Filters in topology#Convergent filter

Archetypical example of a filter

{{See also|Filter (set theory)}}

The archetypical example of a filter is the Neighbourhood filter $\backslash mathcal\{N\}(x)$ at a point $x$ in a topological space $(X,\; \backslash tau),$ which is the family of sets consisting of all neighborhoods of $x.$

By definition, a neighborhood of some given point $x$ is any subset $B\; \backslash subseteq\; X$ whose topological interior contains this point; that is, such that $x\; \backslash in\; \backslash operatorname\{Int\}\_X\; B.$ Importantly, neighborhoods are {{em|not}} required to be open sets; those are called {{em|open neighborhoods}}.

Listed below are those fundamental properties of neighborhood filters that ultimately became the definition of a "filter."

A {{em|filter on $X$}} is a set $\backslash mathcal\{B\}$ of subsets of $X$ that satisfies all of the following conditions:

1. {{em|Not empty}}:   $X\; \backslash in\; \backslash mathcal\{B\}$  –  just as $X\; \backslash in\; \backslash mathcal\{N\}(x),$ since $X$ is always a neighborhood of $x$ (and of anything else that it contains);
2. {{em|Does not contain the empty set}}:   $\backslash varnothing\; \backslash not\backslash in\; \backslash mathcal\{B\}$  –  just as no neighborhood of $x$ is empty;
3. {{em|Closed under finite intersections}}:   If $B,\; C\; \backslash in\; \backslash mathcal\{B\}\; \backslash text\{\; then\; \}\; B\; \backslash cap\; C\; \backslash in\; \backslash mathcal\{B\}$  –  just as the intersection of any two neighborhoods of $x$ is again a neighborhood of $x$;
4. {{em|Upward closed}}:   If $B\; \backslash in\; \backslash mathcal\{B\}\; \backslash text\{\; and\; \}\; B\; \backslash subseteq\; S\; \backslash subseteq\; X$ then $S\; \backslash in\; \backslash mathcal\{B\}$  –  just as any subset of $X$ that includes a neighborhood of $x$ will necessarily {{em|be}} a neighborhood of $x$ (this follows from $\backslash operatorname\{Int\}\_X\; B\; \backslash subseteq\; \backslash operatorname\{Int\}\_X\; S$ and the definition of "a neighborhood of $x$").

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

{{See also|Limit of a sequence|Net (mathematics)}}

A {{em|sequence in $X$}} is by definition a map $\backslash N\; \backslash to\; X$ from the natural numbers into the space $X.$

The original notion of convergence in a topological space was that of a sequence converging to some given point in a space, such as a metric space.

With metrizable spaces (or more generally first-countable spaces or Fréchet–Urysohn spaces), sequences usually suffices to characterize, or "describe", most topological properties, such as the closures of subsets or continuity of functions.

But there are many spaces where sequences can {{em|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 {{em|never}} fail to characterize topological properties.

Nets directly generalize the notion of a sequence since nets are, by definition, maps $I\; \backslash to\; X$ from an arbitrary directed set $(I,\; \backslash leq)$ into the space $X.$ A sequence is just a net whose domain is $I\; =\; \backslash N$ 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 {{em|only}} the values of a sequence.

To see how this is done, consider a sequence $x\_\backslash bull\; =\; \backslash left(x\_i\backslash right)\_\{i=1\}^\backslash infty\; \backslash text\{\; in\; \}\; X,$ which is by definition just a function $x\_\{\backslash bullet\}\; :\; \backslash N\; \backslash to\; X$ whose value at $i\; \backslash in\; \backslash N$ is denoted by $x\_i$ rather than by the usual parentheses notation $x\_\backslash bull(i)$ that is commonly used for arbitrary functions.

Knowing only the image (sometimes called "the range") $\backslash operatorname\{Im\}\; x\_\backslash bull\; :=\; \backslash left\backslash \{x\_i\; :\; i\; \backslash in\; \backslash N\backslash right\backslash \}\; =\; \backslash left\backslash \{x\_1,\; x\_2,\; \backslash ldots\backslash right\backslash \}$ of the sequence is not enough to characterize its convergence; multiple sets are needed.

It turns out that the needed sets are the following,Technically, any infinite subfamily of this set of tails is enough to characterize this sequence's convergence. But in general, unless indicated otherwise, the set of {{em|all}} tails is taken unless there is some reason to do otherwise. which are called the {{em|tails}} of the sequence $x\_\backslash bull$:

$$\backslash begin\{alignat\}\{8\}$$

x_{\geq 1} =\; &\{&&x_1, &&x_2, &&x_3, &&x_4, &&\ldots&& \,\} \\[0.3ex]

x_{\geq 2} =\; &\{&&x_2, &&x_3, &&x_4, &&x_5, &&\ldots&& \,\} \\[0.3ex]

x_{\geq 3} =\; &\{&&x_3, &&x_4, &&x_5, &&x_6, &&\ldots&& \,\} \\[0.3ex]

& && && &&\;\,\vdots && && && \\[0.3ex]

x_{\geq n} =\; &\{&&x_n, \;\;\,&&x_{n+1}, \;&&x_{n+2}, \;&&x_{n+3},&&\ldots&& \,\} \\[0.3ex]

& && && &&\;\,\vdots && && && \\[0.3ex]

\end{alignat}

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 $x\_n,\; x\_\{n+1\},\; \backslash ldots\; .$ This can be reworded as:

{{in5}}every neighborhood $U$ must contain some set of the form $\backslash \{x\_n,\; x\_\{n+1\},\; \backslash ldots\backslash \}$ as a subset.

Or more briefly: every neighborhood must contain some tail $x\_\{\backslash geq\; n\}$ as a subset.

It is this characterization that can be used with the above family of tails to determine convergence (or non-convergence) of the sequence $x\_\backslash bull\; :\; \backslash N\; \backslash to\; X.$

Specifically, with the family of {{em|sets}} $\backslash \{x\_\{\backslash geq\; 1\},\; x\_\{\backslash geq\; 2\},\; \backslash ldots\backslash \}$ in hand, the {{em|function}} $x\_\backslash bull\; :\; \backslash N\; \backslash to\; X$ is no longer needed to determine convergence of this sequence (no matter what topology is placed on $X$).

By generalizing this observation, the notion of "convergence" can be extended from sequences/functions to families of sets.

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

Nets versus filters − advantages and disadvantages

Filters and nets each have their own advantages and drawbacks and there's no reason to use one notion exclusively over the other.Indeed, net convergence is defined using neighborhood filters while (pre)filters are directed sets with respect to $\backslash ,\backslash supseteq\backslash ,,$ so it is difficult to keep these notions completely separate.

Depending on what is being proved, a proof may be made significantly easier by using one of these notions instead of the other.{{sfn|Wilansky|2013|p=44}} Both filters and nets can be used to completely characterize any given topology.

Nets are direct generalizations of sequences and can often be used similarly to sequences, so the learning curve for nets is typically much less steep than that for filters.

However, filters, and especially ultrafilters, have many more uses outside of topology, such as in set theory, mathematical logic, model theory (ultraproducts, for example), abstract algebra,{{sfn|Schechter|1996|pp=155-171}} combinatorics,{{cite journal|last=Fernández-Bretón|first=David J.|title=Using Ultrafilters to Prove Ramsey-type Theorems|journal=The American Mathematical Monthly|publisher=Informa UK Limited|volume=129|issue=2|date=2021-12-22|issn=0002-9890|doi=10.1080/00029890.2022.2004848|pages=116–131|arxiv=1711.01304|s2cid=231592954 }} dynamics, order theory, generalized convergence spaces, Cauchy spaces, and in the definition and use of hyperreal numbers.

Like sequences, nets are {{em|functions}} and so they have the {{em|advantages of functions}}.

For example, like sequences, nets can be "plugged into" other functions, where "plugging in" is just function composition.

Theorems related to functions and function composition may then be applied to nets.

One example is the universal property of inverse limits, which is defined in terms of composition of functions rather than sets and it is more readily applied to functions like nets than to sets like filters (a prominent example of an inverse limit is the Cartesian product).

Filters may be awkward to use in certain situations, such as when switching between a filter on a space $X$ and a filter on a dense subspace $S\; \backslash subseteq\; X.${{sfn|Howes|1995|pp=83-92}}

In contrast to nets, filters (and prefilters) are families of {{em|sets}} and so they have the {{em|advantages of sets}}.

For example, if $f$ is surjective then the {{em|image}} $f^\{-1\}(\backslash mathcal\{B\})\; :=\; \backslash left\backslash \{f^\{-1\}(B)\; ~:~\; B\; \backslash in\; \backslash mathcal\{B\}\backslash right\backslash \}$ under $f^\{-1\}$ of an arbitrary filter or prefilter $\backslash mathcal\{B\}$ is both easily defined and guaranteed to be a prefilter on $f$'s domain, whereas it is less clear how to pullback (unambiguously/without choice) an arbitrary sequence (or net) $y\_\backslash bull$ so as to obtain a sequence or net in the domain (unless $f$ is also injective and consequently a bijection, which is a stringent requirement). Similarly, the intersection of any collection of filters is once again a filter whereas it is not clear what this could mean for sequences or nets.

Because filters are composed of subsets of the very topological space $X$ that is under consideration, topological set operations (such as closure or interior) may be applied to the sets that constitute the filter.

Taking the closure of all the sets in a filter is sometimes useful in functional analysis for instance.

Theorems and results about images or preimages of sets under a function may also be applied to the sets that constitute a filter; an example of such a result might be one of continuity's characterizations in terms of preimages of open/closed sets or in terms of the interior/closure operators.

Special types of filters called {{em|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 {{em|any}} cardinality.

In contrast, the collection of all filters (and of all prefilters) on $X$ is a set whose cardinality is no larger than that of $\backslash wp(\backslash wp(X)).$

Similar to a topology on $X,$ a filter on $X$ is "intrinsic to $X$" in the sense that both structures consist {{em|entirely}} of subsets of $X$ and neither definition requires any set that cannot be constructed from $X$ (such as $\backslash N$ or other directed sets, which sequences and nets require).

{{Main|Filter (set theory)}}

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

Whenever these assumptions are needed, then it should be assumed that $X$ is non-empty and that $\backslash mathcal\{B\},\; \backslash mathcal\{F\},$ 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 as they are used.

Unfortunately, not all notation related to filters is well established and some notation varies greatly across the literature (for example, the notation for the set of all prefilters on a set) so in such cases this article uses whatever notation is most self describing or easily remembered.

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

Their important properties are described later.

