tries to limit assigning symbols to sets in ${\mathcal {F}}.$ Because of characterization (b), it would not be beneficial to attempt this with sets in ${\mathcal {C}}.$

Statement (a) defines ${\mathcal {C}}\leq {\mathcal {F}},$ where by definition, ${\mathcal {C}}\leq {\mathcal {F}}$ if and only if

every set in

{\mathcal {C}}

is larger than some set in

{\mathcal {F}}.

(def)

If $C$ is a set then $\{C\}\leq {\mathcal {F}}$ if and only if $C$ is larger than some set in ${\mathcal {F}}.$ The equivalence of (b) and (def) follows immediately. The corollaries of part (b) given in the proposition's statement now hold and will be used later.

If (def) is true, then it will remain true if ${\mathcal {C}}$ is replaced by a smaller sub-family. For this reason, ${\mathcal {C}}^{\uparrow X}\leq {\mathcal {F}}$ implies ${\mathcal {C}}\leq {\mathcal {F}},$ which is exactly (d) ⇒ (def). Similarly, (e) ⇒ (c).

If (def) is true, then it will remain true if ${\mathcal {F}}$ is enlarged. For this reason, ${\mathcal {C}}\leq {\mathcal {F}}$ implies ${\mathcal {C}}\leq {\mathcal {F}}^{\uparrow X},$ which is exactly (def) ⇒ (c). Similarly, (d) ⇒ (e).

If $C$ is a set that is larger than some set in ${\mathcal {F}}$ then so is every superset of $C.$ By definition, $\{C\}^{\uparrow X}$ consists exactly of all supersets of $C.$ For this reason, by using the corollary of (b), we conclude that $\{C\}\leq {\mathcal {F}}$ implies $\{C\}^{\uparrow X}\leq {\mathcal {F}}.$ Consequently, if (def) holds then $\{C\}^{\uparrow X}\leq {\mathcal {F}}$ for every $C\in {\mathcal {C}}$ so by taking the union of these families $\{C\}^{\uparrow X}$ as $C$ ranges over ${\mathcal {C}},$ the corollary of (b) gives

{\mathcal {C}}^{\uparrow X}=\bigcup _{C\in {\mathcal {C}}}\{C\}^{\uparrow X}\leq {\mathcal {F}}

This proves that (def) ⇒ (d). Applying (def) ⇒ (d) with ${\mathcal {F}}^{\uparrow X}$ in place of ${\mathcal {F}}$ proves (c) ⇒ (e). We have so far established that (d) ⇔ (a) ⇔ (b) and (c) ⇔ (e) as well as (a) ⇒ (c). It remains to show (c) ⇒ (a) and to justify why $\leq$ can be replaced with $\subseteq$ in statements (c) and (e).

By definition, the upward closure ${\mathcal {F}}^{\uparrow X}$ consists of all sets larger than some set in ${\mathcal {F}}.$ Said differently, if $C$ is a set then

C \in ℱ ↑ X

⇔

C

is larger than some set in

{\mathcal {F}}.

(↑X def)

Restricting $C$ to range over ${\mathcal {C}},$ it follows from (↑X def) that $𝒞 \subseteq ℱ ↑ X$ ⇔ (def) holds, where the left hand side of this equivalence is statement (c). We have just shown that

𝒞 \subseteq ℱ ↑ X

\Leftrightarrow

{\mathcal {C}}\leq {\mathcal {F}}

.

By definition, ${\mathcal {F}}$ is upward closed if and only if $ℱ = ℱ ↑ X$ , in which case the above equivalence becomes: (f) ⇔ (a). In particular, because the family ${\mathcal {F}}^{\uparrow X}$ is always upward closed, this immediately gives:

𝒞 \subseteq ℱ ↑ X

\Leftrightarrow

𝒞 \leq ℱ ↑ X

, and also that

{\mathcal {C}}^{\uparrow X}\subseteq {\mathcal {F}}^{\uparrow X}

\Leftrightarrow

𝒞 ↑ X \leq ℱ ↑ X

It remains to show (c) ⇒ (a). By (↑X def), if a set $S$ is in ${\mathcal {F}}^{\uparrow X}$ then $S$ is larger than some set in ${\mathcal {F}}.$ So in particular, if some set $C$ is larger than some set in ${\mathcal {F}}^{\uparrow X}$ (call it $S$ ) then $C$ will necessarily be larger than some set in ${\mathcal {F}}.$ In short, ${C} \leq ℱ ↑ X$ implies ${C} \leq ℱ$ . The corollary of (b) allows us to conclude that $𝒞 \leq ℱ ↑ X$ implies ${\mathcal {C}}\leq {\mathcal {F}},$ which is (c) ⇒ (a). ∎

Assume that ${\mathcal {C}}$ and ${\mathcal {F}}$ are families of sets that satisfy ${\mathcal {C}}\leq {\mathcal {F}}.$ Then $ker ℱ \subseteq ker 𝒞$ , and ${\mathcal {C}}\neq \varnothing$ implies ${\mathcal {F}}\neq \varnothing ,$ and also $\varnothing \in {\mathcal {C}}$ implies $\varnothing \in {\mathcal {F}}.$ If in addition to ${\mathcal {C}}\leq {\mathcal {F}},$ ${\mathcal {F}}$ is a filter subbase and ${\mathcal {C}}\neq \varnothing ,$ then ${\mathcal {C}}$ is a filter subbase[9] and also ${\mathcal {C}}$ and ${\mathcal {F}}$ mesh.[18][proof 2] Every filter subbase is coarser than both the $π$ -system that it generates and the filter that it generates.[9]

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

Relational properties of subordination

The comparison/subordination relation $\leq$ is reflexive and transitive, which makes it into a preorder on $℘(℘(X))$ .[31]

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

Antisymmetry: If $ℬ \subseteq 𝒞$ then ${\mathcal {B}}\leq {\mathcal {C}}$ but while the converse does not hold in general, it does hold if ${\mathcal {C}}$ is upward closed (such as if ${\mathcal {C}}$ is a filter). Two filters are equivalent if and only if they are equal, which makes the restriction of $\leq$ to $\operatorname {Filters} (X)$ antisymmetric. But in general, $\leq$ is not antisymmetric on $\operatorname {Prefilters} (X)$ nor on $℘(℘(X))$ ; that is, $𝒞 \leq ℬ$ and ${\mathcal {B}}\leq {\mathcal {C}}$ does not necessarily imply $ℬ = 𝒞$ ; not even if both ${\mathcal {C}}$ and ${\mathcal {B}}$ are prefilters.[12] For instance, if ${\mathcal {B}}$ is a prefilter but not a filter then $ℬ \leq ℬ ↑ X$ and $ℬ ↑ X \leq ℬ$ but $ℬ \neq ℬ ↑ X$ .

Equivalent families of sets

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

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

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

Properties preserved between equivalent families

Let $ℬ, 𝒞 \in ℘(℘(X))$ be arbitrary and let ${\mathcal {F}}$ be any family of sets. If ${\mathcal {B}}$ and ${\mathcal {C}}$ are equivalent (which implies that $ker ℬ = ker 𝒞$ ) then for each of the statements/properties listed below, either it is true of both ${\mathcal {B}}$ and ${\mathcal {C}}$ or else it is false of both ${\mathcal {B}}$ and ${\mathcal {C}}$ :[31]

Not empty
Proper
- Moreover, any two degenerate families are necessarily equivalent.
Filter subbase
Prefilter
- In which case ${\mathcal {B}}$ and ${\mathcal {C}}$ generate the same filter on $X$ (i.e. their upward closures in $X$ are equal).
Free/Fixed
Principal
Ultra
Is equal to the trivial filter $\{X\}$ $\text{[math]}$
- In words, this means that the only subset of $℘(X)$ that is equivalent to the trivial filter is the trivial filter. In general, this conclusion of equality does not extend to non-trivial filters.
Meshes with ${\mathcal {F}}$
Is finer than ${\mathcal {F}}$
Is coarser than ${\mathcal {F}}$
Is equivalent to ${\mathcal {F}}$

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

Equivalence of prefilters and filter subbases

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

${\mathcal {B}}$ ;
the $π$ –system generated by ${\mathcal {B}}$ ;
the filter on $X$ generated by ${\mathcal {B}}$ ;

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.[9][proof 3] Every prefilter is equivalent to exactly one filter on $X,$ which is the filter that it generates (i.e. the prefilter's upward closure). Said differently, every equivalence class of prefilters contains exactly one representative that is a filter. In this way, filters can be considered as just being distinguished elements of these equivalence classes of prefilters.[9]

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

Examples of determining equivalence/non–equivalence

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

ℬ = { [e, \infty) : e \in E}

and

𝒞 open = { (-\infty, e) \cup (1 + e, \infty) : e \in E}

and

𝒞 closed = { (-\infty, e] \cup [1 + e, \infty) : e \in E}.

All three sets are filter subbases but none are filters on $X$ and only ${\mathcal {B}}$ is prefilter (in fact, ${\mathcal {B}}$ is even a free and closed under finite intersections). The set $𝒞 closed$ is fixed while $𝒞 open$ is free (unless $E = ℕ$ ). They satisfy $𝒞 closed \leq 𝒞 open \leq ℬ$ , but no two of these sets are equivalent; moreover, no two of the filters generated by these three filter subbases are equivalent/equal. This conclusion can be reached by showing that the $π$ –systems that they generate are not equivalent. Unlike with $𝒞 open$ , every set in the $π$ –systems generated by $𝒞 closed$ contains $ℤ$ as a subset,[note 7] which is what prevents their generated $π$ –systems (and hence their generated filters) from being equivalent. If $E$ was instead $ℚ$ or $ℝ$ then all three families would be free and although the sets $𝒞 closed$ and $𝒞 open$ would remain not equivalent to each other, their generated $π$ –systems would be equivalent and consequently, they would generate the same filter on $X$ ; however, this common filter would still be strictly coarser than the filter generated by ${\mathcal {B}}.$

Summary of filter limits and cluster points definitions

The application of (pre)filters to Topology has at its foundations the following basic and fundamental definitions, all of which can defined entirely in terms of the subordination relation. They are what make the relation $\leq$ of such fundamental importance in applying filters to Topology. In essence, the preorder $\leq$ is incapable of distinguishing between equivalent families. For example, two equivalent families ${\mathcal {B}}$ and ${\mathcal {C}}$ can be used interchangeably in the definition of "limit" given below because their equivalency guarantees that

{\mathcal {N}}\leq {\mathcal {B}}

if and only if

{\mathcal {N}}\leq {\mathcal {C}}

.

Throughout, $(X,\tau )$ is a topological space, ${\mathcal {B}}$ is a family of sets, and $𝒩 = 𝒩(x)$ is the neighborhood filter at a given point $x\in X.$

If

{\mathcal {N}}\leq {\mathcal {B}}

then

{\mathcal {B}}

is said to converge in

(X,\tau )

to

x

,[8] written "

{\mathcal {B}}\to x

in

X,

" and

x

is called a limit or limit point of

{\mathcal {B}}.

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

A point

x

is a cluster point or accumulation point of

{\mathcal {B}}

[8] if

{\mathcal {B}}

meshes with the neighborhood filter

{\mathcal {N}}

at

x

(i.e. if

ℬ # 𝒩

).

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

If

f:X\to Y

is a map a topological space

Y

and

y\in Y

then

y

is a limit point or limit (resp. a cluster point) of $f$ with respect to ${\mathcal {B}}$ [32] if

y

is a limit point (resp. a cluster point) of

f (ℬ)

in

Y.

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

Set theoretic properties, examples, and constructions involving prefilters

Supremum, trace, and meshing

Supremum and least upper bound

If $𝔽 \neq \emptyset$ is a set of filters on $X$ then the supremum or least upper bound of $\mathbb {F}$ in $\operatorname {Filters} (X)$ , if it exists, is the smallest filter on $X$ containing every element of $\mathbb {F}$ as a subset; that is, it is the smallest filter on $X$ containing $\cup ℱ \in 𝔽 ℱ$ as a subset. As with any non–empty family of sets, $\cup ℱ \in 𝔽 ℱ$ is contained in some filter if and only if it is a filter subbase (meaning it has the finite intersection property). If ${\mathcal {B}}$ denotes the $π$ –system generated by $\cup ℱ \in 𝔽 ℱ$ , which is the set

ℬ ≝ {F 1 \cap \cdot\cdot\cdot \cap F n : n \in ℕ and every F i belongs to some ℱ \in 𝔽 },

then there exists a filter on $X$ containing every $ℱ \in 𝔽$ as a subset if and only if $\varnothing \not \in {\mathcal {B}},$ in which case ${\mathcal {B}}$ is a prefilter and ${\mathcal {B}}^{\uparrow X}$ is the smallest filter on $X$ containing every $ℱ \in 𝔽$ as a subset; this makes the filter ${\mathcal {B}}^{\uparrow X}$ the supremum and the least upper bound of $\mathbb {F}$ in $\operatorname {Filters} (X)$ [10] and ${\mathcal {B}}^{\uparrow X}$ is equal to the intersection of all filters on $X$ containing $\cup ℱ \in 𝔽 ℱ$ .

The least upper bound of a family of filters $\mathbb {F}$ may fail to be a filter.[10] Indeed, if $X$ contains at least 2 distinct elements then there exist filters ${\mathcal {B}}$ and ${\mathcal {C}}$ on $X$ for which there does not exist a filter ${\mathcal {F}}$ on $X$ that contains both ${\mathcal {B}}$ and ${\mathcal {C}}.$

Trace and meshing

If ${\mathcal {B}}$ is a prefilter (resp. filter) on $X$ and $S\subseteq X$ then the trace of ${\mathcal {B}}$ on $S$ , which is the family $ℬ | S ≝ ℬ (\cap) {S}$ , is a prefilter (resp. a filter) if and only if ${\mathcal {B}}$ and $S$ mesh (i.e. $\emptyset \notin ℬ (\cap) {S}$ [10]), in which case the trace of ${\mathcal {B}}$ on $S$ is said to be induced by $S$ . If ${\mathcal {B}}$ is ultra and if ${\mathcal {B}}$ and $S$ mesh then the trace $ℬ | S$ is ultra. If ${\mathcal {B}}$ is an ultrafilter on $X$ then the trace of ${\mathcal {B}}$ on $S$ is a filter on $S$ if and only if $S\in {\mathcal {B}}.$

For example, suppose that ${\mathcal {B}}$ is a filter on $X$ and $S\subseteq X$ is such that $S \neq X$ and $X ∖ S \notin ℬ$ . Then ${\mathcal {B}}$ and $S$ mesh and $ℬ \cup {S}$ generates a filter on $X$ that is strictly finer than ${\mathcal {B}}.$ [10]

When prefilters mesh

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

ℬ (\cap) 𝒞 ≝ {B \cap C : B \in ℬ and C \in 𝒞 }

satisfies $𝒞 \leq ℬ (\cap) 𝒞$ and $ℬ \leq ℬ (\cap) 𝒞$ . If $ℬ (\cap) 𝒞$ is proper (resp. a prefilter, a filter subbase) then this is also true of both ${\mathcal {B}}$ and ${\mathcal {C}}.$ In order to make any meaningful deductions about $ℬ (\cap) 𝒞$ from ${\mathcal {B}}$ and ${\mathcal {C}},$ $ℬ (\cap) 𝒞$ needs to be proper (i.e. $\emptyset \notin ℬ (\cap) 𝒞$ ), 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 ${\mathcal {B}}$ and ${\mathcal {C}}.$ Said differently, if ${\mathcal {B}}$ and ${\mathcal {C}}$ 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 (i.e. $\leq$ ):

Two prefilters (resp. filter subbases)

{\mathcal {B}}

and

{\mathcal {C}}

mesh if and only if there exists a prefilter (resp. filter subbase)

ℬ (\cap) 𝒞

such that

𝒞 \leq ℬ (\cap) 𝒞

and

ℬ \leq ℬ (\cap) 𝒞

.

If the least upper bound of two filters ${\mathcal {B}}$ and ${\mathcal {C}}$ exists in $\operatorname {Filters} (X)$ then this least upper bound is equal to $ℬ (\cap) 𝒞$ .[27]

Products and other examples

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

\operatorname {Pr} _{X_{i}}:\prod X_{\bullet }\to X_{i}

denote the canonical projection. Let ${\mathcal {B}}_{\bullet }:=\left({\mathcal {B}}_{i}\right)_{i\in I}$ be non-empty families, also indexed by $I$ , such that ${\mathcal {B}}_{i}\subseteq \wp \left(X_{i}\right)$ for each $i\in I.$ The product of the families ${\mathcal {B}}_{\bullet }$ [10] is defined identically to how the basic open subsets of the product topology are defined (had all of these ${\mathcal {B}}_{i}$ been topologies). That is, both the notations

\prod _{}{\mathcal {B}}_{\bullet }=\prod _{i\in I}{\mathcal {B}}_{i}

denote the family of all 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 for any one of these finitely many $i$ that satisfy $S_{i}\neq X_{i},$ it is necessarily true that $S_{i}\in {\mathcal {B}}_{i}.$ This family is also equal to[10]

\prod _{}{\mathcal {B}}_{\bullet }=\bigcup _{i\in I}\operatorname {Pr} _{X_{i}}^{-1}\left({\mathcal {B}}_{i}\right).

If $\prod {\mathcal {B}}_{\bullet }$ is a filter subbase then the filter on $\prod X_{\bullet }$ that it generates is called the filter generated by ${\mathcal {B}}_{\bullet }$ .[10] If every ${\mathcal {B}}_{i}$ is a prefilter on $X_{i}$ then $\prod {\mathcal {B}}_{\bullet }$ will be a prefilter on $\prod X_{\bullet }$ and moreover, this prefilter is equal to the coarsest prefilter ${\mathcal {F}}$ on $\prod X_{\bullet }$ such that $\operatorname {Pr} _{X_{i}}\left({\mathcal {F}}\right)={\mathcal {B}}_{i}$ for every $i\in I.$ [10] However, $\prod {\mathcal {B}}_{\bullet }$ may fail to be a filter on $\prod X_{\bullet }$ even if every ${\mathcal {B}}_{i}$ is a filter on $X_{i}.$ [10]

Set subtracting away a subset of the kernel

If ${\mathcal {B}}$ is a prefilter on $X,$ $S \subseteq ker ℬ$ , and $S \notin ℬ$ then ${B ∖ S : B \in ℬ }$ is a prefilter, where this latter set is a filter if and only if ${\mathcal {B}}$ is a filter and $S=\varnothing .$ In particular, if ${\mathcal {B}}$ is a neighborhood basis at a point $x$ in a topological space $X$ having at least 2 points, then ${B ∖ {x} : B \in ℬ }$ is a prefilter on $X.$ This construction is used to define $\lim _{\stackrel {x\to x_{0}}{x\neq x_{0}}}f(x)\to y$ in terms of prefilter convergence.

Dual relation and downward closure

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

Using duality between ideals and dual ideals

Let $f:X\to Y$ be a map and suppose that $Ξ \subseteq ℘(Y)$ . Define

Ξ f ≝ {I \subseteq X : f (I) \in Ξ }

which contains the empty set if and only if $Ξ$ does. It is possible for $Ξ$ to be an ultrafilter and for $Ξ f$ to be empty or not closed under finite intersections (see footnote for example).[note 8] Although $Ξ f$ does not preserve properties of filters very well, if $Ξ$ is downward closed (resp. closed under finite unions, an ideal) then this will also be true for $Ξ f$ . Using the duality between ideals and dual ideals allows for a construction of the following filter.

Suppose

{\mathcal {B}}

is a filter on

Y

and let

Ξ ≝ Y ∖ ℬ

be its dual in

Y.

If

X \notin Ξ f

then

Ξ f

's dual

X ∖ Ξ f

will be a filter.

Other topological examples

Example: The set ${\mathcal {B}}$ of all dense open subsets of a topological space is a proper $π$ –system and a prefilter. If the space is a Baire space, then the set of all countable intersections of dense open subsets is a $π$ –system and a prefilter that is finer than ${\mathcal {B}}.$

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

Images and preimages of filters and prefilters

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

Images of prefilters

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

Explicitly, if one of the following properties is true of ${\mathcal {B}}$ on $Y,$ then it will necessarily also be true of $g (ℬ)$ on $g (Y)$ (although possibly not on the codomain $Z$ unless $g$ is surjective):[10][13][33][34][35][30]

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

Moreover, if ${\mathcal {B}}\subseteq \wp (Y)$ is a prefilter then so are both $g (ℬ)$ and $g -1 (g (ℬ))$ .[10] The image under a map $f:X\to Y$ of an ultra set ${\mathcal {B}}\subseteq \wp (X)$ is again ultra and if ${\mathcal {B}}$ is an ultra prefilter then so is $f (ℬ)$ .

If ${\mathcal {B}}$ 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.[33] 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 $g (ℬ)$ in $Z$ is

g (ℬ) ↑ Z = {S \subseteq Z : B \subseteq g -1 (S) for some B \in ℬ}

where if ${\mathcal {B}}$ is upward closed in $Y$ (i.e. a filter) then this simplifies to:

g (ℬ) ↑ Z = {S \subseteq Z : g -1 (S) \in ℬ

}.

If $S \subseteq Y$ and if $In : S \to Y$ denotes the natural inclusion then the trace of ${\mathcal {B}}$ on $S$ is equal to the preimage $In -1 (ℬ)$ .[10] This observation allows the results in this subsection to be applied to investigating the trace on a set. If $X \subseteq Y$ then taking $g$ to be the natural inclusion $X \to Y$ shows that any prefilter (resp. ultra prefilter, filter subbase) on $S$ is also a prefilter (resp. ultra prefilter, filter subbase) on $Y.$ [10]

Preimages of prefilters

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

f -1 (ℬ)

is a prefilter (resp. filter subbase,

π

–system, closed under finite unions, proper) if and only if this is true of

{\mathcal {B}}.

However, if ${\mathcal {B}}$ 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 $X$ [34] (see this[note 9] footnote for an example).

If $f:X\to Y$ is not surjective then denote the trace of ${\mathcal {B}}$ 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 (ℬ | f (X))

.

This 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 ${\mathcal {B}}$ and the surjection $f : X \to f (X)$ can be used in place of $f:X\to Y$ . For example:[13][10][35]

f -1 (ℬ)

is a prefilter (resp. filter subbase,

π

–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.

Even if ${\mathcal {B}}$ is an ultrafilter on $Z$ , if $f$ is not surjective then it is nevertheless possible that $\emptyset \in ℬ | f (X)$ , which would make $f -1 (ℬ)$ degenerate as well. The next characterization shows that degeneracy is the only obstacle. If ${\mathcal {B}}$ is a prefilter then the following are equivalent:[13][10][35]

$f -1 (ℬ)$ is a prefilter;
$ℬ | f (X)$ is a prefilter;
$\emptyset \notin ℬ | f (X)$ ;
${\mathcal {B}}$ meshes with ${f (X)$ }

and moreover, if $f -1 (ℬ)$ is a prefilter then so is $f (f -1 (ℬ))$ .[13][10]

Bijections, injections, and surjections

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

A map $g:Y\to Z$ is injective if and only if for all prefilters ${\mathcal {B}}$ on $X,$ ${\mathcal {B}}$ is equivalent to $g -1 (g (ℬ))$ .[27] The image of an ultra family of sets under an injection is again ultra.

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

Subordination is preserved by images and preimages

The relation $\leq$ is preserved under both images and preimages of families of sets.[10] This means that for any families ${\mathcal {C}}$ and ${\mathcal {F}},$

{\mathcal {C}}\leq {\mathcal {F}}

implies

g({\mathcal {C}})\leq g({\mathcal {F}})

[35] and

f^{-1}({\mathcal {C}})\leq f^{-1}({\mathcal {F}}).

[35]

and moreover, the following relations always hold for any family of sets ${\mathcal {C}}$ :

𝒞 \leq f (f -1 (𝒞))

[35]

where equality will hold if $f$ is surjective.[35] Moreover,

f -1 (𝒞) = f -1 (f (f -1 (𝒞)))

and

g (𝒞) = g (g -1 (g (𝒞)))

.

If $𝒞 \subseteq ℘(Y)$ then

g -1 (g (𝒞)) \leq 𝒞

[35]

where equality will hold if $g$ is injective.[35]

Limits, cluster points, and nets

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

A note on intuition

Suppose that ${\mathcal {F}}$ is a non–principal filter on an infinite set $X.$ ${\mathcal {F}}$ 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 proper subset of $F_{0}$ ; this may be continued ad infinitum to get a sequence $F 0 \supset F 1 \supset F 2 \supset \cdot\cdot\cdot$ of sets in ${\mathcal {F}}$ with each $F_{i+1}$ being a proper subset of $F_{i}.$ The same is not true going "upward", for if $F 0 = X \in ℱ$ then there is no set in ${\mathcal {F}}$ 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 subbasis is contained in a unique smallest filter but there may not exist a unique smallest prefilter containing it.

Prefilters vs. filters

With respect to maps and subsets, the property of being a prefilter is in general more well behaved and better preserved than the property of being a filter. For instance, the image of a prefilter under some map is again a prefilter; but the image of a filter under a non–surjective map is never a filter on the codomain, although it will be a prefilter. The situation is the same with preimages under non–injective maps (even if the map is surjective). If $S\subseteq X$ 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 $\leq$ ), meaning that any equivalence class of prefilters contains a unique filter. This property may be useful when dealing with equivalence classes of prefilters (for instance, they are useful in construct completions using Cauchy filters). The many properties that characterize ultrafilters are also often useful. They are used to, for example, construct the Stone–Čech compactification. The use of ultrafilters generally requires that the ultrafilter lemma be assumed. But in the many fields where the axiom of choice (or the Hahn–Banach theorem) is assumed, the ultrafilter lemma necessarily holds and does not require an addition assumption.