Sets operations

The {{em|Upward closed set}} or {{em|isotonization}} in $X${{sfn|Dolecki|Mynard|2016|pp=27–29}}{{sfn|Dolecki|Mynard|2016|pp=33–35}} of a family of sets $\backslash mathcal\{B\}\; \backslash subseteq\; \backslash wp(X)$ is

{{Quote frame|1=$\backslash mathcal\{B\}^\{\backslash uparrow\; X\}\; :=\; \backslash \{S\; \backslash subseteq\; X\; ~:~\; B\; \backslash subseteq\; S\; \backslash text\{\; for\; some\; \}\; B\; \backslash in\; \backslash mathcal\{B\}\; \backslash ,\backslash \}\; =\; \{\backslash textstyle\backslash bigcup\backslash limits\_\{B\; \backslash in\; \backslash mathcal\{B\}\}\}\; \backslash \{S\; ~:~\; B\; \backslash subseteq\; S\; \backslash subseteq\; X\backslash \}$

}}

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

{{Quote box|align=center|1=For any two families $\backslash mathcal\{C\}\; \backslash text\{\; and\; \}\; \backslash mathcal\{F\},$ declare that $\backslash mathcal\{C\}\; \backslash leq\; \backslash mathcal\{F\}$ if and only if for every $C\; \backslash in\; \backslash mathcal\{C\}$ there exists some $F\; \backslash in\; \backslash mathcal\{F\}\; \backslash text\{\; such\; that\; \}\; F\; \backslash subseteq\; C,$ in which case it is said that $\backslash mathcal\{C\}$ is {{em|coarser than}} $\backslash mathcal\{F\}$ and that $\backslash mathcal\{F\}$ is {{em|finer than}} (or {{em|subordinate to}}) $\backslash mathcal\{C\}.${{sfn|Bourbaki|1989|p=58}}{{sfn|Schubert|1968|pp=48–71}}{{sfn|Narici|Beckenstein|2011|pp=3–4}} The notation $\backslash mathcal\{F\}\; \backslash vdash\; \backslash mathcal\{C\}\; \backslash text\{\; or\; \}\; \backslash mathcal\{F\}\; \backslash geq\; \backslash mathcal\{C\}$ may also be used in place of $\backslash mathcal\{C\}\; \backslash leq\; \backslash mathcal\{F\}.$

If $\backslash mathcal\{C\}\; \backslash leq\; \backslash mathcal\{F\}$ and $\backslash mathcal\{F\}\; \backslash leq\; \backslash mathcal\{C\}$ then $\backslash mathcal\{C\}\; \backslash text\{\; and\; \}\; \backslash mathcal\{F\}$ are said to be {{em|{{visible anchor|equivalent families|equivalent|text=equivalent}}}} (with respect to subordination).

Two families $\backslash mathcal\{B\}\; \backslash text\{\; and\; \}\; \backslash mathcal\{C\}$ {{em|mesh}},{{sfn|Narici|Beckenstein|2011|pp=2–7}} written $\backslash mathcal\{B\}\; \backslash \#\; \backslash mathcal\{C\},$ if $B\; \backslash cap\; C\; \backslash neq\; \backslash varnothing\; \backslash text\{\; for\; all\; \}\; B\; \backslash in\; \backslash mathcal\{B\}\; \backslash text\{\; and\; \}\; C\; \backslash in\; \backslash mathcal\{C\}.$

}}

Throughout, $f$ is a map.

Topology notation

Denote the set of all topologies on a set $X\; \backslash text\{\; by\; \}\; \backslash operatorname\{Top\}(X).$

Suppose $\backslash tau\; \backslash in\; \backslash operatorname\{Top\}(X),$ $S\; \backslash subseteq\; X$ is any subset, and $x\; \backslash in\; X$ is any point.

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

{{anchor|Definition of nets|Definition of tails}}

Nets and their tails

A {{em|directed set}} is a set $I$ together with a preorder, which will be denoted by $\backslash ,\backslash leq\backslash ,$ (unless explicitly indicated otherwise), that makes $(I,\; \backslash leq)$ into an ({{em|upward}}) {{em|directed set}};{{sfn|Wilansky|2013|p=5}} this means that for all $i,\; j\; \backslash in\; I,$ there exists some $k\; \backslash in\; I$ such that $i\; \backslash leq\; k\; \backslash text\{\; and\; \}\; j\; \backslash leq\; k.$ For any indices $i\; \backslash text\{\; and\; \}\; j,$ the notation $j\; \backslash geq\; i$ is defined to mean $i\; \backslash leq\; j$ while is defined to mean that $i\; \backslash leq\; j$ holds but it is {{em|not}} true that $j\; \backslash leq\; i$ (if $\backslash ,\backslash leq\backslash ,$ is antisymmetric then this is equivalent to $i\; \backslash leq\; j\; \backslash text\{\; and\; \}\; i\; \backslash neq\; j$).

A {{em|net in $X$}}{{sfn|Wilansky|2013|p=5}} is a map from a non-empty directed set into $X.$

The notation $x\_\backslash bull\; =\; \backslash left(x\_i\backslash right)\_\{i\; \backslash in\; I\}$ will be used to denote a net with domain $I.$

Warning about using strict comparison

If $x\_\backslash bull\; =\; \backslash left(x\_i\backslash right)\_\{i\; \backslash in\; I\}$ is a net and $i\; \backslash in\; I$ then it is possible for the set $x\_\{>\; i\}\; =\; \backslash left\backslash \{x\_j\; ~:~\; j\; >\; i\; \backslash text\{\; and\; \}\; j\; \backslash in\; I\backslash right\backslash \},$ which is called {{em|the tail of $x\_\backslash bull$ {{em|after}} $i$}}, to be empty (for example, this happens if $i$ is an upper bound of the directed set $I$).

In this case, the family $\backslash left\backslash \{x\_\{>\; i\}\; ~:~\; i\; \backslash in\; I\backslash right\backslash \}$ would contain the empty set, which would prevent it from being a prefilter (defined later).

This is the (important) reason for defining $\backslash operatorname\{Tails\}\backslash left(x\_\backslash bull\backslash right)$ as $\backslash left\backslash \{x\_\{\backslash geq\; i\}\; ~:~\; i\; \backslash in\; I\backslash right\backslash \}$ rather than $\backslash left\backslash \{x\_\{>\; i\}\; ~:~\; i\; \backslash in\; I\backslash right\backslash \}$ or even $\backslash left\backslash \{x\_\{>\; i\}\; ~:~\; i\; \backslash in\; I\backslash right\backslash \}\backslash cup\; \backslash left\backslash \{x\_\{\backslash geq\; i\}\; ~:~\; i\; \backslash in\; I\backslash right\backslash \}$ 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 $\backslash ,\backslash leq.$

{{Families of sets}}

{{Main|Filter (set theory)}}

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

{{Quote frame|1=The family of sets $\backslash mathcal\{B\}$ is:

1. {{em|{{visible anchor|Proper|Proper filter}}}} or {{em|{{visible anchor|nondegenerate|Nondegenerate filter}}}} if $\backslash varnothing\; \backslash not\backslash in\; \backslash mathcal\{B\}.$ Otherwise, if $\backslash varnothing\; \backslash in\; \backslash mathcal\{B\},$ then it is called {{em|improper}}{{sfn|Schechter|1996|pp=100–130}} or {{em|degenerate}}.
2. {{em|{{visible anchor|Directed downward}}}}{{sfn|Wilansky|2013|p=5}} if whenever $A,\; B\; \backslash in\; \backslash mathcal\{B\}$ then there exists some $C\; \backslash in\; \backslash mathcal\{B\}$ such that $C\; \backslash subseteq\; A\; \backslash cap\; B.$
   • This property can be characterized in terms of directedness, which explains the word "directed": A binary relation $\backslash ,\backslash preceq\backslash ,$ on $\backslash mathcal\{B\}$ is called {{em|(upward) directed}} if for any two $A\; \backslash text\{\; and\; \}\; B,$ there is some $C$ satisfying $A\; \backslash preceq\; C\; \backslash text\{\; and\; \}\; B\; \backslash preceq\; C.$ Using $\backslash ,\backslash supseteq\backslash ,$ in place of $\backslash ,\backslash preceq\backslash ,$ gives the definition of {{em|directed downward}} whereas using $\backslash ,\backslash subseteq\backslash ,$ instead gives the definition of {{em|directed upward}}. Explicitly, $\backslash mathcal\{B\}$ is {{em|directed downward}} (resp. {{em|directed upward}}) if and only if for all $A,\; B\; \backslash in\; \backslash mathcal\{B\},$ there exists some "greater" $C\; \backslash in\; \backslash mathcal\{B\}$ such that $A\; \backslash supseteq\; C\; \backslash text\{\; and\; \}\; B\; \backslash supseteq\; C$ (resp. such that $A\; \backslash subseteq\; C\; \backslash text\{\; and\; \}\; B\; \backslash subseteq\; C$) − where the "greater" element is always on the right hand side, − which can be rewritten as $A\; \backslash cap\; B\; \backslash supseteq\; C$ (resp. as $A\; \backslash cup\; B\; \backslash subseteq\; C$).

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

4. {{em|Upward closed set}} or {{em|Isotone}} in $X${{sfn|Dolecki|Mynard|2016|pp=27–29}} if $\backslash mathcal\{B\}\; \backslash subseteq\; \backslash wp(X)\; \backslash text\{\; and\; \}\; \backslash mathcal\{B\}\; =\; \backslash mathcal\{B\}^\{\backslash uparrow\; X\},$ or equivalently, if whenever $B\; \backslash in\; \backslash mathcal\{B\}$ and some set $C$ satisfies $B\; \backslash subseteq\; C\; \backslash subseteq\; X,\; \backslash text\{\; then\; \}\; C\; \backslash in\; \backslash mathcal\{B\}.$ Similarly, $\backslash mathcal\{B\}$ is {{em|downward closed}} if $\backslash mathcal\{B\}\; =\; \backslash mathcal\{B\}^\{\backslash downarrow\}.$ An upward (respectively, downward) closed set is also called an {{em|upper set}} or {{em|upset}} (resp. a {{em|lower set}} or {{em|down set}}).
   • The family $\backslash mathcal\{B\}^\{\backslash uparrow\; X\},$ which is the upward closure of $\backslash mathcal\{B\}\; \backslash text\{\; in\; \}\; X,$ is the unique {{em|smallest}} (with respect to $\backslash ,\backslash subseteq$) isotone family of sets over $X$ having $\backslash mathcal\{B\}$ as a subset.