Limits and cluster points of prefilters

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

A family ${\mathcal {B}}$ is said to converge to a point $x\in X$ in $(X,\tau )$ [8] (in which case $x$ is said to be a limit or limit point of ${\mathcal {B}}$ ), written ${\mathcal {B}}\to x$ or $lim ℬ \to x$ in $(X,\tau ),$ [28] if $ℬ \geq 𝒩(x)$ (i.e. if ${\mathcal {B}}$ is finer than $𝒩(x)$ ). Explicitly, this means that every neighborhood $N$ of $x$ contains some element of ${\mathcal {B}}$ as a subset. More generally, given $S\subseteq X,$ if $ℬ \geq 𝒩(S)$ then ${\mathcal {B}}$ is said to converge to $S$ in $(X,\tau ),$ and $S$ is called a limit of ${\mathcal {B}},$ where this is expressed as ${\mathcal {B}}\to S$ in $(X,\tau ).$
Notation: $lim ℬ$ will denote[8] the set of all limit points of ${\mathcal {B}}$ in $(X,\tau ).$

Notation: As usual, $lim ℬ = x$ is defined to mean that ${\mathcal {B}}\to x$ in $(X,\tau )$ and $x$ is the only limit point of ${\mathcal {B}}$ in $(X,\tau )$ (i.e. if $ℬ \to z$ in $(X,\tau )$ then $z=x$ ).[28] (If the notation " $lim ℬ = x$ " did not also require that the limit $x$ be unique then the equals sign = would no longer be guaranteed to be transitive).

In the above definitions, it suffices to check that ${\mathcal {B}}$ is finer than some (or equivalently, than every) neighborhood base in $(X,\tau )$ of $x$ or $S.$ The neighborhood filter ${\mathcal {N}}(x)$ is the smallest (i.e. coarsest) filter on $X$ that converges to $x$ in $(X,\tau )$ ; any filter converging to $x$ in $(X,\tau )$ must contain ${\mathcal {N}}(x)$ as a subset. Said differently, the family of filters that converge to $x$ are exactly those filter on $X$ that contain ${\mathcal {N}}(x)$ as a subset. The finer the topology on $X$ then the fewer prefilters exist that have any limit points in $X.$ If ${\mathcal {B}}$ is a prefilter and $B\in {\mathcal {B}}$ then ${\mathcal {B}}$ converges to a point (or subset) of $X$ if and only if this is true of the trace $ℬ | B$ .[36]

If $x\in X$ and ${\mathcal {B}}$ is a family of sets, then call $x$ a cluster point or accumulation point of ${\mathcal {B}}$ [8] if ${\mathcal {B}}$ meshes with the neighborhood filter at $x$ ; that is, if $B\cap N\neq \varnothing$ for every $B\in {\mathcal {B}}$ and every neighborhood $N$ of $x$ in $X.$ The set of all cluster points of ${\mathcal {B}}$ is denoted by $\operatorname {cl} _{X}{\mathcal {B}}$ or $cl X ℬ$ . More generally, if $S\subseteq X$ and ${\mathcal {B}}\subseteq \wp (X)$ then ${\mathcal {B}}$ is said to clusters at $S$ if ${\mathcal {B}}$ meshes with the neighborhood filter of $S$ ; that is, if $B\cap N\neq \varnothing$ for every $B\in {\mathcal {B}}$ and every neighborhood $N$ of $S$ in $X.$

In the above definitions, it suffices to check that ${\mathcal {B}}$ meshes with some (or equivalently, with every) neighborhood base in $(X,\tau )$ of $x$ or $S.$

Just like sequences and nets, it is possible for a prefilter on an topological space of infinite cardinality to not have any cluster points or limit points.[32]

Limit and cluster point relationships and sufficient conditions

Every limit point of a prefilter ${\mathcal {B}}$ is also a cluster point of ${\mathcal {B}},$ since if $x$ is a limit point of a prefilter ${\mathcal {B}}$ then ${\mathcal {N}}(x)$ and ${\mathcal {B}}$ mesh,[18][32] which makes $x$ a cluster point of ${\mathcal {B}}.$ [8] Every accumulation point of an ultrafilter is also a limit point.

If $x$ is a limit point of ${\mathcal {B}}$ then $x$ is necessarily a limit point of any family ${\mathcal {C}}$ finer than ${\mathcal {B}}$ (i.e. if ${\mathcal {N}}(x)\leq {\mathcal {B}}$ and ${\mathcal {B}}\leq {\mathcal {C}}$ then $𝒩(x) \leq 𝒞$ ).[32] In contrast, if $x$ is a cluster point of ${\mathcal {B}}$ then $x$ is necessarily a cluster point of any family ${\mathcal {C}}$ coarser than ${\mathcal {B}}$ (i.e. if ${\mathcal {N}}(x)$ and ${\mathcal {B}}$ mesh and $𝒞 \leq ℬ$ then ${\mathcal {N}}(x)$ and ${\mathcal {C}}$ mesh).

Closure and cluster points

The set $\operatorname {cl} _{X}{\mathcal {B}}$ of all cluster points of a prefilter ${\mathcal {B}}$ in a topological space $X$ is a closed subset of $X$ and moreover,[8]

\operatorname {cl} _{X}{\mathcal {B}}=\bigcap _{B\in {\mathcal {B}}}\operatorname {cl} _{X}B

which justifies the notation $\operatorname {cl} _{X}{\mathcal {B}}$ for the set of cluster points. Consequently, the set of all cluster points of any prefilter ${\mathcal {B}}$ is a closed subset of $X.$ [32]

If $S\subseteq X$ and if ${\mathcal {B}}$ is a prefilter on $S$ then every cluster point of ${\mathcal {B}}$ in $X$ belongs to $\operatorname {cl} _{X}S$ and any point in $\operatorname {cl} _{X}S$ is a limit point of a filter on $S.$ [32]

Characterizations

The following are equivalent for a prefilter ${\mathcal {B}}$ :

${\mathcal {B}}$ converges to (resp. clusters at) $x.$
${\mathcal {B}}$ converges to (resp. clusters at) the set $\{x\}.$
The family ${\mathcal {B}}^{\uparrow X}$ generated by ${\mathcal {B}}$ converges to (resp. clusters at) $x.$
There exists a family equivalent to ${\mathcal {B}}$ that converges to (resp. clusters at) $x.$

The following are equivalent for a filter ${\mathcal {F}}$ on $X$ :

${\mathcal {F}}$ clusters at $x.$
For every neighborhood $N$ of $x,$
$x\in \bigcap _{F\in {\mathcal {F}}}\operatorname {cl} _{X}F.$
There exists a filter ${\mathcal {B}}$ on $X$ such that ${\mathcal {B}}\supseteq {\mathcal {F}}$ and ${\mathcal {B}}$ converges to $x.$

If ${\mathcal {B}}$ is an ultra prefilter on $X$ and $x\in X,$ then $x$ is a cluster point of ${\mathcal {B}}$ if and only if ${\mathcal {B}}\to x$ in $(X,\tau ).$ [29]

Primitive sets

A subset $P\subseteq X$ is called primitive[37] if it is the set of limit points of some ultrafilter on $X.$ That is, if there exists an ultrafilter ${\mathcal {B}}$ on $X$ such that $P$ is equal to $\operatorname {lim} _{X}{\mathcal {B}},$ which recall denotes the set of limit points of ${\mathcal {B}}$ in $(X,\tau ).$

Any closed singleton subset of $X$ is a primitive subset of $X.$ [37] The image of a primitive subset of $X$ under a continuous map $f:X\to Y$ is contained in a primitive subset of $Y.$ [37]

Assume that $P,Q\subseteq X$ are two primitive subset of $X.$ If $U$ is an open subset of $X$ such that $P\cap Q\neq \varnothing ,$ then $U\in {\mathcal {B}}$ for any ultrafilter ${\mathcal {B}}$ on $X$ such that $P=\operatorname {lim} _{X}{\mathcal {B}}.$ [37] In addition, if $P$ and $Q$ are distinct then there exists some $S\subseteq X$ and some ultrafilters ${\mathcal {B}}_{P}$ and ${\mathcal {B}}_{Q}$ on $X$ such that $P=\operatorname {lim} _{X}{\mathcal {B}}_{P},$ $Q=\operatorname {lim} _{X}{\mathcal {B}}_{Q},$ $S\in {\mathcal {B}}_{P},$ and $X\setminus S\in {\mathcal {B}}_{Q}.$ [37]

Other results

If $X$ is a complete lattice then:

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

If

f:X\to Y

is a map from a set into a topological space

Y,

y\in Y,

and

{\mathcal {B}}\subseteq \wp (X)

then

y

is a limit point or limit (respectively, a cluster point) of $f$ with respect to ${\mathcal {B}}$ [32] if

y

is a limit point (resp. a cluster point) of

f (ℬ)

in

Y,

in which case this may be expressed by writing

f (ℬ) \to y

, or

lim f (ℬ) \to y

, in

Y.

If the limit

y

is unique then the arrow

\to

may be replaced with an equals sign =.[28]

If $x\in X$ and ${\displaystyle \chi$ is a net in $X$ then $\chi \to x$ in $X$ (i.e. $\chi$ converges as a net to $x$ ) if and only if $\chi \left(\operatorname {Tails} (I,\leq )\right)\to x$ (i.e. $x$ is a limit of the function $\chi$ with respect to $\operatorname {Tails} (I,\leq )$ ). In this way, the definition of a convergent net is a special case of the above definition of a limit of a function.

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.[36] 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, $x_{0}\in X,$ and $y\in Y.$ If $Y$ is Hausdorff then all arrows " $\to y$ " in the table may be replaced with equal signs " $=y$ " and " $\lim f\left({\mathcal {B}}\right)\to y$ " may be replaced with " $\lim f\left({\mathcal {B}}\right)=y$ ".[28]


Type of limit		Definition in terms of prefilters[36]				Assumptions
$\lim _{x\to x_{0}}f(x)\to y$	$\Leftrightarrow$	$f\left({\mathcal {B}}\right)\to y$	where	${\mathcal {B}}\,:=\,$	${\mathcal {N}}\left(x_{0}\right)$
$\lim _{\stackrel {x\to x_{0}}{x\neq x_{0}}}f(x)\to y$	⇔	$f\left({\mathcal {B}}\right)\to y$	where	${\mathcal {B}}\,:=\,$	$\left\{N\setminus \left\{x_{0}\right\}:N\in {\mathcal {N}}\left(x_{0}\right)\right\}$
$\lim _{\stackrel {x\to x_{0}}{x\in S}}f(x)\to y$ or $\lim _{x\to x_{0}}f{\big \vert }_{S}(x)\to y$	⇔	$f\left({\mathcal {B}}\right)\to y$	where	${\mathcal {B}}\,:=\,$	$\left\{N\cap S:N\in {\mathcal {N}}\left(x_{0}\right)\right\}$	$S\subseteq X$ and $x_{0}\in \operatorname {cl} _{X}S$
$\lim _{\stackrel {x\to x_{0}}{x\neq x_{0}}}f(x)\to y$	⇔	$f\left({\mathcal {B}}\right)\to y$	where	${\mathcal {B}}\,:=\,$	$\left\{\left(x_{0}-r,x_{0}\right)\cup \left(x_{0},x_{0}+r\right):0<r\in \mathbb {R} \right\}$	$x_{0}\in X=\mathbb {R}$
$\lim _{\stackrel {x\to x_{0}}{x<x_{0}}}f(x)\to y$	⇔	$f\left({\mathcal {B}}\right)\to y$	where	${\mathcal {B}}\,:=\,$	$\left\{\left(x,x_{0}\right):x<x_{0}\right\}$	$x_{0}\in X=\mathbb {R}$
$\lim _{\stackrel {x\to x_{0}}{x\leq x_{0}}}f(x)\to y$	⇔	$f\left({\mathcal {B}}\right)\to y$	where	${\mathcal {B}}\,:=\,$	$\left\{\left(x,x_{0}\right]:x<x_{0}\right\}$	$x_{0}\in X=\mathbb {R}$
$\lim _{\stackrel {x\to x_{0}}{x>x_{0}}}f(x)\to y$	⇔	$f\left({\mathcal {B}}\right)\to y$	where	${\mathcal {B}}\,:=\,$	$\left\{\left(x_{0},x\right):x_{0}<x\right\}$	$x_{0}\in X=\mathbb {R}$
$\lim _{\stackrel {x\to x_{0}}{x\geq x_{0}}}f(x)\to y$	⇔	$f\left({\mathcal {B}}\right)\to y$	where	${\mathcal {B}}\,:=\,$	$\left\{\left[x_{0},x\right):x_{0}\leq x\right\}$	$x_{0}\in X=\mathbb {R}$
$\lim _{n\to \infty }f(n)\to y$	⇔	$f\left({\mathcal {B}}\right)\to y$	where	${\mathcal {B}}\,:=\,$	$\{\{n,n+1,\ldots \}~:~n\in \mathbb {N} \}\}$	$X=\mathbb {N}$ so $f:\mathbb {N} \to Y$ is a sequence in $Y$
$\lim _{x\to \infty }f(x)\to y$	⇔	$f\left({\mathcal {B}}\right)\to y$	where	${\mathcal {B}}\,:=\,$	$\{(x,\infty ):x\in \mathbb {R} \}$	$X=\mathbb {R}$
$\lim _{x\to -\infty }f(x)\to y$	⇔	$f\left({\mathcal {B}}\right)\to y$	where	${\mathcal {B}}\,:=\,$	$\{(-\infty ,x):x\in \mathbb {R}$	$X=\mathbb {R}$
$\lim _{\|x\|\to \infty }f(x)\to y$	⇔	$f\left({\mathcal {B}}\right)\to y$	where	${\mathcal {B}}\,:=\,$	$\{X\cap [(-\infty ,x)\cup (x,\infty )]:x\in \mathbb {R} \}$	$X=\mathbb {R}$ or $X=\mathbb {Z}$ for a double-ended sequence
$\lim _{\\|x\\|\to \infty }f(x)\to y$	⇔	$f\left({\mathcal {B}}\right)\to y$	where	${\mathcal {B}}\,:=\,$	$\{\{x\in X:\\|x\\|>r\}~:~0<r\in \mathbb {R} \}$	$(X,\\|\cdot \\|)$ is a seminormed space (e.g. a Banach space like $X=\mathbb {C}$ )

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

Filters and nets

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

Nets to prefilters

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

A net $x_{\bullet }=\left(x_{i}\right)_{i\in I}$ in $X$ is said to converge in $(X,\tau )$ to a point $x\in X,$ written $x_{\bullet }\to x$ in $(X,\tau ),$ and $x$ is called a limit or limit point of $x_{\bullet },$ [38] if any of the following equivalent conditions hold:

For every $N\in {\mathcal {N}}_{\tau }(x)$ , there exists some $i\in I$ such that if $i\leq j\in I$ then $x_{j}\in N.$

For every $N\in {\mathcal {N}}_{\tau }(x)$ , there exists some $i\in I$ such that the tail of $x_{\bullet }$ starting at $i$ is contained in $N.$

For every $N\in {\mathcal {N}}_{\tau }(x)$ , there exists some $B\in \operatorname {Tails} \left(x_{\bullet }\right)$ such that $B \subseteq N$ .

${\mathcal {N}}_{\tau }(x)\leq \operatorname {Tails} \left(x_{\bullet }\right).$

$\operatorname {Tails} \left(x_{\bullet }\right)\to x$ in $(X,\tau )$ ; that is, the prefilter $\operatorname {Tails} \left(x_{\bullet }\right)$ converges to $x$ in $(X,\tau ).$

Notation: As usual, $\lim x_{\bullet }=x$ is defined to mean that $x_{\bullet }\to x$ in $(X,\tau )$ and $x$ is the only limit point of $x_{\bullet }$ in $(X,\tau )$ (i.e. if $x_{\bullet }\to z$ in $(X,\tau )$ then $z=x.$ [38]

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

For every $N\in {\mathcal {N}}_{\tau }(x)$ and every $i\in I,$ there exists some $i\leq j\in I$ such that $x j \in N$ .

For every $N\in {\mathcal {N}}_{\tau }(x)$ and every $i\in I,$ the tail of $x_{\bullet }$ starting at $i$ intersects $N$ ("intersects" means that the intersection is not empty).

For every $N\in {\mathcal {N}}_{\tau }(x)$ and every $B \in Tails(x •)$ , $B \cap N \neq \emptyset$ .

${\mathcal {N}}_{\tau }(x)$ and $\operatorname {Tails} \left(x_{\bullet }\right)$ mesh (by definition of "mesh").

$x$ is a cluster point of $\operatorname {Tails} \left(x_{\bullet }\right)$ in $(X,\tau ).$

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

Prefilters to nets

A pointed set is a pair $(S,s)$ consisting of a non–empty set $S$ and an element $s\in S.$ Define a canonical preorder $\leq$ on pointed sets by declaring
$(R,r)\leq (S,s)$ if and only if $R\supseteq S.$

For any family ${\mathcal {B}},$ let

$\operatorname {PointedSets} ({\mathcal {B}})$

denote the set of all pointed sets $(B,b)$ such that $B\in {\mathcal {B}}$ and $b\in B.$

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

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

If ${\mathcal {B}}$ is a prefilter on $X$ then the net associated with ${\mathcal {B}}$ is the map
$Net ℬ : (PointedSets(ℬ), \leq) \to X$ defined by $Net ℬ (B, b) = b$ .

If ${\mathcal {B}}$ is a prefilter on $X$ then $\operatorname {Net} _{\mathcal {B}}$ is a net in $X$ and the prefilter associated with $\operatorname {Net} _{\mathcal {B}}$ is ${\mathcal {B}}$ ; that is:

Tails(Net ℬ) = ℬ

.[note 10]

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

However, if $x_{\bullet }=\left(x_{i}\right)_{i\in I}$ is a net in $X$ then it is not in general true that $\operatorname {Net} _{\operatorname {Tails} \left(x_{\bullet }\right)}$ is equal to $x_{\bullet }$ because, for example, the domain of a net in $X$ (i.e. the directed set $I$ ) may have any cardinality (so the class of nets in $X$ is not even a set) whereas the cardinality of the set of prefilters on $X,$ which is a subset of $\wp (\wp (X)),$ is bounded above.

Proposition — If ${\mathcal {B}}$ is a prefilter on $X$ and $x\in X$ then

${\mathcal {B}}\to x$ in $(X,\tau )$ if and only if $Net ℬ \to x$ in $(X,\tau ).$
$x$ is a cluster point of ${\mathcal {B}}$ if and only if $x$ is a cluster point of $\operatorname {Net} _{\mathcal {B}}.$

Proof —

Recall that $ℬ = Tails(Net ℬ)$ and that if $x_{\bullet }$ is a net in $X$ then (1) $x_{\bullet }\to x$ if and only if $\operatorname {Tails} \left(x_{\bullet }\right)\to x,$ and (2) $x$ is a cluster point of $x_{\bullet }$ if and only if $x$ is a cluster point of $\operatorname {Tails} \left(x_{\bullet }\right)$ . By using $x • := Net ℬ$ and $ℬ = Tails(Net ℬ)$ , it follows that ${\mathcal {B}}\to x$ $\Leftrightarrow$ $Tails(Net ℬ) \to x$ $\Leftrightarrow$ $Net ℬ \to x$ . It also follows that $x$ is a cluster point of ${\mathcal {B}}$ $\Leftrightarrow$ $x$ is a cluster point of $Tails(Net ℬ)$ $\Leftrightarrow$ $x$ is a cluster point of $\operatorname {Net} _{\mathcal {B}}.$

Ultranets and ultra prefilters

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

Partially ordered net

The domain of the canonical net $\operatorname {Net} _{\mathcal {B}}$ is in general not partially ordered. However, in 1955 Bruns and Schmidt discovered[39] a construction, similar to a lexicographical order, that allows for the canonical net to have a domain that is both partially ordered and directed; this was independently rediscovered by Albert Wilansky in 1970.[4] Let $Poset ℬ ≝ { (B, b, m) : B \in ℬ, b \in B, and m \in ℕ }$ and for any two elements $i=(B,b,m)$ and $j=(C,c,n),$ declare that $i<j$ if and only if $B\supseteq C$ and either: (1) $B\neq C$ or else (2) $B=C$ and $m<n$ (or equivalently, $i<j$ if and only if (1) $B\supseteq C,$ and (2) $B=C$ implies $m<n$ ). This defines a strict partial order whose corresponding non-strict partial order, denoted by $\leq ,$ is defined by declaring that $i\leq j$ if and only if $i<j$ or $i=j.$ Both $<$ and $\leq$ are serial and neither possesses a greatest element or a maximal element. Let $PosetNet ℬ : Poset ℬ \to X$ be the map defined by $i=(B,b,m)\mapsto b.$ If $i_{0}=\left(B_{0},b_{0},m_{0}\right)\in$ $Poset ℬ$ then just as with $\operatorname {Net} _{\mathcal {B}}$ before, the tail of the $PosetNet ℬ$ starting at $i_{0}$ is equal to $B_{0}.$ If ${\mathcal {B}}$ is a prefilter on $X$ then $PosetNet ℬ$ is a net in $X$ whose domain $Poset ℬ$ is a partially ordered set and moreover, $Tails(PosetNet ℬ) = ℬ$ .[4] Because the tails of $PosetNet ℬ$ and $\operatorname {Net} _{\mathcal {B}}$ are identical (since both are equal to the prefilter ${\mathcal {B}}$ ), there is typically nothing lost by assuming that the domain of the net associated with a prefilter is both directed and partially ordered.[4] If the set $ℕ$ is replaced with the positive rational numbers then the strict partial order $<$ will also be a dense order.

Subordinate filters and subnets

The notion of " ${\mathcal {B}}$ is subordinate to ${\mathcal {C}}$ " (written $ℬ ⊢ 𝒞$ ) is for filters and prefilters what " $x_{n_{\bullet }}=\left(x_{n_{i}}\right)_{i=1}^{\infty }$ is a subsequence of $x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }$ " is for sequences.[23] For example, if $\operatorname {Tails} \left(x_{\bullet }\right)=\left\{x_{\geq i}:i\in \mathbb {N} \right\}$ denotes the set of tails of $x_{\bullet }$ and if $\operatorname {Tails} \left(x_{n_{\bullet }}\right)=\left\{x_{n_{\geq i}}:i\in \mathbb {N} \right\}$ denotes the set of tails of the subsequence $x_{n_{\bullet }}$ (where $x_{n_{\geq i}}:=\left\{x_{n_{i}}~:~i\in \mathbb {N} \right\}$ ) then $\operatorname {Tails} \left(x_{n_{\bullet }}\right)~\vdash ~\operatorname {Tails} \left(x_{\bullet }\right)$ (i.e. $\operatorname {Tails} \left(x_{\bullet }\right)\leq \operatorname {Tails} \left(x_{n_{\bullet }}\right)$ ) is true but $\operatorname {Tails} \left(x_{\bullet }\right)~\vdash ~\operatorname {Tails} \left(x_{n_{\bullet }}\right)$ is in general false. If $x_{\bullet }=\left(x_{i}\right)_{i\in I}$ is a net in a topological space $X$ and if ${\mathcal {N}}(x)$ is the neighborhood filter at a point $x\in X,$ then $x_{\bullet }\to x$ in $X$ if and only if ${\mathcal {N}}(x)\leq \operatorname {Tails} \left(x_{\bullet }\right).$

Subordination analogs of results involving subsequences

The following results are the prefilter analogs of statements involving subsequences.[40] The condition " $𝒞 \geq ℬ$ ," which is also written $𝒞 ⊢ ℬ$ , is the analog of " ${\mathcal {C}}$ is a subsequence of ${\mathcal {B}}.$ " So "finer than" and "subordinate to" is the prefilter analog of "subsequence of." Some people prefer saying "subordinate to" instead of "finer than" because it is more reminiscent of "subsequence of."

Proposition[40][32] — Let ${\mathcal {B}}$ be a prefilter on $X$ and let $x\in X.$

Suppose ${\mathcal {C}}$ $\text{[math]}$ is a prefilter such that 𝒞 ≥ ℬ.
1. If ${\mathcal {B}}\to x$ $\text{[math]}$ in $X$ $\text{[math]}$ then 𝒞 → x in $X.$ $\text{[math]}$ [proof 4]
  - This is the analog of "if a sequence converges to $x$ then so does every subsequence."
2. If $x$ $\text{[math]}$ is a cluster point of ${\mathcal {C}}$ $\text{[math]}$ in $X$ $\text{[math]}$ then $x$ $\text{[math]}$ is a cluster point of ${\mathcal {B}}$ $\text{[math]}$ in $X.$ $\text{[math]}$
  - This is the analog of "if $x$ is a cluster point of some subsequence, then $x$ is a cluster point of the original sequence."
${\mathcal {B}}\to x$ $\text{[math]}$ in $X$ $\text{[math]}$ if and only if for any finer prefilter 𝒞 ≥ ℬ there exists some even more fine prefilter ℱ ≥ 𝒞 such that ℱ → x in $X.$ $\text{[math]}$ [32]
- This is the analog of "a sequence converges to $x$ if and only if every subsequence has a sub–subsequence that converges to $x.$ "
$x$ $\text{[math]}$ is a cluster point of ${\mathcal {B}}$ $\text{[math]}$ in $X$ $\text{[math]}$ if and only if there exists some finer prefilter 𝒞 ≥ ℬ such that 𝒞 → x in $X.$ $\text{[math]}$
- This is the analog of " $x$ is a cluster point of a sequence if and only if it has a subsequence that converges to $x.$ "

Non–equivalence of subnets and subordinate filters

A subset $R\subseteq I$ of a preordered space $(I,\leq )$ is frequent or cofinal in $I$ if for every $i\in I$ there exists some $r\in R$ such that $i\leq r.$ If $R\subseteq I$ contains a tail of $I$ then $R$ is said to be eventual in $I$ ; explicitly, this means that there exists some $i\in I$ such that $I_{\geq i}\subseteq R$ (that is, $j\in R$ for all $j\in I$ such that $i\leq j$ ). A subset is eventual if and only if its complement is not frequent (i.e. infrequent).[41] A map $h:A\to I$ between two preordered sets is order–preserving if $h(a)\leq h(b)$ whenever $a\leq b$ for $a,b\in A.$

Subnets in the sense of Willard and subnets in the sense of Kelley are the most commonly used definitions of "subnet."[41] The first definition of a subnet was introduced by John L. Kelley in 1955.[41] Stephen Willard introduced his own variant of Kelley's definition of subnet in 1970.[41] 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.[41]

Let $S=S_{\bullet }~:~(A,\leq )\to X$ and $N=N_{\bullet }~:~(I,\leq )\to X$ be nets. Then[41]

$S_{\bullet }$ is a Willard–subnet of $N_{\bullet }$ or a subnet in the sense of Willard if there exists an order–preserving map $h:A\to I$ such that $S=N\circ h$ and $h(A)$ is cofinal in $I$ .