}}

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

{{Quote frame|1=A family $\backslash mathcal\{B\}$ is/is a(n):

1. {{em|Ideal (set theory)}}{{sfn|Schechter|1996|pp=100–130}}{{sfn|Császár|1978|pp=82-91}} if $\backslash mathcal\{B\}\; \backslash neq\; \backslash varnothing$ is downward closed and closed under finite unions.
2. {{em|{{visible anchor|Dual ideal}}}} on $X${{sfn|Dugundji|1966|pp=211–213}} if $\backslash mathcal\{B\}\; \backslash neq\; \backslash varnothing$ is upward closed in $X$ and also closed under finite intersections. Equivalently, $\backslash mathcal\{B\}\; \backslash neq\; \backslash varnothing$ is a dual ideal if for all $R,\; S\; \backslash subseteq\; X,$ $R\; \backslash cap\; S\; \backslash in\; \backslash mathcal\{B\}\; \backslash ;\backslash text\{\; if\; and\; only\; if\; \}\backslash ;\; R,\; S\; \backslash in\; \backslash mathcal\{B\}.${{sfn|Dolecki|Mynard|2016|pp=27-54}}
   • Explanation of the word "dual": A family $\backslash mathcal\{B\}$ is a dual ideal (resp. an ideal) on $X$ if and only if the {{anchor|Dual of a family}}{{em|dual of $\backslash mathcal\{B\}\; \backslash text\{\; in\; \}\; X,$}} which is the family $$X\; \backslash setminus\; \backslash mathcal\{B\}\; :=\; \backslash \{X\; \backslash setminus\; B\; ~:~\; B\; \backslash in\; \backslash mathcal\{B\}\backslash \},$$ is an ideal (resp. a dual ideal) on $X.$ In other words, {{em|dual ideal}} means "{{em|dual of an ideal}}". The dual of the dual is the original family, meaning $X\; \backslash setminus\; (X\; \backslash setminus\; \backslash mathcal\{B\})\; =\; \backslash mathcal\{B\}.${{sfn|Schechter|1996|pp=100–130}}

3. {{em|Filter (set theory)}} on $X${{sfn|Dugundji|1966|pp=211–213}}{{sfn|Narici|Beckenstein|2011|pp=2–7}} if $\backslash mathcal\{B\}$ is a proper dual ideal on $X.$ That is, a filter on $X$ is a non−empty subset of $\backslash wp(X)\; \backslash setminus\; \backslash \{\backslash varnothing\backslash \}$ 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.
   • {{em|Warning}}: Some authors, particularly algebrists, use "filter" to mean a dual ideal; others, particularly topologists, use "filter" to mean a {{em|proper}}/{{em|non-degenerate}} dual ideal.{{sfn|Schechter|1996|p=100}} It is recommended that readers always check how "filter" is defined when reading mathematical literature. However, the definitions of "ultrafilter," "prefilter," and "filter subbase" always require non-degeneracy. This article uses Henri Cartan's original definition of "filter",{{sfn|Cartan|1937a|p=}}{{sfn|Cartan|1937b|p=}} which required non-degeneracy.
   • The power set $\backslash wp(X)$ is the one and only dual ideal on $X$ that is not also a filter. Excluding $\backslash wp(X)$ from the definition of "filter" in topology has the same benefit as excluding $1$ from the definition of "prime number": it obviates the need to specify "non-degenerate" (the analog of "non-unital" or "non-$1$") in many important results, thereby making their statements less awkward.

4. {{em|{{visible anchor|Prefilter}}}} or {{em|filter base}}{{sfn|Narici|Beckenstein|2011|pp=2–7}}{{sfn|Császár|1978|pp=53-65, 82-91}} if $\backslash mathcal\{B\}\; \backslash neq\; \backslash varnothing$ is proper and directed downward. Equivalently, $\backslash mathcal\{B\}$ is called a prefilter if its upward closure $\backslash mathcal\{B\}^\{\backslash uparrow\; X\}$ is a filter. It can also be defined as any family that is equivalent to {{em|some}} filter.{{sfn|Császár|1978|pp=53-65}} A proper family $\backslash mathcal\{B\}\; \backslash neq\; \backslash varnothing$ is a prefilter if and only if $\backslash mathcal\{B\}\; \backslash ,(\backslash cap)\backslash ,\; \backslash mathcal\{B\}\; \backslash leq\; \backslash mathcal\{B\}.${{sfn|Császár|1978|pp=53-65}} A family is a prefilter if and only if the same is true of its upward closure.
   • If $\backslash mathcal\{B\}$ is a prefilter then its upward closure $\backslash mathcal\{B\}^\{\backslash uparrow\; X\}$ is the unique smallest (relative to $\backslash subseteq$) filter on $X$ containing $\backslash mathcal\{B\}$ and it is called {{em|the filter generated by}} $\backslash mathcal\{B\}.$ A filter $\backslash mathcal\{F\}$ is said to be {{em|generated by}} a prefilter $\backslash mathcal\{B\}$ if $\backslash mathcal\{F\}\; =\; \backslash mathcal\{B\}^\{\backslash uparrow\; X\},$ in which $\backslash mathcal\{B\}$ is called a {{em|filter base for $\backslash mathcal\{F\}.$}}
   • Unlike a filter, a prefilter is {{em|not}} necessarily closed under finite intersections.

5. {{em|{{pi}}-system}} if $\backslash mathcal\{B\}\; \backslash neq\; \backslash varnothing$ is closed under finite intersections. Every non-empty family $\backslash mathcal\{B\}$ is contained in a unique smallest {{pi}}-system called {{em|the {{pi}}-system generated by}} $\backslash mathcal\{B\},$ which is sometimes denoted by $\backslash pi(\backslash mathcal\{B\}).$ It is equal to the intersection of all {{pi}}-systems containing $\backslash mathcal\{B\}$ and also to the set of all possible finite intersections of sets from {{nowrap|$\backslash mathcal\{B\}$:}}

   $$\backslash pi(\backslash mathcal\{B\})\; =\; \backslash left\backslash \{B\_1\; \backslash cap\; \backslash cdots\; \backslash cap\; B\_n\; ~:~\; n\; \backslash geq\; 1\; \backslash text\{\; and\; \}\; B\_1,\; \backslash ldots,\; B\_n\; \backslash in\; \backslash mathcal\{B\}\backslash right\backslash \}.$$

   • A {{pi}}-system is a prefilter if and only if it is proper. Every filter is a proper {{pi}}-system and every proper {{pi}}-system is a prefilter but the converses do not hold in general.
   • A prefilter is equivalent to the {{pi}}-system generated by it and both of these families generate the same filter on $X.$

6. {{em|{{visible anchor|Filter subbase}}}}{{sfn|Narici|Beckenstein|2011|pp=2–7}}{{sfn|Arkhangel'skii|Ponomarev|1984|pp=7–8}} and {{em|centered}}{{sfn|Császár|1978|pp=53-65}} if $\backslash mathcal\{B\}\; \backslash neq\; \backslash varnothing$ and $\backslash mathcal\{B\}$ satisfies any of the following equivalent conditions:

   1. $\backslash mathcal\{B\}$ has the {{em|finite intersection property}}, which means that the intersection of any finite family of (one or more) sets in $\backslash mathcal\{B\}$ is not empty; explicitly, this means that whenever $n\; \backslash geq\; 1\; \backslash text\{\; and\; \}\; B\_1,\; \backslash ldots,\; B\_n\; \backslash in\; \backslash mathcal\{B\}$ then $\backslash varnothing\; \backslash neq\; B\_1\; \backslash cap\; \backslash cdots\; \backslash cap\; B\_n.$
   2. The {{pi}}-system generated by $\backslash mathcal\{B\}$ is proper; that is, $\backslash varnothing\; \backslash not\backslash in\; \backslash pi(\backslash mathcal\{B\}).$
   3. The {{pi}}-system generated by $\backslash mathcal\{B\}$ is a prefilter.
   4. $\backslash mathcal\{B\}$ is a subset of {{em|some}} prefilter.
   5. $\backslash mathcal\{B\}$ is a subset of {{em|some}} filter.{{sfn|Bourbaki|1989|p=58}}

   • Assume that $\backslash mathcal\{B\}$ is a filter subbase. Then there is a unique smallest (relative to $\backslash subseteq$) filter $\backslash mathcal\{F\}\_\{\backslash mathcal\{B\}\}\; \backslash text\{\; on\; \}\; X$ containing $\backslash mathcal\{B\}$ called the {{em|{{visible anchor|filter generated by}} $\backslash mathcal\{B\}$}}, and $\backslash mathcal\{B\}$ is said to {{em|be a filter subbase for}} this filter. This filter is equal to the intersection of all filters on $X$ that are supersets of $\backslash mathcal\{B\}.$ The {{pi}}-system generated by $\backslash mathcal\{B\},$ denoted by $\backslash pi(\backslash mathcal\{B\}),$ will be a prefilter and a subset of $\backslash mathcal\{F\}\_\{\backslash mathcal\{B\}\}.$ Moreover, the filter generated by $\backslash mathcal\{B\}$ is equal to the upward closure of $\backslash pi(\backslash mathcal\{B\}),$ meaning $\backslash pi(\backslash mathcal\{B\})^\{\backslash uparrow\; X\}\; =\; \backslash mathcal\{F\}\_\{\backslash mathcal\{B\}\}.${{sfn|Császár|1978|pp=53-65}} However, $\backslash mathcal\{B\}^\{\backslash uparrow\; X\}\; =\; \backslash mathcal\{F\}\_\{\backslash mathcal\{B\}\}$ if {{em|and only if}} $\backslash mathcal\{B\}$ is a prefilter (although $\backslash mathcal\{B\}^\{\backslash uparrow\; X\}$ is always an upward closed filter {{em|sub}}base for $\backslash mathcal\{F\}\_\{\backslash mathcal\{B\}\}$).
   • A $\backslash subseteq${{hsp}}-smallest (meaning smallest relative to $\backslash subseteq${{hsp}}) {{em|pre}}filter containing a filter subbase $\backslash mathcal\{B\}$ will exist only under certain circumstances. It exists, for example, if the filter subbase $\backslash mathcal\{B\}$ happens to also be a prefilter. It also exists if the filter (or equivalently, the {{pi}}-system) generated by $\backslash mathcal\{B\}$ is principal, in which case $\backslash mathcal\{B\}\; \backslash cup\; \backslash \{\backslash ker\; \backslash mathcal\{B\}\backslash \}$ is the unique smallest prefilter containing $\backslash mathcal\{B\}.$ Otherwise, in general, a $\backslash subseteq${{hsp}}-smallest {{em|pre}}filter containing $\backslash mathcal\{B\}$ might not exist. For this reason, some authors may refer to the {{pi}}-system generated by $\backslash mathcal\{B\}$ as {{em|{{visible anchor|the prefilter generated by|Prefilter generated by a filter subbase}} $\backslash mathcal\{B\}.$}} However, if a $\backslash subseteq${{hsp}}-smallest prefilter does exist (say it is denoted by $\backslash operatorname\{minPre\}\; \backslash mathcal\{B\}$) then contrary to usual expectations, it is {{em|not}} necessarily equal to "the prefilter generated by $\backslash mathcal\{B\}$" (that is, $\backslash operatorname\{minPre\}\; \backslash mathcal\{B\}\; \backslash neq\; \backslash pi(\backslash mathcal\{B\})$ is possible). And if the filter subbase $\backslash mathcal\{B\}$ happens to also be a prefilter but not a {{pi}}-system then unfortunately, "the prefilter generated by this prefilter" (meaning $\backslash pi(\backslash mathcal\{B\})$) will not be $\backslash mathcal\{B\}\; =\; \backslash operatorname\{minPre\}\; \backslash mathcal\{B\}$ (that is, $\backslash pi(\backslash mathcal\{B\})\; \backslash neq\; \backslash mathcal\{B\}$ is possible even when $\backslash mathcal\{B\}$ is a prefilter), which is why this article will prefer the accurate and unambiguous terminology of "the {{pi}}-system generated by $\backslash mathcal\{B\}$".