$S_{\bullet }$ is a Kelley–subnet of $N_{\bullet }$ or a subnet in the sense of Kelly if there exists a map $h~:~A\to I$ such that $S=N\circ h$ and whenever $E\subseteq I$ is eventual in $I$ then $h^{-1}(E)$ is eventual in $A.$

$S_{\bullet }$ is an AA–subnet of $N_{\bullet }$ or a subnet in the sense of Aarnes and Andenaes if any of the following equivalent conditions are satisfied:

$\operatorname {Tails} \left(N_{\bullet }\right)\leq \operatorname {Tails} \left(S_{\bullet }\right).$

$\operatorname {TailsFilter} \left(N_{\bullet }\right)\subseteq \operatorname {TailsFilter} \left(S_{\bullet }\right).$

If $J$ is eventual in $I$ then $S^{-1}\left(N(J)\right)$ is eventual in $A$ .

For any subset $R\subseteq X,$ if $\operatorname {Tails} \left(S_{\bullet }\right)$ and $\{R\},$ then so do $\operatorname {Tails} \left(N_{\bullet }\right)$ and $\{R\}.$

For any subset $R\subseteq X,$ if $\operatorname {Tails} \left(S_{\bullet }\right)\leq \{R\}$ then $\operatorname {Tails} \left(N_{\bullet }\right)\leq \{R\}.$

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 $X$ (i.e. the nets' common codomain). Every Willard–subnet is a Kelley–subnet and both are AA–subnets.[41] 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 $\operatorname {Tails} \left(x_{\bullet }\right)\leq \operatorname {Tails} \left(y_{\bullet }\right).$

AA–subnets have a defining characterization that immediately shows that they are fully interchangeable with sub(ordinate)filters.[41][42] Explicitly, what is meant is that the following statement is true for AA–subnets:

If

{\mathcal {B}}

and

{\mathcal {F}}

are prefilters then

{\mathcal {B}}\leq {\mathcal {F}}

if and only if

Net ℱ

is an AA–subset of

\operatorname {Net} _{\mathcal {B}}.

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

False statement: If

{\mathcal {B}}

and

{\mathcal {F}}

are prefilters such that

{\mathcal {B}}\leq {\mathcal {F}}

then

Net ℱ

is a Kelley–subset of

\operatorname {Net} _{\mathcal {B}}.

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

Example: For all $n\in \mathbb {N} ,$ let $B_{n}=\{1\}\cup \mathbb {N} _{\geq n}.$ Let $ℬ = {B n : n \in ℕ$ }, which is a proper $π$ –system, and let $ℱ = { {1} } \cup ℬ$ , where both families are prefilters on the natural numbers $X:=\mathbb {N} =\{1,2,\ldots \}.$ Because ${\mathcal {B}}\leq {\mathcal {F}},$ ${\mathcal {F}}$ is to ${\mathcal {B}}$ as a subsequence is to a sequence. So ideally, $S ≝ Net ℱ$ should be a subnet of $N ≝ Net ℬ$ . Let $I ≝ PointedSets(ℬ)$ be the domain of $\operatorname {Net} _{\mathcal {B}},$ so $I$ contains a cofinal subset that is order isomorphic to $ℕ$ and consequently contains neither a maximal nor greatest element. Let $A ≝ PointedSets(ℱ) = { M } \cup I$ , where $M:=\left(1,\{1\}\right)$ is both a maximal and greatest element of $A$ . The directed set $A$ also contains a subset that is order isomorphic to $\mathbb {N}$ (because it contains $I$ , which contains such subsets) but no such subset can be cofinal in $A$ because of the maximal element $M.$ Consequently, any order–preserving map $h:A\to I$ must be eventually constant (with value $h(M)$ ) where $h(M)$ is then a greatest element of the range $\operatorname {Im} h.$ Because of this, there can be no order preserving map $h:A\to I$ that satisfies the conditions required for $Net ℱ$ to be a Willard–subnet of $\operatorname {Net} _{\mathcal {B}}$ (because the range of such a map $h$ cannot be cofinal in $I$ ). Suppose for the sake of contradiction that there exists a map $h:A\to I$ such that $h^{-1}\left(I_{\geq i}\right)$ is eventual in $A$ for all $i\in I.$ Because $h(M)\in I,$ there exist $n,n_{0}\in \mathbb {N}$ such that $h(M)=\left(n_{0},B_{n}\right)$ with $n_{0}\in B_{n}.$ For every $i\in I,$ because $h -1 (I \geq i)$ is eventual in $A$ , it is necessary that $h (M) \in I \geq i$ . In particular, if $i:=\left(n+2,B_{n+2}\right)$ then $h(M)\geq i=\left(n+2,B_{n+2}\right),$ which by definition is equivalent to $B_{n}\subseteq B_{n+2},$ which is false. Consequently, $Net ℱ$ is not a Kelley–subnet of $\operatorname {Net} _{\mathcal {B}}$ .[42]

If "subnet" is defined to mean Willard–subnet or Kelley–subnet, then nets and filters are not completely interchangeable because there exist a filter–sub(ordinate)filter relationships that cannot be expressed in terms of a net–subnet relationship between the two induced nets. In particular, the problem is that Kelley–subnets and Willard–subnets are not fully interchangeable with subordinate filters. So 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 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.[41][42]

Topologies and prefilters

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

Examples of relationships between filters and topologies

Bases and prefilters

Let ${\mathcal {B}}\neq \varnothing$ 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: ${\mathcal {B}}$ 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 $ℬ \subseteq τ$ then the definitions of ${\mathcal {B}}$ is a basis (resp. subbasis) for $\tau$ can be reworded as:

{\mathcal {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 principle. If $X = ℝ$ has its usual topology and if $x\in X,$ then any neighborhood filter base ${\mathcal {B}}$ of $x$ is fixed by $x$ (in fact, it is even true that $ker ℬ = {x$ }) but ${\mathcal {B}}$ is not principal since ${x} \notin ℬ$ . 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 (e.g. 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 principle but not discrete (i.e. not principal at a single point). A neighborhood basis ${\mathcal {B}}$ of a point $x$ in a topological space is principal if and only if the kernel of ${\mathcal {B}}$ is an open set. If in addition the space is T₁ then $ker ℬ = {x}$ so that this basis ${\mathcal {B}}$ is principal if and only if ${x}$ is an open set.

Generating topologies from filters and prefilters

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

If ${\mathcal {B}}$ is a prefilter (resp. filter subbase, $π$ –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 ${\mathcal {B}}.$ If ${\mathcal {B}}$ is closed under finite intersections then the set $τ ℬ = { \emptyset, X} \cup ℬ \cup$ is a topology on $X$ with both ${X} \cup ℬ \cup$ and ${X} \cup ℬ$ being bases for it. If the $π$ –system ${\mathcal {B}}$ covers $X$ then both $ℬ \cup$ and ${\mathcal {B}}$ are also bases for $τ ℬ$ . If $\tau$ is a topology on $X$ then $τ ∖ { \emptyset }$ is a prefilter (or equivalently, a $π$ –system) if and only if it has the finite intersection property (i.e. it is a filter subbase), in which case a subset $ℬ \subseteq τ$ will be a basis for $\tau$ if and only if $ℬ ∖ { \emptyset }$ is equivalent to $τ ∖ { \emptyset }$ , in which case $ℬ ∖ { \emptyset }$ will be a prefilter.

Topologies on directed sets and net convergence

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

Prefilters and topological properties

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

Neighborhoods and topologies

The neighborhood filter of a non–empty subset $S\subseteq X$ in a topological space $X$ is equal to the intersection of all neighborhood filters of all points in $S$ .[27]

Suppose $σ$ and $\tau$ are topologies on $X.$ Then $\tau$ is finer than $σ$ (i.e. $σ \subseteq τ$ ) if and only if whenever $x\in X$ and ${\mathcal {B}}$ is a filter on $X,$ if ${\mathcal {B}}\to x$ in $(X,\tau )$ then ${\mathcal {B}}\to x$ in $(X, σ)$ .[37] Consequently, $σ = τ$ if and only if for every filter ${\mathcal {B}}$ on $X$ and every $x\in X,$ ${\mathcal {B}}\to x$ in $(X, σ)$ if and only if ${\mathcal {B}}\to x$ in $(X,τ)$ .[28] However, it is possible that $σ \neq τ$ while also for every filter ${\mathcal {B}}$ on $X,$ ${\mathcal {B}}$ converges to some point of $X$ in $(X, σ)$ if and only if ${\mathcal {B}}$ converges to some point of $X$ in $(X,τ)$ .[28]

Closure

If $x\in X$ and $S\subseteq X$ with $S \neq \emptyset$ then the following are equivalent:

$x \in cl S$
$x$ is a limit point of the prefilter $\{S\}$ (i.e. ${S} \to x$ in $X$ ).
There exists a prefilter $ℱ \subseteq ℘(X)$ on $X$ such that $S \in ℱ$ and $ℱ \to x$ in $X$ .
There exists a prefilter $ℱ \subseteq ℘(S)$ on $S$ such that $ℱ \to x$ in $X$ .[40]
$x$ is a cluster point of the prefilter $\{S\}.$
The prefilter $\{S\}$ meshes with the neighborhood filter ${\mathcal {N}}(x).$
The prefilter $\{S\}$ meshes with some (or equivalently, with every) prefilter of ${\mathcal {N}}(x).$

The following are equivalent:

$x$ is a limit points of $S$ in $X$ .
There exists a prefilter $ℱ \subseteq ℘(S)$ on $S ∖ {x}$ such that $ℱ \to x$ in $X$ .[40]

Closed sets

If $S\subseteq X$ with $S \neq \emptyset$ then the following are equivalent:

$S$ is a closed subset of $X$ .
If $x\in X$ and $ℱ \subseteq ℘(S)$ is a prefilter on $S$ such that $ℱ \to x$ in $X$ , then $x \in S$ .
If $x\in X$ and $ℱ \subseteq ℘(S)$ is a prefilter on $S$ such that $x$ is an accumulation points of ${\mathcal {F}}$ in $X$ , then $x \in S$ .[40]
If $x\in X$ $\text{[math]}$ is such that the neighborhood filter ${\mathcal {N}}(x)$ $\text{[math]}$ meshes with { S } then x ∈ S.
- The proof of this characterization depends the ultrafilter lemma, which depends on the axiom of choice.

Hausdorff

The following are equivalent:

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

Compactness

The following are equivalent:

$(X,\tau )$ is a compact space.
Every ultrafilter on $X$ $\text{[math]}$ converges to at least one point in $X.$ $\text{[math]}$
- 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).
The above statement but with the words "prefilter" replaced by any one of the following: filter, ultrafilter.[43]
For every filter ${\mathcal {C}}$ on $X$ there exists a filter ${\mathcal {F}}$ on $X$ such that ${\mathcal {C}}\leq {\mathcal {F}}$ and ${\mathcal {F}}$ converges to some point of $X.$
For every prefilter ${\mathcal {C}}$ on $X$ there exists a prefilter ${\mathcal {F}}$ on $X$ such that ${\mathcal {C}}\leq {\mathcal {F}}$ and ${\mathcal {F}}$ converges to some point of $X.$
Every maximal (i.e. ultra) prefilter on $X$ converges to at least one point in $X.$ [8]
The above statement but with the words "maximal prefilter" replaced by any one of the following: prefilter, filter, ultra prefilter, ultrafilter.
Every prefilter on $X$ $\text{[math]}$ has at least one cluster point in $X.$ $\text{[math]}$ [8]
- That this condition is equivalent to compactness can be proven by using only the ultrafilter lemma.
Alexander subbase theorem: There exists a subbase ${\mathcal {S}}$ $\text{[math]}$ for $\tau$ $\text{[math]}$ such that every cover of $X$ $\text{[math]}$ by sets in ${\mathcal {S}}$ $\text{[math]}$ has a finite subcover.
- That this condition is equivalent to compactness can be proven by using only the ultrafilter lemma.

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

Theorem[43] — If ${\mathcal {B}}$ is a filter on a compact space $X$ and $C$ is the set of cluster points of $X,$ then every neighborhood of $C$ belongs to ${\mathcal {B}}.$ Thus a filter on a compact Hausdorff space converges if and only if it has a single cluster point.

Continuity

Let $f:X\to Y$ is a map between topological spaces $(X, τ X)$ and $(Y, τ Y)$ .

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

$f:X\to Y$ is continuous at $x.$
Definition: For every neighborhood $V$ of $f (x)$ in $Y$ there exists some neighborhood $N$ of $x$ in $X$ such that $f (N) \subseteq V$ .
$f({\mathcal {N}}(x))$ is a filter base for ${\mathcal {N}}(f(x))$ ; that is, the upward closure of $f({\mathcal {N}}(x))$ in $Y$ is equal to ${\mathcal {N}}(f(x)).$ [40]
$f({\mathcal {N}}(x))\to f(x)$ in $Y.$ [40]
If ${\mathcal {B}}$ is a filter on $X$ such that ${\mathcal {B}}\to x$ in $X$ then $f({\mathcal {B}})\to f(x)$ in $Y.$
The above statement but with the word "filter" replaced by "prefilter".

The following are equivalent:

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

If ${\mathcal {B}}$ is a prefilter on $X,$ $x\in X$ is a cluster point of ${\mathcal {B}},$ and $f:X\to Y$ is continuous, then $f (x)$ is a cluster point in $Y$ of the prefilter $f (ℬ)$ .[37]

Products

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

If $x • = (x i) i \in I \in \prod X •$ , then $ℬ • \to x •$ in $\prod X •$ if and only if ${\mathcal {B}}_{i}\to x_{i}$ in $X_{i}$ for every $i\in I.$

Suppose $X$ and $Y$ are topological spaces, ${\mathcal {B}}$ is a prefilter on $X$ having $x\in X$ as a cluster point, and ${\mathcal {C}}$ is a prefilter on $Y$ having $y\in Y$ as a cluster point. Then $(x,y)$ is a cluster point of ${\mathcal {B}}\times {\mathcal {C}}$ in the product space $X \times Y$ .[37] However, if $X=Y=\mathbb {Q}$ then there exist sequences $x_{\bullet }:=\left(x_{i}\right)_{i=1}^{\infty }\subseteq X$ and $y_{\bullet }:=\left(y_{i}\right)_{i=1}^{\infty }\subseteq Y$ such that both of these sequences have a cluster point in $\mathbb {Q}$ but the sequence $\left(x_{i},y_{i}\right)_{i=1}^{\infty }\subseteq X\times Y$ does not have a cluster point in $X\times Y.$ [37]

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

Proof

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

Let ${\mathcal {B}}$ be an ultrafilter on $X$ and for every $i,$ let ${\mathcal {B}}_{i}$ denote the ultrafilter on $X_{i}$ generated by the ultra prefilter $\operatorname {Pr} _{i}({\mathcal {B}}).$ Because $X_{i}$ is compact and Hausdorff, the ultrafilter ${\mathcal {B}}_{i}$ converges to a unique limit point $x i \in X i$ (because of $x_{i}$ 's uniqueness, this definition doesn't require the axiom of choice). Let $x:=\left(x_{i}\right)_{i\in I}$ where $x$ satisifes $\operatorname {Pr} _{i}(x)=x_{i}$ for every $i.$ The characterization of convergence in the product topology that was given above implies that ${\mathcal {B}}\to x$ 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). ∎

Examples of applications of prefilters

Convergence of nets of sets

Often, people prefer nets over filters or filters over nets. This example shows that the choice between nets and filters is not a dichotomy by combining them together.

If $S$ is a subset of a topological space $(X, τ X)$ then the set $τ X (S)$ of open neighborhoods of $S$ in $(X, τ X)$ is a prefilter if and only if $S \neq \emptyset$ . The same is true of the set $𝒩 τ (S) ≝ τ X (S) ↑ X$ of all neighborhoods of $S$ in $(X, τ X)$ . The following definition generalizes the notion of the set of tails of a net of points in $X$ to nets of subsets of $X.$

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

(Nets of points $\leftrightarrow$ Nets of singleton sets): Every net $x_{\bullet }=\left(x_{i}\right)_{i\in I}$ of points in $X$ can be uniquely associated with the following canonical nets of singleton sets in $X$

\left\{x_{\bullet }\right\}:=\left(\left\{x_{i}\right\}\right)_{i\in I}

which is called the canonical net of (singleton) sets in $X$ associated with or induced by $x_{\bullet }.$ Conversely, every net of singleton sets in $X$ is uniquely associated with a canonical net of points in $X$ (defined in the obvious way).

The following definition is completely analogous to the definition of the prefilter of tails of a net (of points) in $X$ that was given above.

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

and define the set of tails generated by $S_{\bullet }$ to be the set

$\operatorname {Tails} \left(S_{\bullet }\right):=\left\{S_{\geq i}:i\in I\right\}$

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

Given any net $x_{\bullet }=\left(x_{i}\right)_{i\in I}$ of points in $X$ it is readily seen that $\operatorname {Tails} \left(x_{\bullet }\right)=\operatorname {Tails} \left(\left\{x_{\bullet }\right\}\right),$ where $\left\{x_{\bullet }\right\}$ is the canonical net of singleton sets associated with $x_{\bullet }.$ This makes it apparent that the following definition of "convergence of a net of sets" in $X$ is indeed a generalization of the definition of "convergence of a net of points" in $X$ that was given above.

If $R$ is any subset of a topological space $(X, τ X)$ , then a net of sets $S_{\bullet }$ in $X$ is said to converges to $R$ in $(X, τ X)$ , written $S_{\bullet }\to R$ in $(X, τ X)$ , if $\operatorname {Tails} \left(S_{\bullet }\right)\to R$ in $(X, τ X)$ .
Recall that by definition, $\operatorname {Tails} \left(S_{\bullet }\right)\to R$ in $(X, τ X)$ if and only if $\operatorname {Tails} \left(S_{\bullet }\right)$ is finer than $𝒩 τ (R)$ (i.e. $𝒩 τ (R) \leq Tails(S •)$ ).

The next subsection illustrates how the above definitions may be used to make rigorous certain intuitive/geometric ideas of convergence involving sets.

Prefilters and spaces of functions

Throughout, $Y\neq \varnothing$ will be a topological space, $X\neq \varnothing$ will be a set, and the graph of a function $g:X\to Y$ will be denoted by $\operatorname {Gr} (g)\subseteq X\times Y.$ Whenever it is needed, it should be automatically assumed that $X$ is also a topological space.

Convergence of maps in any one of the most well–known function space topologies (e.g. the topology of uniform convergence on $X,$ or the topology of pointwise convergence, which are defined below) is often imagined by visualizing the graphs of these maps as "moving towards the limit function's graph" in some way; this visualization is dependent on the particular function space topology. For example, suppose that $Y$ is a metric space with metric $d.$ Then a net $g_{\bullet }=\left(g_{i}\right)_{i\in I}$ of $Y$ –valued maps on $X$ converges uniformly on $X$ to a map $g$ if and only if the prefilter of tails generated by $\operatorname {Gr} \left(g_{\bullet }\right):=\left(\operatorname {Gr} \left(g_{i}\right)\right)_{i\in I}$ is finer than the filter on $X\times Y$ generated by the family of all sets $\operatorname {Gr} \left(g,r\right):=\{(x,y)\in X\times Y\;:\;d\left(y,g(x)\right)<r\}$ as $r$ ranges over the positive reals. This section generalizes this example and shows how prefilters together with nets may be used to directly translate such intuitive geometric notions of "convergence of graphs" into a definition that is equivalent to the usual open–set definition of function space topologies (of uniform convergence).

Let $G\neq \varnothing$ be a family of maps from $X$ into $Y.$ Let ${\mathcal {C}}$ be a family of sets over $X$ that is closed under finite unions (e.g. all finite subsets of $X,$ or all compact subsets of $X$ ), which implies[note 11] that $\varnothing \in {\mathcal {C}}.$ [note 12] Whenever it is necessary, it will be assumed that the union of all sets in $𝒞$ is equal to $X$ and/or that $𝒞$ is downward closed.[note 13]

Let $τ G,𝒞$ denote the topology (on $G$ ) of uniform convergence on sets in ${\mathcal {C}}$ , which recall is defined by the subbasis consisting of all sets of the form

\{g\in G:g(A)\subseteq V\}

where $A$ ranges over ${\mathcal {C}}$ and $V$ ranges over the open subset of $Y.$ If ${\mathcal {C}}$ is the set of all finite subsets of $X$ then this topology is called the topology of pointwise convergence while if $X$ is a topological space and ${\mathcal {C}}$ is the set of all compact subsets of $X$ then this topology is commonly called the compact–open topology on $G$ or the topology of uniform convergence on compact sets of $G.$

The topology on $X\times Y$ that will now be defined, which is in general different from the product topology, allows for a characterization of convergence of nets in $(G, τ G,𝒞)$ in terms of convergence of the graphs (which are sets) of maps in $G$ .

Let

τ 𝒞

denote the topology for convergence of graphs on

X\times Y

that is generated by the subbasis consisting of all sets of the form

(X \times Y) ∖ [A \times (Y ∖ V)] = [(X ∖ A) \times Y] \cup (A \times V)

where

A

ranges over

{\mathcal {C}}

and

V

ranges over the open subset of

Y.

If $A$ is a topological space and if each $A \in 𝒞$ is closed in $X$ then $τ 𝒞$ is weaker than the product topology on $X\times Y.$ More importantly, if $g \in G$ and if $g • = (g i) i \in I$ is a net in $G,$ then $g • \to g$ in $(G, τ G,𝒞)$ if and only if its net of graphs $\operatorname {Gr} \left(g_{\bullet }\right):=\left(\operatorname {Gr} \left(g_{i}\right)\right)_{i\in I}$ converges to $Gr(g)$ in $(X \times Y, τ 𝒞)$ . In particular, when $X$ is a topological space and ${\mathcal {C}}$ is the set of all compact subsets (resp. finite subsets) of $X$ then this characterizes the compact–open topology (resp. topology of pointwise convergence) on $G.$

In general, there is a much larger variety of filters on $X\times Y$ than there are subsets of $G$ so there are many more generalizations of the above notions of convergence. For example, the above notions of convergence of graphs can be extended to maps that are defined merely on subsets of $X.$ [note 14]

Uniformities and Cauchy prefilters

A uniform space is a set $X$ equipped with a filter on $X \times X$ that has certain properties. A base or fundamental system of entourages is a prefilter on $X \times X$ whose upward closure is a uniform space. A prefilter ${\mathcal {B}}$ on a uniform space $X$ with uniformity ${\mathcal {F}}$ is called a Cauchy prefilter if for every entourages $N \in ℱ$ , there exists some $B\in {\mathcal {B}}$ such that $B \times B \subseteq N$ . A uniform space $(X, ℱ)$ is called complete (resp. sequentially complete) if every Cauchy prefilter (resp. every elementary Cauchy prefilter) on $X$ converges to at least one point of $X.$

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

Topologizing the set of prefilters

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

ℙ := Prefilters(X)

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

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

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

For every $S\subseteq X,$ let

𝕆(S) ≝ { ℬ \in ℙ : S \in ℬ ↑ X}

where $𝕆(X) = ℙ$ and $𝕆(\emptyset) = \emptyset$ .[note 15] These sets will be the basic open subsets of the Stone topology. If $R \subseteq S \subseteq X$ then

{ ℬ \in ℘(℘(X)) : R \in ℬ ↑ X} \subseteq { ℬ \in ℘(℘(X)) : S \in ℬ ↑ X

}.

From this inclusion, it is possible to deduce all of the subset inclusions displayed below with the exception of $𝕆(R \cap S) \supseteq 𝕆(R) \cap 𝕆(S)$ .[note 16] For all $R, S \subseteq X$ ,

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 $π$ -system that forms a basis for a topology on $ℙ$ , where it is henceforth assumed that $ℙ$ carries this topology and that any subset of $ℙ$ carries the induced subspace topology.

In contrast to most other general constructions of topologies (e.g. the product, quotient, subspace topologies, etc.), this topology on $ℙ$ was defined without using anything other than the set $X$ ; there were no 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 $𝔹 \subseteq ℙ$ and $ℱ \in ℙ$ then:

Closure in $ℙ$ : ${\mathcal {F}}$ belongs to the closure of $𝔹$ in $ℙ$ if and only if $ℱ \subseteq \cup ℬ \in 𝔹 ℬ ↑ X$ .
Neighborhoods in $ℙ$ : $𝔹$ is a neighborhood of ${\mathcal {F}}$ in $ℙ$ if and only if there exists some $F\in {\mathcal {F}}$ such that $𝕆(F) = { ℬ \in ℙ : F \in ℬ ↑ X} \subseteq 𝔹$ (i.e. for all $ℬ \in ℙ$ , if $F \in ℬ ↑ X$ then $ℬ \in 𝔹$ ).

It will be henceforth assumed that $X\neq \varnothing$ because otherwise $ℙ = \emptyset$ and the topology is ${ \emptyset$ }, 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 $β : X \to UltraFilters(X)$ , defined by sending $x\in X$ to the principal ultrafilter at $x,$ is a topological embedding whose image is a dense subset of $UltraFilters(X)$ (see the article Stone–Čech compactification for more details).