7. {{em|{{visible anchor|Subfilter}}}} of a filter $\backslash mathcal\{F\}$ and that $\backslash mathcal\{F\}$ is a {{em|{{visible anchor|superfilter}}}} of $\backslash mathcal\{B\}${{sfn|Schechter|1996|pp=100–130}}{{sfn|Joshi|1983|p=244}} if $\backslash mathcal\{B\}$ is a filter and $\backslash mathcal\{B\}\; \backslash subseteq\; \backslash mathcal\{F\}$ where for filters, $\backslash mathcal\{B\}\; \backslash subseteq\; \backslash mathcal\{F\}\; \backslash text\{\; if\; and\; only\; if\; \}\; \backslash mathcal\{B\}\; \backslash leq\; \backslash mathcal\{F\}.$
   • Importantly, the expression "is a {{em|super}}filter of" is for filters the analog of "is a {{em|sub}}sequence of". So despite having the prefix "sub" in common, "is a {{em|sub}}filter of" is actually the {{em|reverse}} of "is a {{em|sub}}sequence of." However, $\backslash mathcal\{B\}\; \backslash leq\; \backslash mathcal\{F\}$ can also be written $\backslash mathcal\{F\}\; \backslash vdash\; \backslash mathcal\{B\}$ which is described by saying "$\backslash mathcal\{F\}$ is subordinate to $\backslash mathcal\{B\}.$" With this terminology, "is {{em|sub}}ordinate to" becomes for filters (and also for prefilters) the analog of "is a {{em|sub}}sequence of,"{{sfn|Dugundji|1966|p=212}} which makes this one situation where using the term "subordinate" and symbol $\backslash ,\backslash vdash\backslash ,$ may be helpful.

}}

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

Named examples

• The singleton set $\backslash mathcal\{B\}\; =\; \backslash \{X\backslash \}$ is called the {{em|indiscrete}} or {{em|{{visible anchor|trivial filter}} on $X.$}}{{sfn|Wilansky|2013|pp=44–46}}{{sfn|Bourbaki|1989|pp=57–68}} It is the unique {{em|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 $X.$
• The dual ideal $\backslash wp(X)$ is also called {{em|the degenerate filter on $X$}}{{sfn|Dolecki|Mynard|2016|pp=27-54}} (despite not actually being a filter). It is the only dual ideal on $X$ that is not a filter on $X.$
• If $(X,\; \backslash tau)$ is a topological space and $x\; \backslash in\; X,$ then the neighborhood filter $\backslash mathcal\{N\}(x)$ at $x$ is a filter on $X.$ By definition, a family $\backslash mathcal\{B\}\; \backslash subseteq\; \backslash wp(X)$ is called a {{em|neighborhood basis}} (resp. a {{em|neighborhood subbase}}) at $x\; \backslash text\{\; for\; \}\; (X,\; \backslash tau)$ if and only if $\backslash mathcal\{B\}$ is a prefilter (resp. $\backslash mathcal\{B\}$ is a filter subbase) and the filter on $X$ that $\backslash mathcal\{B\}$ generates is equal to the neighborhood filter $\backslash mathcal\{N\}(x).$ The subfamily $\backslash tau(x)\; \backslash subseteq\; \backslash mathcal\{N\}(x)$ of open neighborhoods is a filter base for $\backslash mathcal\{N\}(x).$ Both prefilters $\backslash mathcal\{N\}(x)\; \backslash text\{\; and\; \}\; \backslash tau(x)$ also form a bases for topologies on $X,$ with the topology generated $\backslash tau(x)$ being coarser than $\backslash tau.$ This example immediately generalizes from neighborhoods of points to neighborhoods of non-empty subsets $S\; \backslash subseteq\; X.$
• $\backslash mathcal\{B\}$ is an {{em|{{visible anchor|elementary prefilter}}}}{{Citation|last1=Castillo|first1=Jesus M. F.|last2=Montalvo|first2=Francisco|title=A Counterexample in Semimetric Spaces|journal=Extracta Mathematicae|volume=5|issue=1|pages=38–40|date=January 1990|url=https://dialnet.unirioja.es/descarga/articulo/118277.pdf}} if $\backslash mathcal\{B\}\; =\; \backslash operatorname\{Tails\}\backslash left(x\_\backslash bull\backslash right)$ for some sequence of points $x\_\backslash bull\; =\; \backslash left(x\_i\backslash right)\_\{i=1\}^\backslash infty.$
• $\backslash mathcal\{B\}$ is an {{em|{{visible anchor|elementary filter}}}} or a {{em|{{visible anchor|sequential filter}}}} on $X${{sfn|Schaefer|Wolff|1999|pp=1–11}} if $\backslash mathcal\{B\}$ 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 {{em|not}} an ultrafilter.{{sfn|Bourbaki|1989|pp=129–133}} Every principal filter on a countable set is sequential as is every cofinite filter on a countably infinite set.{{sfn|Dolecki|Mynard|2016|pp=27-54}} The intersection of finitely many sequential filters is again sequential.{{sfn|Dolecki|Mynard|2016|pp=27-54}}
• The set $\backslash mathcal\{F\}$ of all cofinite subsets of $X$ (meaning those sets whose complement in $X$ is finite) is proper if and only if $\backslash mathcal\{F\}$ is infinite (or equivalently, $X$ is infinite), in which case $\backslash mathcal\{F\}$ is a filter on $X$ known as the Fréchet filter on $X.${{sfn|Bourbaki|1989|pp=57–68}}{{sfn|Wilansky|2013|pp=44–46}} If $X$ is finite then $\backslash mathcal\{F\}$ is equal to the dual ideal $\backslash wp(X),$ which is not a filter. If $X$ is infinite then the family $\backslash \{X\; \backslash setminus\; \backslash \{x\backslash \}\; ~:~\; x\; \backslash in\; X\backslash \}$ 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 $\backslash \{X\; \backslash setminus\; \backslash \{x\backslash \}\; ~:~\; x\; \backslash in\; X\backslash \},$ the kernel of the Fréchet filter on $X$ is the empty set: $\backslash ker\; \backslash mathcal\{F\}\; =\; \backslash varnothing.$
• The intersection of all elements in any non-empty family $\backslash mathbb\{F\}\; \backslash subseteq\; \backslash operatorname\{Filters\}(X)$ is itself a filter on $X$ called the {{em|infimum}} or {{em|greatest lower bound}} of $\backslash mathbb\{F\}\; \backslash text\{\; in\; \}\; \backslash operatorname\{Filters\}(X),$ which is why it may be denoted by $\{\backslash textstyle\backslash bigwedge\backslash limits\_\{\backslash mathcal\{F\}\; \backslash in\; \backslash mathbb\{F\}\}\}\; \backslash mathcal\{F\}.$ Said differently, $\backslash ker\; \backslash mathbb\{F\}\; =\; \{\backslash textstyle\backslash bigcap\backslash limits\_\{\backslash mathcal\{F\}\; \backslash in\; \backslash mathbb\{F\}\}\}\; \backslash mathcal\{F\}\; \backslash in\; \backslash operatorname\{Filters\}(X).$ Because every filter on $X$ has $\backslash \{X\backslash \}$ as a subset, this intersection is never empty. By definition, the infimum is the finest/largest (relative to $\backslash ,\backslash subseteq\backslash ,\; \backslash text\{\; and\; \}\; \backslash ,\backslash leq\backslash ,$) filter contained as a subset of each member of $\backslash mathbb\{F\}.${{sfn|Bourbaki|1989|pp=57–68}}
  • If $\backslash mathcal\{B\}\; \backslash text\{\; and\; \}\; \backslash mathcal\{F\}$ are filters then their infimum in $\backslash operatorname\{Filters\}(X)$ is the filter $\backslash mathcal\{B\}\; \backslash ,(\backslash cup)\backslash ,\; \backslash mathcal\{F\}.${{sfn|Császár|1978|pp=53-65}} If $\backslash mathcal\{B\}\; \backslash text\{\; and\; \}\; \backslash mathcal\{F\}$ are prefilters then $\backslash mathcal\{B\}\; \backslash ,(\backslash cup)\backslash ,\; \backslash mathcal\{F\}$ is a prefilter that is coarser than both $\backslash mathcal\{B\}\; \backslash text\{\; and\; \}\; \backslash mathcal\{F\}$ (that is, $\backslash mathcal\{B\}\; \backslash ,(\backslash cup)\backslash ,\; \backslash mathcal\{F\}\; \backslash leq\; \backslash mathcal\{B\}\; \backslash text\{\; and\; \}\; \backslash mathcal\{B\}\; \backslash ,(\backslash cup)\backslash ,\; \backslash mathcal\{F\}\; \backslash leq\; \backslash mathcal\{F\}$); indeed, it is one of the finest such prefilters, meaning that if $\backslash mathcal\{S\}$ is a prefilter such that $\backslash mathcal\{S\}\; \backslash leq\; \backslash mathcal\{B\}\; \backslash text\{\; and\; \}\; \backslash mathcal\{S\}\; \backslash leq\; \backslash mathcal\{F\}$ then necessarily $\backslash mathcal\{S\}\; \backslash leq\; \backslash mathcal\{B\}\; \backslash ,(\backslash cup)\backslash ,\; \backslash mathcal\{F\}.${{sfn|Császár|1978|pp=53-65}} More generally, if $\backslash mathcal\{B\}\; \backslash text\{\; and\; \}\; \backslash mathcal\{F\}$ are non−empty families and if $\backslash mathbb\{S\}\; :=\; \backslash \{\backslash mathcal\{S\}\; \backslash subseteq\; \backslash wp(X)\; ~:~\; \backslash mathcal\{S\}\; \backslash leq\; \backslash mathcal\{B\}\; \backslash text\{\; and\; \}\; \backslash mathcal\{S\}\; \backslash leq\; \backslash mathcal\{F\}\backslash \}$ then $\backslash mathcal\{B\}\; \backslash ,(\backslash cup)\backslash ,\; \backslash mathcal\{F\}\; \backslash in\; \backslash mathbb\{S\}$ and $\backslash mathcal\{B\}\; \backslash ,(\backslash cup)\backslash ,\; \backslash mathcal\{F\}$ is a greatest element of $(\backslash mathbb\{S\},\; \backslash leq).${{sfn|Császár|1978|pp=53-65}}