Relationships between topologies on $X$ and the Stone topology on $ℙ$

Every $τ \in Top(X)$ induces a canonical map $𝒩 τ : X \to Filters(X)$ defined by $x \mapsto 𝒩 τ (x)$ , which sends $x\in X$ to the neighborhood filter of $x$ in $(X,\tau ).$ Clearly, $𝒩 τ : X \to Filters(X)$ is injective if and only if $\tau$ is $T 0$ (i.e. a Kolmogorov space) and moreover, if $τ, σ \in Top(X)$ then $τ = σ$ if and only if $𝒩 τ = 𝒩 σ$ . Thus every $τ \in Top(X)$ can be identified with the canonical map $𝒩 τ$ , which means that $Top(X)$ can be canonically identified as a subset of $Func(X; ℙ)$ (as a side note, it is now possible to place on $Func(X; ℙ)$ (and thus also on $Top(X)$ ) the topology of pointwise convergence or the topology of uniform convergence on $X,$ as examples). For every $τ \in Top(X)$ , the surjection $𝒩 τ : (X, τ) \to Im 𝒩 τ$ is continuous, closed, and open.[note 17] In particular, for every $T 0$ topology $\tau$ on $X,$ $𝒩 τ : (X, τ) \to ℙ$ is a topological embedding.

In addition, if $𝔉 : X \to Filter(X)$ is a map such that $x \in ker 𝔉(x) = \cap F \in 𝔉(x) F$ for every $x\in X,$ then for every $x\in X$ and every $F \in 𝔉(x)$ , $𝔉(F)$ is a neighborhood of $𝔉(x)$ in $Im 𝔉$ (where $Im 𝔉$ has the subspace topology inherited from $ℙ$ ).

Notes

Sequences and nets in a space $X$ are maps from directed sets like the natural number, which in general maybe entirely unrelated to the set $X$ and so they, and consequently also their notions of convergence, are not intrinsic to $X.$
Technically, any infinite subfamily of this set of tails is enough to characterization this sequences convergence. But in general, the set of all tails is taken unless there is some reason to do otherwise.
Indeed, net convergence is defined using neighborhood filters while (pre)filters are directed sets with respect to $\supseteq$ , so it is difficult to keep these notions completely separate.
The terms "Filter base" and "Filter" are used if and only if $S\neq \varnothing .$
More generally, for any real numbers satisfying $r\leq s$ and $u\leq v,$ $B_{r,s}\cap B_{u,v}=B_{m,\max(s,v)}$ where $m:=\min(s,v,\max(r,u)).$
If $R, S \subseteq ℝ$ then $ℬ R \cap ℬ S = ℬ R \cap S$ . This property and the fact that $ℬ R$ is nonempty and proper if and only if $R\neq \varnothing$ actually allows for the construction of even more examples of prefilters, because if $𝒮 \subseteq ℘(ℝ)$ is any prefilter (resp. filter subbase, $π$ -system) then so is ${ ℬ S : S \in 𝒮$ }.
The $π$ –system generated by $𝒞 open$ (resp. by $𝒞 closed$ ) is a prefilter whose elements are finite unions of open (resp. closed) intervals having endpoints in $E \cup { -\infty, \infty }$ with two of these intervals being of the forms $(-\infty, e 1)$ and $(e 2, \infty)$ (resp. $(-\infty, e 1]$ and $[e 2, \infty)$ ) where $e 1 \leq 1 + e 2$ ; in the case of $𝒞 closed$ , it is possible for one or more of these closed intervals to be singleton sets (i.e. degenerate closed intervals).
Suppose $X$ has more than one point, $f:X\to Y$ is a constant map, and $Ξ = {f (X) }$ then $Ξ f$ will consist of all non–empty subsets of $Y.$
For an example of how this failure can happen, consider the case where there exists some $B\in {\mathcal {B}}$ and some $y \in Y ∖ B$ such that both $f -1 (y)$ and its complement in $X$ contains at least two distinct points.
The set equality $Tails(Net ℬ) = ℬ$ holds more generally: if the family of sets ${\mathcal {B}}\neq \varnothing$ satisfies $\varnothing \not \in {\mathcal {B}}$ then the family of tails of the map $PointedSets(ℬ) \to X$ (defined by $(B,b)\mapsto b$ ) is equal to ${\mathcal {B}}.$
Here, the nullary union convention is assumed, which is the convention that the union of an empty family of sets is equal to the empty set. Since the union of $0$ sets is a finite union and since ${\mathcal {C}}$ is assumed to be closed under all finite unions, ${\mathcal {C}}$ must contain $\varnothing .$
In the definition of the "topology of uniform convergence," this allows us to take $A:=\varnothing$ where now $V:=Y$ implies that ${g \in G : g (A) \subseteq V} = G$ . This guarantees that the family described in this definition satisfy one of the requirements of being subbase on $G$ .
If needed, assume also that $𝒞$ has additional properties that will guarantee that the sets ${g \in G : g (A) \subseteq V}$ in the definition of the "topology of uniform convergence on sets in ${\mathcal {C}}$ " do indeed form a subbasis for a topology $τ G,𝒞$ .
For instance, it is possible to rigorously define (often without great difficulty) notions of what it could mean for a net of maps on $X$ to converge to a set of maps (e.g. such as convergence to a germ of maps in $G$ at some given point of $X$ ).
As a side note, had the definitions of "filter" and "prefilter" not required propriety then the degenerate dual ideal $℘(X)$ would have been a prefilter on $X$ so that in particular, $𝕆(\emptyset) = { ℘(X) } \neq \emptyset$ with $℘(X) \in 𝕆(S)$ for every $S\subseteq X.$
This is because the inclusion $𝕆(R \cap S) \supseteq 𝕆(R) \cap 𝕆(S)$ is the only one in the sequence below whose proof uses the defining assumption that $𝕆(S) \subseteq ℙ$ .
Here, $Im 𝒩 τ$ is the set of all $\tau$ –neighborhood filters of all points in $X$ and it is endowed with the subspace topology inherited from $ℙ$ .

Proofs

Suppose ${\mathcal {B}}$ is filter subbase that is ultra. Let $C,D\in {\mathcal {B}}$ and define $S=C\cap D.$ Because ${\mathcal {B}}$ is ultra, there exists some $B\in {\mathcal {B}}$ such that $B\cap S$ equals $B$ or $\varnothing .$ The finite intersection property implies that $B\cap S\neq \varnothing$ so necessarily $B\cap S=B,$ which is equivalent to $B\subseteq C\cap D.$ ∎
To prove that ${\mathcal {C}}$ and ${\mathcal {F}}$ mesh, let $C\in {\mathcal {C}}$ and $F\in {\mathcal {F}}$ . Because ${\mathcal {C}}\leq {\mathcal {F}},$ there exists some $G \in ℱ$ such that $G \subseteq C$ , so ${\mathcal {F}}$ having the finite intersection property guarantees $\emptyset \neq G \cap F \subseteq C \cap F$ . ∎ If $C 1, ..., C n \in 𝒞$ then there are $F 1, ..., F n \in ℱ$ such that $F i \subseteq C i$ and now $\emptyset \neq F 1 \cap ... \cap F n \subseteq C 1 \cap ... \cap C n$ . This shows that ${\mathcal {C}}$ is a filter subbasis. ∎
This is because if ${\mathcal {C}}$ and ${\mathcal {F}}$ are prefilters on $X$ then ${\mathcal {C}}\leq {\mathcal {F}}$ if and only if $𝒞 ↑ X \subseteq ℱ ↑ X$ .
By definition, ${\mathcal {B}}\to x$ if and only if $ℬ \geq 𝒩(x)$ . Since $𝒞 \geq ℬ$ and $ℬ \geq 𝒩(x)$ , transitivity implies $𝒞 \geq 𝒩(x)$ .∎

Citations

H. Cartan, "Théorie des filtres", CR Acad. Paris, 205, (1937) 595–598.
H. Cartan, "Filtres et ultrafiltres", CR Acad. Paris, 205, (1937) 777–779.
Wilansky 2013, p. 44.
Schechter 1996, pp. 155-171.
Howes 1995, pp. 83-92.
Dolecki & Mynard 2016, pp. 27–29.
Dolecki & Mynard 2016, pp. 33–35.
Narici & Beckenstein 2011, pp. 2–7.
Császár 1978, pp. 53-65.
Bourbaki 1987, pp. 57–68.
Schubert 1968, pp. 48–71.
Narici & Beckenstein 2011, pp. 3–4.
Dugundji 1966, pp. 215–221.
Wilansky 2013, p. 5.
Dolecki & Mynard 2016, p. 10.
Schechter 1996, pp. 100–130.
Császár 1978, pp. 82-91.
Dugundji 1966, pp. 211–213.
Schechter 1996, p. 100.
Császár 1978, pp. 53-65, 82-91.
Arkhangel'skii & Ponomarev 1984, pp. 7–8.
Joshi 1983, p. 244.
Dugundji 1966, p. 212.
Wilansky 2013, pp. 44–46.
Castillo, Jesus M. F.; Montalvo, Francisco (January 1990), "A Counterexample in Semimetric Spaces" (PDF), Extracta Mathematicae, 5 (1): 38–40
Schaefer & Wolff 1999, pp. 1–11.
Bourbaki 1987, pp. 129–133.
Wilansky 2008, pp. 32-35.
Dugundji 1966, pp. 219–221.
Jech 2006, pp. 73-89.
Császár 1978, pp. 53-65, 82-91, 102-120.
Bourbaki 1987, pp. 68–74.
Dolecki & Mynard 2016, pp. 37–39.
Arkhangel'skii & Ponomarev 1984, pp. 20–22.
Császár 1978, pp. 102-120.
Dixmier 1984, pp. 13–18.
Bourbaki 1987, pp. 132–133.
Kelley 1975, pp. 65–72.
Bruns G., Schmidt J.,Zur Aquivalenz von Moore-Smith-Folgen und Filtern, Math. Nachr. 13 (1955), 169-186.
Dugundji 1966, pp. 211–221.
Schechter 1996, pp. 157–168.
Clark, Pete L. (October 18, 2016). "Convergence" (PDF). math.uga.edu/. Retrieved August 18, 2020.
Bourbaki 1987, pp. 83–85.

References

Adams, Colin; Franzosa, Robert (2009). Introduction to Topology: Pure and Applied. New Delhi: Pearson Education. ISBN 978-81-317-2692-1. OCLC 789880519.
Arkhangel'skii, Alexander Vladimirovich; Ponomarev, V.I. (1984). Fundamentals of General Topology: Problems and Exercises. Mathematics and Its Applications. 13. Dordrecht Boston: D. Reidel. ISBN 978-90-277-1355-1. OCLC 9944489.
Berberian, Sterling K. (1974). Lectures in Functional Analysis and Operator Theory. Graduate Texts in Mathematics. 15. New York: Springer. ISBN 978-0-387-90081-0. OCLC 878109401.
Bourbaki, Nicolas (1989) [1966]. General Topology: Chapters 1–4 [Topologie Générale]. Éléments de mathématique. Berlin New York: Springer Science & Business Media. ISBN 978-3-540-64241-1. OCLC 18588129.
Bourbaki, Nicolas (1989) [1967]. General Topology 2: Chapters 5–10 [Topologie Générale]. Éléments de mathématique. 4. Berlin New York: Springer Science & Business Media. ISBN 978-3-540-64563-4. OCLC 246032063.
Bourbaki, Nicolas (1987) [1981]. Sur certains espaces vectoriels topologiques [Topological Vector Spaces: Chapters 1–5]. Annales de l'Institut Fourier. Éléments de mathématique. 2. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag. ISBN 978-3-540-42338-6. OCLC 17499190.
Burris, Stanley N., and H.P. Sankappanavar, H. P., 2012. A Course in Universal Algebra. Springer-Verlag. ISBN 9780988055209.
Comfort, William Wistar; Negrepontis, Stylianos (1974). The Theory of Ultrafilters. 211. Berlin Heidelberg New York: Springer-Verlag. ISBN 978-0-387-06604-2. OCLC 1205452.
Császár, Ákos (1978). General topology. Translated by Császár, Klára. Bristol England: Adam Hilger Ltd. ISBN 0-85274-275-4. OCLC 4146011.
Dixmier, Jacques (1984). General Topology. Undergraduate Texts in Mathematics. Translated by Berberian, S. K. New York: Springer-Verlag. ISBN 978-0-387-90972-1. OCLC 10277303.
Dolecki, Szymon; Mynard, Frederic (2016). Convergence Foundations Of Topology. New Jersey: World Scientific Publishing Company. ISBN 978-981-4571-52-4. OCLC 945169917.
Dugundji, James (1966). Topology. Boston: Allyn and Bacon. ISBN 978-0-697-06889-7. OCLC 395340485.
Dunford, Nelson; Schwartz, Jacob T. (1988). Linear Operators. Pure and applied mathematics. 1. New York: Wiley-Interscience. ISBN 978-0-471-60848-6. OCLC 18412261.
Edwards, Robert E. (1995). Functional Analysis: Theory and Applications. New York: Dover Publications. ISBN 978-0-486-68143-6. OCLC 30593138.
Howes, Norman R. (23 June 1995). Modern Analysis and Topology. Graduate Texts in Mathematics. New York: Springer-Verlag Science & Business Media. ISBN 978-0-387-97986-1. OCLC 31969970. OL 1272666M.CS1 maint: date and year (link)
Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.
Jech, Thomas (2006). Set Theory: The Third Millennium Edition, Revised and Expanded. Berlin New York: Springer Science & Business Media. ISBN 978-3-540-44085-7. OCLC 50422939.
Joshi, K. D. (1983). Introduction to General Topology. New York: John Wiley and Sons Ltd. ISBN 978-0-85226-444-7. OCLC 9218750.
Kelley, John L. (1975). General Topology. Graduate Texts in Mathematics. 27. New York: Springer Science & Business Media. ISBN 978-0-387-90125-1. OCLC 338047.
Köthe, Gottfried (1969). Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704.
David MacIver, Filters in Analysis and Topology (2004) (Provides an introductory review of filters in topology and in metric spaces.)
Munkres, James R. (2000). Topology (Second ed.). Upper Saddle River, NJ: Prentice Hall, Inc. ISBN 978-0-13-181629-9. OCLC 42683260.
Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
Robertson, Alex P.; Robertson, Wendy J. (1980). Topological Vector Spaces. Cambridge Tracts in Mathematics. 53. Cambridge England: Cambridge University Press. ISBN 978-0-521-29882-7. OCLC 589250.
Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
Schechter, Eric (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN 978-0-12-622760-4. OCLC 175294365.
Schubert, Horst (1968). Topology. London: Macdonald & Co. ISBN 978-0-356-02077-8. OCLC 463753.
Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.
Wilansky, Albert (17 October 2008) [1970]. Topology for Analysis. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-46903-4. OCLC 227923899.CS1 maint: date and year (link)
Willard, Stephen (2004) [1970]. General Topology. Dover Books on Mathematics (First ed.). Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240.