• Let $\backslash varnothing\; \backslash neq\; \backslash mathbb\{F\}\; \backslash subseteq\; \backslash operatorname\{DualIdeals\}(X)$ and let $\backslash cup\; \backslash mathbb\{F\}\; =\; \{\backslash textstyle\backslash bigcup\backslash limits\_\{\backslash mathcal\{F\}\; \backslash in\; \backslash mathbb\{F\}\}\}\; \backslash mathcal\{F\}.$

  The {{em|supremum}} or {{em|least upper bound}} of $\backslash mathbb\{F\}\; \backslash text\{\; in\; \}\; \backslash operatorname\{DualIdeals\}(X),$ denoted by $\{\backslash textstyle\backslash bigvee\backslash limits\_\{\backslash mathcal\{F\}\; \backslash in\; \backslash mathbb\{F\}\}\}\; \backslash mathcal\{F\},$ is the smallest (relative to $\backslash subseteq$) dual ideal on $X$ containing every element of $\backslash mathbb\{F\}$ as a subset; that is, it is the smallest (relative to $\backslash subseteq$) dual ideal on $X$ containing $\backslash cup\; \backslash mathbb\{F\}$ as a subset.

  This dual ideal is $\{\backslash textstyle\backslash bigvee\backslash limits\_\{\backslash mathcal\{F\}\; \backslash in\; \backslash mathbb\{F\}\}\}\; \backslash mathcal\{F\}\; =\; \backslash pi\backslash left(\backslash cup\; \backslash mathbb\{F\}\backslash right)^\{\backslash uparrow\; X\},$ where $\backslash pi\backslash left(\backslash cup\; \backslash mathbb\{F\}\backslash right)\; :=\; \backslash left\backslash \{F\_1\; \backslash cap\; \backslash cdots\; \backslash cap\; F\_n\; ~:~\; n\; \backslash in\; \backslash N\; \backslash text\{\; and\; every\; \}\; F\_i\; \backslash text\{\; belongs\; to\; some\; \}\; \backslash mathcal\{F\}\; \backslash in\; \backslash mathbb\{F\}\backslash right\backslash \}$ is the {{pi}}-system generated by $\backslash cup\; \backslash mathbb\{F\}.$

  As with any non-empty family of sets, $\backslash cup\; \backslash mathbb\{F\}$ is contained in {{em|some}} filter on $X$ if and only if it is a filter subbase, or equivalently, if and only if $\{\backslash textstyle\backslash bigvee\backslash limits\_\{\backslash mathcal\{F\}\; \backslash in\; \backslash mathbb\{F\}\}\}\; \backslash mathcal\{F\}\; =\; \backslash pi\backslash left(\backslash cup\; \backslash mathbb\{F\}\backslash right)^\{\backslash uparrow\; X\}$ is a filter on $X,$ in which case this family is the smallest (relative to $\backslash subseteq$) filter on $X$ containing every element of $\backslash mathbb\{F\}$ as a subset and necessarily $\backslash mathbb\{F\}\; \backslash subseteq\; \backslash operatorname\{Filters\}(X).$

• Let $\backslash varnothing\; \backslash neq\; \backslash mathbb\{F\}\; \backslash subseteq\; \backslash operatorname\{Filters\}(X)$ and let $\backslash cup\; \backslash mathbb\{F\}\; =\; \{\backslash textstyle\backslash bigcup\backslash limits\_\{\backslash mathcal\{F\}\; \backslash in\; \backslash mathbb\{F\}\}\}\; \backslash mathcal\{F\}.$

  The {{em|supremum}} or {{em|least upper bound}} of $\backslash mathbb\{F\}\; \backslash text\{\; in\; \}\; \backslash operatorname\{Filters\}(X),$ denoted by $\{\backslash textstyle\backslash bigvee\backslash limits\_\{\backslash mathcal\{F\}\; \backslash in\; \backslash mathbb\{F\}\}\}\; \backslash mathcal\{F\}$ if it exists, is by definition the smallest (relative to $\backslash subseteq$) filter on $X$ containing every element of $\backslash mathbb\{F\}$ as a subset.

  If it exists then necessarily $\{\backslash textstyle\backslash bigvee\backslash limits\_\{\backslash mathcal\{F\}\; \backslash in\; \backslash mathbb\{F\}\}\}\; \backslash mathcal\{F\}\; =\; \backslash pi\backslash left(\backslash cup\; \backslash mathbb\{F\}\backslash right)^\{\backslash uparrow\; X\}${{sfn|Bourbaki|1989|pp=57–68}} (as defined above) and $\{\backslash textstyle\backslash bigvee\backslash limits\_\{\backslash mathcal\{F\}\; \backslash in\; \backslash mathbb\{F\}\}\}\; \backslash mathcal\{F\}$ will also be equal to the intersection of all filters on $X$ containing $\backslash cup\; \backslash mathbb\{F\}.$

  This supremum of $\backslash mathbb\{F\}\; \backslash text\{\; in\; \}\; \backslash operatorname\{Filters\}(X)$ exists if and only if the dual ideal $\backslash pi\backslash left(\backslash cup\; \backslash mathbb\{F\}\backslash right)^\{\backslash uparrow\; X\}$ is a filter on $X.$

  The least upper bound of a family of filters $\backslash mathbb\{F\}$ may fail to be a filter.{{sfn|Bourbaki|1989|pp=57–68}} Indeed, if $X$ contains at least two distinct elements then there exist filters $\backslash mathcal\{B\}\; \backslash text\{\; and\; \}\; \backslash mathcal\{C\}\; \backslash text\{\; on\; \}\; X$ for which there does {{em|not}} exist a filter $\backslash mathcal\{F\}\; \backslash text\{\; on\; \}\; X$ that contains both $\backslash mathcal\{B\}\; \backslash text\{\; and\; \}\; \backslash mathcal\{C\}.$

  If $\backslash cup\; \backslash mathbb\{F\}$ is not a filter subbase then the supremum of $\backslash mathbb\{F\}\; \backslash text\{\; in\; \}\; \backslash operatorname\{Filters\}(X)$ does not exist and the same is true of its supremum in $\backslash operatorname\{Prefilters\}(X)$ but their supremum in the set of all dual ideals on $X$ will exist (it being the degenerate filter $\backslash wp(X)$).{{sfn|Dolecki|Mynard|2016|pp=27-54}}

  • If $\backslash mathcal\{B\}\; \backslash text\{\; and\; \}\; \backslash mathcal\{F\}$ are prefilters (resp. filters on $X$) then $\backslash mathcal\{B\}\; \backslash ,(\backslash cap)\backslash ,\; \backslash mathcal\{F\}$ is a prefilter (resp. a filter) if and only if it is non-degenerate (or said differently, if and only if $\backslash mathcal\{B\}\; \backslash text\{\; and\; \}\; \backslash mathcal\{F\}$ mesh), in which case it is {{em|one of the}} coarsest prefilters (resp. {{em|the}} coarsest filter) on $X$ that is finer (with respect to $\backslash ,\backslash leq$) than both $\backslash mathcal\{B\}\; \backslash text\{\; and\; \}\; \backslash mathcal\{F\};$ this means that if $\backslash mathcal\{S\}$ is any prefilter (resp. any filter) such that $\backslash mathcal\{B\}\; \backslash leq\; \backslash mathcal\{S\}\; \backslash text\{\; and\; \}\; \backslash mathcal\{F\}\; \backslash leq\; \backslash mathcal\{S\}$ then necessarily $\backslash mathcal\{B\}\; \backslash ,(\backslash cap)\backslash ,\; \backslash mathcal\{F\}\; \backslash leq\; \backslash mathcal\{S\},${{sfn|Császár|1978|pp=53-65}} in which case it is denoted by $\backslash mathcal\{B\}\; \backslash vee\; \backslash mathcal\{F\}.${{sfn|Dolecki|Mynard|2016|pp=27-54}}

Other examples

• Let $X\; =\; \backslash \{p,1,2,3\backslash \}$ and let $\backslash mathcal\{B\}\; =\; \backslash \{\backslash \{p\backslash \},\; \backslash \{p,1,2\backslash \},\; \backslash \{p,1,3\backslash \}\backslash \},$ which makes $\backslash mathcal\{B\}$ a prefilter and a filter subbase that is not closed under finite intersections. Because $\backslash mathcal\{B\}$ is a prefilter, the smallest prefilter containing $\backslash mathcal\{B\}$ is $\backslash mathcal\{B\}.$ The {{pi}}-system generated by $\backslash mathcal\{B\}$ is $\backslash \{\backslash \{p,1\backslash \}\backslash \}\; \backslash cup\; \backslash mathcal\{B\}.$ In particular, the smallest prefilter containing the filter subbase $\backslash mathcal\{B\}$ is {{em|not}} equal to the set of all finite intersections of sets in $\backslash mathcal\{B\}.$ The filter on $X$ generated by $\backslash mathcal\{B\}$ is $\backslash mathcal\{B\}^\{\backslash uparrow\; X\}\; =\; \backslash \{S\; \backslash subseteq\; X\; :\; p\; \backslash in\; S\backslash \}\; =\; \backslash \{\backslash \{p\backslash \}\; \backslash cup\; T\; ~:~\; T\; \backslash subseteq\; \backslash \{1,2,3\backslash \}\backslash \}.$ All three of $\backslash mathcal\{B\},$ the {{pi}}-system $\backslash mathcal\{B\}$ generates, and $\backslash mathcal\{B\}^\{\backslash uparrow\; X\}$ are examples of fixed, principal, ultra prefilters that are principal at the point $p;\; \backslash mathcal\{B\}^\{\backslash uparrow\; X\}$ is also an ultrafilter on $X.$
• Let $(X,\; \backslash tau)$ be a topological space, $\backslash mathcal\{B\}\; \backslash subseteq\; \backslash wp(X),$ and define $\backslash overline\{\backslash mathcal\{B\}\}\; :=\; \backslash left\backslash \{\backslash operatorname\{cl\}\_X\; B\; ~:~\; B\; \backslash in\; \backslash mathcal\{B\}\backslash right\backslash \},$ where $\backslash mathcal\{B\}$ is necessarily finer than $\backslash overline\{\backslash mathcal\{B\}\}.${{sfn|Wilansky|2008|pp=32-35}} If $\backslash mathcal\{B\}$ is non-empty (resp. non-degenerate, a filter subbase, a prefilter, closed under finite unions) then the same is true of $\backslash overline\{\backslash mathcal\{B\}\}.$ If $\backslash mathcal\{B\}$ is a filter on $X$ then $\backslash overline\{\backslash mathcal\{B\}\}$ is a prefilter but not necessarily a filter on $X$ although $\backslash left(\backslash overline\{\backslash mathcal\{B\}\}\backslash right)^\{\backslash uparrow\; X\}$ is a filter on $X$ equivalent to $\backslash overline\{\backslash mathcal\{B\}\}.$
• The set $\backslash mathcal\{B\}$ of all dense open subsets of a (non-empty) topological space $X$ is a proper {{pi}}-system and so also a prefilter. If the space is a Baire space, then the set of all countable intersections of dense open subsets is a {{pi}}-system and a prefilter that is finer than $\backslash mathcal\{B\}.$ If $X\; =\; \backslash R^n$ (with $1\; \backslash leq\; n\; \backslash in\; \backslash N$) then the set $\backslash mathcal\{B\}\_\{\backslash operatorname\{LebFinite\}\}$ of all $B\; \backslash in\; \backslash mathcal\{B\}$ such that $B$ has finite Lebesgue measure is a proper {{pi}}-system and a free prefilter that is also a proper subset of $\backslash mathcal\{B\}.$ The prefilters $\backslash mathcal\{B\}\_\{\backslash operatorname\{LebFinite\}\}$ and $\backslash mathcal\{B\}$ are equivalent and so generate the same filter on $X.$

  Since $X$ is a Baire space, every countable intersection of sets in $\backslash mathcal\{B\}\_\{\backslash operatorname\{LebFinite\}\}$ is dense in $X$ (and also comeagre and non-meager) so the set of all countable intersections of elements of $\backslash mathcal\{B\}\_\{\backslash operatorname\{LebFinite\}\}$ is a prefilter and {{pi}}-system; it is also finer than, and not equivalent to, $\backslash mathcal\{B\}\_\{\backslash operatorname\{LebFinite\}\}.$

{{Main|Ultrafilter (set theory)|Ultrafilter}}

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.

{{Quote frame|1=A non-empty family $\backslash mathcal\{B\}\; \backslash subseteq\; \backslash wp(X)$ of sets is/is an:

8. {{em|Ultra}}{{sfn|Narici|Beckenstein|2011|pp=2–7}}{{sfn|Dugundji|1966|pp=219–221}} if $\backslash varnothing\; \backslash not\backslash in\; \backslash mathcal\{B\}$ and any of the following equivalent conditions are satisfied:

   1. For every set $S\; \backslash subseteq\; X$ there exists some set $B\; \backslash in\; \backslash mathcal\{B\}$ such that $B\; \backslash subseteq\; S\; \backslash text\{\; or\; \}\; B\; \backslash subseteq\; X\; \backslash setminus\; S$ (or equivalently, such that $B\; \backslash cap\; S\; \backslash text\{\; equals\; \}\; B\; \backslash text\{\; or\; \}\; \backslash varnothing$).
   2. For every set $S\; \backslash subseteq\; \{\backslash textstyle\backslash bigcup\backslash limits\_\{B\; \backslash in\; \backslash mathcal\{B\}\}\}\; B$ there exists some set $B\; \backslash in\; \backslash mathcal\{B\}$ such that $B\; \backslash cap\; S\; \backslash text\{\; equals\; \}\; B\; \backslash text\{\; or\; \}\; \backslash varnothing.$
      • This characterization of "$\backslash mathcal\{B\}$ is ultra" does not depend on the set $X,$ so mentioning the set $X$ is optional when using the term "ultra."

   3. For {{em|every}} set $S$ (not necessarily even a subset of $X$) there exists some set $B\; \backslash in\; \backslash mathcal\{B\}$ such that $B\; \backslash cap\; S\; \backslash text\{\; equals\; \}\; B\; \backslash text\{\; or\; \}\; \backslash varnothing.$

9. {{em|Ultra prefilter}}{{sfn|Narici|Beckenstein|2011|pp=2–7}}{{sfn|Dugundji|1966|pp=219–221}} if it is a prefilter that is also ultra. Equivalently, it is a filter subbase that is ultra. A prefilter $\backslash mathcal\{B\}$ is ultra if and only if it satisfies any of the following equivalent conditions:

   1. $\backslash mathcal\{B\}$ is {{em|maximal}} in $\backslash operatorname\{Prefilters\}(X)$ with respect to $\backslash ,\backslash leq,\backslash ,$ which means that $$\backslash text\{For\; all\; \}\; \backslash mathcal\{C\}\; \backslash in\; \backslash operatorname\{Prefilters\}(X),\; \backslash ;\; \backslash mathcal\{B\}\; \backslash leq\; \backslash mathcal\{C\}\; \backslash ;\; \backslash text\{\; implies\; \}\; \backslash ;\; \backslash mathcal\{C\}\; \backslash leq\; \backslash mathcal\{B\}.$$
   2. $\backslash text\{For\; all\; \}\; \backslash mathcal\{C\}\; \backslash in\; \backslash operatorname\{Filters\}(X),\; \backslash ;\; \backslash mathcal\{B\}\; \backslash leq\; \backslash mathcal\{C\}\; \backslash ;\; \backslash text\{\; implies\; \}\; \backslash ;\; \backslash mathcal\{C\}\; \backslash leq\; \backslash mathcal\{B\}.$
      • Although this statement is identical to that given below for ultrafilters, here $\backslash mathcal\{B\}$ is merely assumed to be a prefilter; it need not be a filter.

   3. $\backslash mathcal\{B\}^\{\backslash uparrow\; X\}$ is ultra (and thus an ultrafilter).
   4. $\backslash mathcal\{B\}$ is equivalent to some ultrafilter.

   • A filter subbase that is ultra is necessarily a prefilter. A filter subbase is ultra if and only if it is a maximal filter subbase with respect to $\backslash ,\backslash leq\backslash ,$ (as above).{{sfn|Schechter|1996|pp=100-130}}

10. {{em|Ultrafilter on $X$}}{{sfn|Narici|Beckenstein|2011|pp=2–7}}{{sfn|Dugundji|1966|pp=219–221}} if it is a filter on $X$ that is ultra. Equivalently, an ultrafilter on $X$ is a filter $\backslash mathcal\{B\}\; \backslash text\{\; on\; \}\; X$ that satisfies any of the following equivalent conditions:

    1. $\backslash mathcal\{B\}$ is generated by an ultra prefilter.
    2. For any $S\; \backslash subseteq\; X,\; S\; \backslash in\; \backslash mathcal\{B\}\; \backslash text\{\; or\; \}\; X\; \backslash setminus\; S\; \backslash in\; \backslash mathcal\{B\}.${{sfn|Schechter|1996|pp=100–130}}
    3. $\backslash mathcal\{B\}\; \backslash cup\; (X\; \backslash setminus\; \backslash mathcal\{B\})\; =\; \backslash wp(X).$ This condition can be restated as: $\backslash wp(X)$ is partitioned by $\backslash mathcal\{B\}$ and its dual $X\; \backslash setminus\; \backslash mathcal\{B\}.$
    4. For any $R,\; S\; \backslash subseteq\; X,$ if $R\; \backslash cup\; S\; \backslash in\; \backslash mathcal\{B\}$ then $R\; \backslash in\; \backslash mathcal\{B\}\; \backslash text\{\; or\; \}\; S\; \backslash in\; \backslash mathcal\{B\}$ (a filter with this property is called a {{em|prime filter}}).
       • This property extends to any finite union of two or more sets.

    5. $\backslash mathcal\{B\}$ is a {{em|maximal}} filter on $X$; meaning that if $\backslash mathcal\{C\}$ is a filter on $X$ such that $\backslash mathcal\{B\}\; \backslash subseteq\; \backslash mathcal\{C\}$ then necessarily $\backslash mathcal\{C\}\; =\; \backslash mathcal\{B\}$ (this equality may be replaced by $\backslash mathcal\{C\}\; \backslash subseteq\; \backslash mathcal\{B\}\; \backslash text\{\; or\; by\; \}\; \backslash mathcal\{C\}\; \backslash leq\; \backslash mathcal\{B\}$).
       • If $\backslash mathcal\{C\}$ is upward closed then $\backslash mathcal\{B\}\; \backslash leq\; \backslash mathcal\{C\}\; \backslash text\{\; if\; and\; only\; if\; \}\; \backslash mathcal\{B\}\; \backslash subseteq\; \backslash mathcal\{C\}.$ So this characterization of ultrafilters as maximal filters can be restated as: $$\backslash text\{For\; all\; \}\; \backslash mathcal\{C\}\; \backslash in\; \backslash operatorname\{Filters\}(X),\; \backslash ;\; \backslash mathcal\{B\}\; \backslash leq\; \backslash mathcal\{C\}\; \backslash ;\; \backslash text\{\; implies\; \}\; \backslash ;\; \backslash mathcal\{C\}\; \backslash leq\; \backslash mathcal\{B\}.$$
       • Because subordination $\backslash ,\backslash geq\backslash ,$ is for filters the analog of "is a subnet/subsequence of" (specifically, "subnet" should mean "AA-subnet," which is defined below), this characterization of an ultrafilter as being a "maximally subordinate filter" suggests that an ultrafilter can be interpreted as being analogous to some sort of "maximally deep net" (which could, for instance, mean that "when viewed only from $X$" in some sense, it is indistinguishable from its subnets, as is the case with any net valued in a singleton set for example),For instance, one sense in which a net $u\_\backslash bull$ could be interpreted as being "maximally deep" is if all important properties related to $X$ (such as convergence for example) of any subnet is completely determined by $u\_\backslash bull$ in all topologies on $X.$ In this case $u\_\backslash bull$ and its subnet become effectively indistinguishable (at least topologically) if one's information about them is limited to only that which can be described in solely in terms of $X$ and directly related sets (such as its subsets). which is an idea that is actually made rigorous by ultranets. The ultrafilter lemma is then the statement that every filter ("net") has some subordinate filter ("subnet") that is "maximally subordinate" ("maximally deep").