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[3] Sequences and nets in a space $X$ are maps from directed sets like the natural number, which in general maybe entirely unrelated to the set $X$ and so they, and consequently also their notions of convergence, are not intrinsic to $X.$

[4] Technically, any infinite subfamily of this set of tails is enough to characterization this sequences convergence. But in general, the set of all tails is taken unless there is some reason to do otherwise.

[5] Indeed, net convergence is defined using neighborhood filters while (pre)filters are directed sets with respect to $\supseteq$ , so it is difficult to keep these notions completely separate.

[filterbase_iff_not_empty-17] The terms "Filter base" and "Filter" are used if and only if $S\neq \varnothing .$

[33] More generally, for any real numbers satisfying $r\leq s$ and $u\leq v,$ $B_{r,s}\cap B_{u,v}=B_{m,\max(s,v)}$ where $m:=\min(s,v,\max(r,u)).$

[34] If $R, S \subseteq ℝ$ then $ℬ R \cap ℬ S = ℬ R \cap S$ . This property and the fact that $ℬ R$ is nonempty and proper if and only if $R\neq \varnothing$ actually allows for the construction of even more examples of prefilters, because if $𝒮 \subseteq ℘(ℝ)$ is any prefilter (resp. filter subbase, $π$ -system) then so is ${ ℬ S : S \in 𝒮$ }.

[41] The $π$ –system generated by $𝒞 open$ (resp. by $𝒞 closed$ ) is a prefilter whose elements are finite unions of open (resp. closed) intervals having endpoints in $E \cup { -\infty, \infty }$ with two of these intervals being of the forms $(-\infty, e 1)$ and $(e 2, \infty)$ (resp. $(-\infty, e 1]$ and $[e 2, \infty)$ ) where $e 1 \leq 1 + e 2$ ; in the case of $𝒞 closed$ , it is possible for one or more of these closed intervals to be singleton sets (i.e. degenerate closed intervals).

[43] Suppose $X$ has more than one point, $f:X\to Y$ is a constant map, and $Ξ = {f (X) }$ then $Ξ f$ will consist of all non–empty subsets of $Y.$

[47] For an example of how this failure can happen, consider the case where there exists some $B\in {\mathcal {B}}$ and some $y \in Y ∖ B$ such that both $f -1 (y)$ and its complement in $X$ contains at least two distinct points.

[51] The set equality $Tails(Net ℬ) = ℬ$ holds more generally: if the family of sets ${\mathcal {B}}\neq \varnothing$ satisfies $\varnothing \not \in {\mathcal {B}}$ then the family of tails of the map $PointedSets(ℬ) \to X$ (defined by $(B,b)\mapsto b$ ) is equal to ${\mathcal {B}}.$

[58] Here, the nullary union convention is assumed, which is the convention that the union of an empty family of sets is equal to the empty set. Since the union of $0$ sets is a finite union and since ${\mathcal {C}}$ is assumed to be closed under all finite unions, ${\mathcal {C}}$ must contain $\varnothing .$

[59] In the definition of the "topology of uniform convergence," this allows us to take $A:=\varnothing$ where now $V:=Y$ implies that ${g \in G : g (A) \subseteq V} = G$ . This guarantees that the family described in this definition satisfy one of the requirements of being subbase on $G$ .

[60] If needed, assume also that $𝒞$ has additional properties that will guarantee that the sets ${g \in G : g (A) \subseteq V}$ in the definition of the "topology of uniform convergence on sets in ${\mathcal {C}}$ " do indeed form a subbasis for a topology $τ G,𝒞$ .

[61] For instance, it is possible to rigorously define (often without great difficulty) notions of what it could mean for a net of maps on $X$ to converge to a set of maps (e.g. such as convergence to a germ of maps in $G$ at some given point of $X$ ).

[62] As a side note, had the definitions of "filter" and "prefilter" not required propriety then the degenerate dual ideal $℘(X)$ would have been a prefilter on $X$ so that in particular, $𝕆(\emptyset) = { ℘(X) } \neq \emptyset$ with $℘(X) \in 𝕆(S)$ for every $S\subseteq X.$

[63] This is because the inclusion $𝕆(R \cap S) \supseteq 𝕆(R) \cap 𝕆(S)$ is the only one in the sequence below whose proof uses the defining assumption that $𝕆(S) \subseteq ℙ$ .

[64] Here, $Im 𝒩 τ$ is the set of all $\tau$ –neighborhood filters of all points in $X$ and it is endowed with the subspace topology inherited from $ℙ$ .

[36] Suppose ${\mathcal {B}}$ is filter subbase that is ultra. Let $C,D\in {\mathcal {B}}$ and define $S=C\cap D.$ Because ${\mathcal {B}}$ is ultra, there exists some $B\in {\mathcal {B}}$ such that $B\cap S$ equals $B$ or $\varnothing .$ The finite intersection property implies that $B\cap S\neq \varnothing$ so necessarily $B\cap S=B,$ which is equivalent to $B\subseteq C\cap D.$ ∎

[38] To prove that ${\mathcal {C}}$ and ${\mathcal {F}}$ mesh, let $C\in {\mathcal {C}}$ and $F\in {\mathcal {F}}$ . Because ${\mathcal {C}}\leq {\mathcal {F}},$ there exists some $G \in ℱ$ such that $G \subseteq C$ , so ${\mathcal {F}}$ having the finite intersection property guarantees $\emptyset \neq G \cap F \subseteq C \cap F$ . ∎ If $C 1, ..., C n \in 𝒞$ then there are $F 1, ..., F n \in ℱ$ such that $F i \subseteq C i$ and now $\emptyset \neq F 1 \cap ... \cap F n \subseteq C 1 \cap ... \cap C n$ . This shows that ${\mathcal {C}}$ is a filter subbasis. ∎

[40] This is because if ${\mathcal {C}}$ and ${\mathcal {F}}$ are prefilters on $X$ then ${\mathcal {C}}\leq {\mathcal {F}}$ if and only if $𝒞 ↑ X \subseteq ℱ ↑ X$ .

[54] By definition, ${\mathcal {B}}\to x$ if and only if $ℬ \geq 𝒩(x)$ . Since $𝒞 \geq ℬ$ and $ℬ \geq 𝒩(x)$ , transitivity implies $𝒞 \geq 𝒩(x)$ .∎

[1] H. Cartan, "Théorie des filtres", CR Acad. Paris, 205, (1937) 595–598.

[2] H. Cartan, "Filtres et ultrafiltres", CR Acad. Paris, 205, (1937) 777–779.

[FOOTNOTEWilansky201344-6] Wilansky 2013, p. 44.

[FOOTNOTESchechter1996155-171-7] Schechter 1996, pp. 155-171.

[FOOTNOTEHowes199583-92-8] Howes 1995, pp. 83-92.

[FOOTNOTEDoleckiMynard201627–29-9] Dolecki & Mynard 2016, pp. 27–29.

[FOOTNOTEDoleckiMynard201633–35-10] Dolecki & Mynard 2016, pp. 33–35.

[FOOTNOTENariciBeckenstein20112–7-11] Narici & Beckenstein 2011, pp. 2–7.

[FOOTNOTECsászár197853-65-12] Császár 1978, pp. 53-65.

[FOOTNOTEBourbaki198757–68-13] Bourbaki 1987, pp. 57–68.

[FOOTNOTESchubert196848–71-14] Schubert 1968, pp. 48–71.

[FOOTNOTENariciBeckenstein20113–4-15] Narici & Beckenstein 2011, pp. 3–4.

[FOOTNOTEDugundji1966215–221-16] Dugundji 1966, pp. 215–221.

[FOOTNOTEWilansky20135-18] Wilansky 2013, p. 5.

[FOOTNOTEDoleckiMynard201610-19] Dolecki & Mynard 2016, p. 10.

[FOOTNOTESchechter1996100–130-20] Schechter 1996, pp. 100–130.

[FOOTNOTECsászár197882-91-21] Császár 1978, pp. 82-91.

[FOOTNOTEDugundji1966211–213-22] Dugundji 1966, pp. 211–213.

[FOOTNOTESchechter1996100-23] Schechter 1996, p. 100.

[FOOTNOTECsászár197853-65,_82-91-24] Császár 1978, pp. 53-65, 82-91.

[FOOTNOTEArkhangel'skiiPonomarev19847–8-25] Arkhangel'skii & Ponomarev 1984, pp. 7–8.

[FOOTNOTEJoshi1983244-26] Joshi 1983, p. 244.

[FOOTNOTEDugundji1966212-27] Dugundji 1966, p. 212.

[FOOTNOTEWilansky201344–46-28] Wilansky 2013, pp. 44–46.

[Castillo_1990-29] Castillo, Jesus M. F.; Montalvo, Francisco (January 1990), "A Counterexample in Semimetric Spaces" (PDF), Extracta Mathematicae, 5 (1): 38–40

[FOOTNOTESchaeferWolff19991–11-30] Schaefer & Wolff 1999, pp. 1–11.

[FOOTNOTEBourbaki1987129–133-31] Bourbaki 1987, pp. 129–133.

[FOOTNOTEWilansky200832-35-32] Wilansky 2008, pp. 32-35.

[FOOTNOTEDugundji1966219–221-35] Dugundji 1966, pp. 219–221.

[FOOTNOTEJech200673-89-37] Jech 2006, pp. 73-89.

[FOOTNOTECsászár197853-65,_82-91,_102-120-39] Császár 1978, pp. 53-65, 82-91, 102-120.

[FOOTNOTEBourbaki198768–74-42] Bourbaki 1987, pp. 68–74.

[FOOTNOTEDoleckiMynard201637–39-44] Dolecki & Mynard 2016, pp. 37–39.

[FOOTNOTEArkhangel'skiiPonomarev198420–22-45] Arkhangel'skii & Ponomarev 1984, pp. 20–22.

[FOOTNOTECsászár1978102-120-46] Császár 1978, pp. 102-120.

[FOOTNOTEDixmier198413–18-48] Dixmier 1984, pp. 13–18.

[FOOTNOTEBourbaki1987132–133-49] Bourbaki 1987, pp. 132–133.

[FOOTNOTEKelley197565–72-50] Kelley 1975, pp. 65–72.

[52] Bruns G., Schmidt J.,Zur Aquivalenz von Moore-Smith-Folgen und Filtern, Math. Nachr. 13 (1955), 169-186.

[FOOTNOTEDugundji1966211–221-53] Dugundji 1966, pp. 211–221.

[FOOTNOTESchechter1996157–168-55] Schechter 1996, pp. 157–168.

[Clark_Convergence_2016-56] Clark, Pete L. (October 18, 2016). "Convergence" (PDF). math.uga.edu/. Retrieved August 18, 2020.

[FOOTNOTEBourbaki198783–85-57] Bourbaki 1987, pp. 83–85.

Filters in topology

Motivation

Preliminaries, notation, and basic notions

Filters and prefilters

Basic examples

Ultrafilters

Free, principal, and kernels

Finer/coarser, subordination, and meshing

Equivalent families of sets

Summary of filter limits and cluster points definitions

Set theoretic properties, examples, and constructions involving prefilters

Supremum, trace, and meshing

Products and other examples

Images and preimages of filters and prefilters

Limits, cluster points, and nets

Limits and cluster points of prefilters

Limits of functions defined as limits of prefilters

Filters and nets

Nets to prefilters

Prefilters to nets

Subordinate filters and subnets

Subordination analogs of results involving subsequences

Non–equivalence of subnets and subordinate filters

Topologies and prefilters

Examples of relationships between filters and topologies

Prefilters and topological properties

Examples of applications of prefilters

Convergence of nets of sets

Prefilters and spaces of functions

Uniformities and Cauchy prefilters

Topologizing the set of prefilters

See also

Notes

Citations

References