}}

The ultrafilter lemma

The following important theorem is due to Alfred Tarski (1930).{{sfn|Jech|2006|pp=73-89}}

{{Math theorem|name=The ultrafilter lemma/principle/theorem{{sfn|Bourbaki|1989|pp=57–68}}|note=Tarski|math_statement=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.{{sfn|Bourbaki|1989|pp=57–68}}

Assuming the axioms of Zermelo–Fraenkel (ZF), the ultrafilter lemma follows from the Axiom of choice (in particular from Zorn's lemma) but is strictly weaker than it. The ultrafilter lemma implies the Axiom of choice for finite sets. If {{em|only}} dealing with Hausdorff spaces, then most basic results (as encountered in introductory courses) in Topology (such as Tychonoff's theorem for compact Hausdorff spaces and the Alexander subbase theorem) and in functional analysis (such as the Hahn–Banach theorem) can be proven using only the ultrafilter lemma; the full strength of the axiom of choice might not be needed.

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

{{Quote frame|1=The {{em|kernel}}{{sfn|Dolecki|Mynard|2016|pp=27–29}} of a family of sets $\backslash mathcal\{B\}$ is the intersection of all sets that are elements of $\backslash mathcal\{B\}:$

$$\backslash ker\; \backslash mathcal\{B\}\; =\; \backslash bigcap\_\{B\; \backslash in\; \backslash mathcal\{B\}\}\; B$$

}}

If $\backslash mathcal\{B\}\; \backslash subseteq\; \backslash wp(X)$ then $\backslash ker\; \backslash left(\backslash mathcal\{B\}^\{\backslash uparrow\; X\}\backslash right)\; =\; \backslash ker\; \backslash mathcal\{B\}$ and this set is also equal to the kernel of the {{pi}}-system that is generated by $\backslash mathcal\{B\}.$

In particular, if $\backslash mathcal\{B\}$ is a filter subbase then the kernels of all of the following sets are equal:

:(1) $\backslash mathcal\{B\},$ (2) the {{pi}}-system generated by $\backslash mathcal\{B\},$ and (3) the filter generated by $\backslash mathcal\{B\}.$

If $f$ is a map then $f(\backslash ker\; \backslash mathcal\{B\})\; \backslash subseteq\; \backslash ker\; f(\backslash mathcal\{B\})\; \backslash text\{\; and\; \}\; f^\{-1\}(\backslash ker\; \backslash mathcal\{B\})\; =\; \backslash ker\; f^\{-1\}(\backslash mathcal\{B\}).$

Equivalent families have equal kernels.

Two principal families are equivalent if and only if their kernels are equal.

{{Quote frame|1=A family $\backslash mathcal\{B\}$ of sets is:

2. {{em|{{visible anchor|Free}}}}{{sfn|Dolecki|Mynard|2016|pp=33–35}} if $\backslash ker\; \backslash mathcal\{B\}\; =\; \backslash varnothing,$ or equivalently, if $\backslash \{X\; \backslash setminus\; \backslash \{x\backslash \}\; ~:~\; x\; \backslash in\; X\backslash \}\; \backslash subseteq\; \backslash mathcal\{B\}^\{\backslash uparrow\; X\};$ this can be restated as $\backslash \{X\; \backslash setminus\; \backslash \{x\backslash \}\; ~:~\; x\; \backslash in\; X\backslash \}\; \backslash leq\; \backslash mathcal\{B\}.$
   • A filter $\backslash mathcal\{F\}\; \backslash text\{\; on\; \}\; X$ is free if and only if $X$ is infinite and $\backslash mathcal\{F\}$ includes the Fréchet filter on $X$ as a subset.

3. {{em|{{visible anchor|Fixed}}}} if $\backslash ker\; \backslash mathcal\{B\}\; \backslash neq\; \backslash varnothing$ in which case, $\backslash mathcal\{B\}$ is said to be {{em|fixed by}} any point $x\; \backslash in\; \backslash ker\; \backslash mathcal\{B\}.$
   • Any fixed family is necessarily a filter subbase.

4. {{em|{{visible anchor|Principal}}}}{{sfn|Dolecki|Mynard|2016|pp=33–35}} if $\backslash ker\; \backslash mathcal\{B\}\; \backslash in\; \backslash mathcal\{B\}.$
   • A proper principal family of sets is necessarily a prefilter.

5. {{em|{{visible anchor|Discrete}}}} or {{em|Principal at}} $x\; \backslash in\; X${{sfn|Wilansky|2013|pp=44–46}} if $\backslash \{x\backslash \}\; =\; \backslash ker\; \backslash mathcal\{B\}\; \backslash in\; \backslash mathcal\{B\}.$
   • The {{em|principal filter at $x\; \backslash text\{\; on\; \}\; X$}} is the filter $\backslash \{x\backslash \}^\{\backslash uparrow\; X\}.$ A filter $\backslash mathcal\{F\}$ is principal at $x$ if and only if $\backslash mathcal\{F\}\; =\; \backslash \{x\backslash \}^\{\backslash uparrow\; X\}.$

6. {{em|Countably deep}} if whenever $\backslash mathcal\{C\}\; \backslash subseteq\; \backslash mathcal\{B\}$ is a countable subset then $\backslash ker\; \backslash mathcal\{C\}\; \backslash in\; \backslash mathcal\{B\}.${{sfn|Dolecki|Mynard|2016|pp=27-54}}

}}

If $\backslash mathcal\{B\}$ is a principal filter on $X$ then $\backslash varnothing\; \backslash neq\; \backslash ker\; \backslash mathcal\{B\}\; \backslash in\; \backslash mathcal\{B\}$ and

$\backslash mathcal\{B\}\; =\; \backslash \{\backslash ker\; \backslash mathcal\{B\}\backslash \}^\{\backslash uparrow\; X\}$

and $\backslash \{\backslash ker\; \backslash mathcal\{B\}\backslash \}$ is also the smallest prefilter that generates $\backslash mathcal\{B\}.$

Family of examples: For any non-empty $C\; \backslash subseteq\; \backslash R,$ the family $\backslash mathcal\{B\}\_C\; =\; \backslash \{\backslash R\; \backslash setminus\; (r\; +\; C)\; ~:~\; r\; \backslash in\; \backslash R\backslash \}$ is free but it is a filter subbase if and only if no finite union of the form $\backslash left(r\_1\; +\; C\backslash right)\; \backslash cup\; \backslash cdots\; \backslash cup\; \backslash left(r\_n\; +\; C\backslash right)$ covers $\backslash R,$ in which case the filter that it generates will also be free. In particular, $\backslash mathcal\{B\}\_C$ is a filter subbase if $C$ is countable (for example, $C\; =\; \backslash Q,\; \backslash Z,$ the primes), a meager set in $\backslash R,$ a set of finite measure, or a bounded subset of $\backslash R.$ If $C$ is a singleton set then $\backslash mathcal\{B\}\_C$ is a subbase for the Fréchet filter on $\backslash R.$

If a family of sets $\backslash mathcal\{B\}$ is fixed (that is, $\backslash ker\; \backslash mathcal\{B\}\; \backslash neq\; \backslash varnothing$) then $\backslash mathcal\{B\}$ is ultra if and only if some element of $\backslash mathcal\{B\}$ is a singleton set, in which case $\backslash mathcal\{B\}$ will necessarily be a prefilter. Every principal prefilter is fixed, so a principal prefilter $\backslash mathcal\{B\}$ is ultra if and only if $\backslash ker\; \backslash mathcal\{B\}$ 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.{{sfn|Dolecki|Mynard|2016|pp=33–35}}

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.

{{math theorem|name=Proposition|math_statement=If $\backslash mathcal\{F\}$ is an ultrafilter on $X$ then the following are equivalent:

1. $\backslash mathcal\{F\}$ is fixed, or equivalently, not free, meaning $\backslash ker\; \backslash mathcal\{F\}\; \backslash neq\; \backslash varnothing.$
2. $\backslash mathcal\{F\}$ is principal, meaning $\backslash ker\; \backslash mathcal\{F\}\; \backslash in\; \backslash mathcal\{F\}.$
3. Some element of $\backslash mathcal\{F\}$ is a finite set.
4. Some element of $\backslash mathcal\{F\}$ is a singleton set.
5. $\backslash mathcal\{F\}$ is principal at some point of $X,$ which means $\backslash ker\; \backslash mathcal\{F\}\; =\; \backslash \{x\backslash \}\; \backslash in\; \backslash mathcal\{F\}$ for some $x\; \backslash in\; X.$
6. $\backslash mathcal\{F\}$ does {{em|not}} contain the Fréchet filter on $X.$
7. $\backslash mathcal\{F\}$ is sequential.{{sfn|Dolecki|Mynard|2016|pp=27-54}}

}}

{{anchor|Comparison of families|Comparison of families of sets|Subordination}}

The preorder $\backslash ,\backslash leq\backslash ,$ 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",{{sfn|Dugundji|1966|p=212}} where "$\backslash mathcal\{F\}\; \backslash geq\; \backslash mathcal\{C\}$" can be interpreted as "$\backslash mathcal\{F\}$ is a subsequence of $\backslash mathcal\{C\}$" (so "subordinate to" is the prefilter equivalent of "subsequence of"). It is also used to define prefilter convergence in a topological space.

The definition of $\backslash mathcal\{B\}$ meshes with $\backslash mathcal\{C\},$ which is closely related to the preorder $\backslash ,\backslash leq,$ is used in topology to define cluster points.

Two families of sets $\backslash mathcal\{B\}\; \backslash text\{\; and\; \}\; \backslash mathcal\{C\}$ {{em|{{visible anchor|mesh}}}}{{sfn|Narici|Beckenstein|2011|pp=2–7}} and are {{em|compatible}}, indicated by writing $\backslash mathcal\{B\}\; \backslash \#\; \backslash mathcal\{C\},$ if $B\; \backslash cap\; C\; \backslash neq\; \backslash varnothing\; \backslash text\{\; for\; all\; \}\; B\; \backslash in\; \backslash mathcal\{B\}\; \backslash text\{\; and\; \}\; C\; \backslash in\; \backslash mathcal\{C\}.$ If $\backslash mathcal\{B\}\; \backslash text\{\; and\; \}\; \backslash mathcal\{C\}$ do not mesh then they are {{em|dissociated}}. If $S\; \backslash subseteq\; X\; \backslash text\{\; and\; \}\; \backslash mathcal\{B\}\; \backslash subseteq\; \backslash wp(X)$ then $\backslash mathcal\{B\}\; \backslash text\{\; and\; \}\; S$ are said to {{em|mesh}} if $\backslash mathcal\{B\}\; \backslash text\{\; and\; \}\; \backslash \{S\backslash \}$ mesh, or equivalently, if the {{em|{{visible anchor|trace|Trace on a set}}}} of $\backslash mathcal\{B\}\; \backslash text\{\; on\; \}\; S,$ which is the family

$$\backslash mathcal\{B\}\backslash big\backslash vert\_S\; =\; \backslash \{B\; \backslash cap\; S\; ~:~\; B\; \backslash in\; \backslash mathcal\{B\}\backslash \},$$

does not contain the empty set, where the trace is also called the {{em|{{visible anchor|restriction|Restriction to a set}}}} of $\backslash mathcal\{B\}\; \backslash text\{\; to\; \}\; S.$

{{Quote frame|1=Declare that $\backslash mathcal\{C\}\; \backslash leq\; \backslash mathcal\{F\},\; \backslash mathcal\{F\}\; \backslash geq\; \backslash mathcal\{C\},\; \backslash text\{\; and\; \}\; \backslash mathcal\{F\}\; \backslash vdash\; \backslash mathcal\{C\},$ stated as $\backslash mathcal\{C\}$ is {{em|coarser than}} $\backslash mathcal\{F\}$ and $\backslash mathcal\{F\}$ is {{em|finer than}} (or {{em|subordinate to}}) $\backslash mathcal\{C\},${{sfn|Bourbaki|1989|pp=57–68}}{{sfn|Schubert|1968|pp=48–71}}{{sfn|Narici|Beckenstein|2011|pp=3–4}}{{sfn|Császár|1978|pp=53-65}}{{sfn|Dolecki|Mynard|2016|pp=27-54}} if any of the following equivalent conditions hold:

1. Definition: Every $C\; \backslash in\; \backslash mathcal\{C\}$ {{em|includes}} some $F\; \backslash in\; \backslash mathcal\{F\}.$ Explicitly, this means that for every $C\; \backslash in\; \backslash mathcal\{C\},$ there is some $F\; \backslash in\; \backslash mathcal\{F\}$ such that $F\; \backslash subseteq\; C$ (thus $\backslash mathcal\{C\}\; \backslash ni\; C\; \backslash supseteq\; F\; \backslash in\; \backslash mathcal\{F\}$ holds).
   • Said more briefly in plain English, $\backslash mathcal\{C\}\; \backslash leq\; \backslash mathcal\{F\}$ if every set in $\backslash mathcal\{C\}$ is {{em|larger}} than some set in $\backslash mathcal\{F\}.$ Here, a "larger set" means a superset.

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

3. $\backslash mathcal\{C\}\; \backslash leq\; \backslash mathcal\{F\}^\{\backslash uparrow\; X\},$ which is equivalent to $\backslash mathcal\{C\}\; \backslash subseteq\; \backslash mathcal\{F\}^\{\backslash uparrow\; X\}$;
4. $\backslash mathcal\{C\}^\{\backslash uparrow\; X\}\; \backslash leq\; \backslash mathcal\{F\}$;
5. $\backslash mathcal\{C\}^\{\backslash uparrow\; X\}\; \backslash leq\; \backslash mathcal\{F\}^\{\backslash uparrow\; X\},$ which is equivalent to $\backslash mathcal\{C\}^\{\backslash uparrow\; X\}\; \backslash subseteq\; \backslash mathcal\{F\}^\{\backslash uparrow\; X\}$;

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

6. $\backslash mathcal\{C\}\; \backslash subseteq\; \backslash mathcal\{F\}.${{sfn|Dolecki|Mynard|2016|pp=27–29}}
   • So in this case, this definition of "$\backslash mathcal\{F\}$ is {{em|finer}} than $\backslash mathcal\{C\}$" would be identical to the topological definition of "finer" had $\backslash mathcal\{C\}\; \backslash text\{\; and\; \}\; \backslash mathcal\{F\}$ been topologies on $X.$

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

Two families are {{em|comparable}} if one of them is finer than the other.{{sfn|Bourbaki|1989|pp=57–68}}

}}

Example: If $x\_\{i\_\backslash bull\}\; =\; \backslash left(x\_\{i\_n\}\backslash right)\_\{n=1\}^\backslash infty$ is a subsequence of $x\_\backslash bull\; =\; \backslash left(x\_i\backslash right)\_\{i=1\}^\backslash infty$ then $\backslash operatorname\{Tails\}\backslash left(x\_\{i\_\backslash bull\}\backslash right)$ is subordinate to $\backslash operatorname\{Tails\}\backslash left(x\_\backslash bull\backslash right);$ in symbols: $\backslash operatorname\{Tails\}\backslash left(x\_\{i\_\backslash bull\}\backslash right)\; \backslash vdash\; \backslash operatorname\{Tails\}\backslash left(x\_\backslash bull\backslash right)$ and also $\backslash operatorname\{Tails\}\backslash left(x\_\backslash bull\backslash right)\; \backslash leq\; \backslash operatorname\{Tails\}\backslash left(x\_\{i\_\backslash bull\}\backslash right).$

Stated in plain English, the prefilter of tails of a subsequence is always subordinate to that of the original sequence.

To see this, let $C\; :=\; x\_\{\backslash geq\; i\}\; \backslash in\; \backslash operatorname\{Tails\}\backslash left(x\_\backslash bull\backslash right)$ be arbitrary (or equivalently, let $i\; \backslash in\; \backslash N$ be arbitrary) and it remains to show that this set contains some $F\; :=\; x\_\{i\_\{\backslash geq\; n\}\}\; \backslash in\; \backslash operatorname\{Tails\}\backslash left(x\_\{i\_\backslash bull\}\backslash right).$

For the set $x\_\{\backslash geq\; i\}\; =\; \backslash left\backslash \{x\_i,\; x\_\{i+1\},\; \backslash ldots\backslash right\backslash \}$ to contain $x\_\{i\_\{\backslash geq\; n\}\}\; =\; \backslash left\backslash \{x\_\{i\_n\},\; x\_\{i\_\{n+1\}\},\; \backslash ldots\backslash right\backslash \},$ it is sufficient to have $i\; \backslash leq\; i\_n.$

Since are strictly increasing integers, there exists $n\; \backslash in\; \backslash N$ such that $i\_n\; \backslash geq\; i,$ and so $x\_\{\backslash geq\; i\}\; \backslash supseteq\; x\_\{i\_\{\backslash geq\; n\}\}$ holds, as desired.

Consequently, $\backslash operatorname\{TailsFilter\}\backslash left(x\_\backslash bull\backslash right)\; \backslash subseteq\; \backslash operatorname\{TailsFilter\}\backslash left(x\_\{i\_\backslash bull\}\backslash right).$

The left hand side will be a {{em|strict/proper}} subset of the right hand side if (for instance) every point of $x\_\backslash bull$ is unique (that is, when $x\_\backslash bull\; :\; \backslash N\; \backslash to\; X$ is injective) and $x\_\{i\_\backslash bull\}$ is the even-indexed subsequence $\backslash left(x\_2,\; x\_4,\; x\_6,\; \backslash ldots\backslash right)$ because under these conditions, every tail $x\_\{i\_\{\backslash geq\; n\}\}\; =\; \backslash left\backslash \{x\_\{2n\},\; x\_\{2n\; +\; 2\},\; x\_\{2n\; +\; 4\},\; \backslash ldots\backslash right\backslash \}$ (for every $n\; \backslash in\; \backslash N$) of the subsequence will belong to the right hand side filter but not to the left hand side filter.

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

A non-empty family that is coarser than a filter subbase must itself be a filter subbase.{{sfn|Császár|1978|pp=53-65}}

Every filter subbase is coarser than both the {{pi}}-system that it generates and the filter that it generates.{{sfn|Császár|1978|pp=53-65}}

If $\backslash mathcal\{C\}\; \backslash text\{\; and\; \}\; \backslash mathcal\{F\}$ are families such that $\backslash mathcal\{C\}\; \backslash leq\; \backslash mathcal\{F\},$ the family $\backslash mathcal\{C\}$ is ultra, and $\backslash varnothing\; \backslash not\backslash in\; \backslash mathcal\{F\},$ then $\backslash mathcal\{F\}$ is necessarily ultra. It follows that any family that is equivalent to an ultra family will necessarily {{em|be}} ultra. In particular, if $\backslash mathcal\{C\}$ is a prefilter then either both $\backslash mathcal\{C\}$ and the filter $\backslash mathcal\{C\}^\{\backslash uparrow\; X\}$ it generates are ultra or neither one is ultra.

The relation $\backslash ,\backslash leq\backslash ,$ is reflexive and transitive, which makes it into a preorder on $\backslash wp(\backslash wp(X)).${{sfn|Császár|1978|pp=53-65, 82-91, 102-120}}

The relation $\backslash ,\backslash leq\backslash ,\; \backslash text\{\; on\; \}\; \backslash operatorname\{Filters\}(X)$ is antisymmetric but if $X$ has more than one point then it is {{em|not}} symmetric.

The preorder $\backslash ,\backslash leq\backslash ,$ induces its canonical equivalence relation on $\backslash wp(\backslash wp(X)),$ where for all $\backslash mathcal\{B\},\; \backslash mathcal\{C\}\; \backslash in\; \backslash wp(\backslash wp(X)),$ $\backslash mathcal\{B\}$ is {{em|equivalent}} to $\backslash mathcal\{C\}$ if any of the following equivalent conditions hold:{{sfn|Császár|1978|pp=53-65}}{{sfn|Dolecki|Mynard|2016|pp=27–29}}

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

Two upward closed (in $X$) subsets of $\backslash wp(X)$ are equivalent if and only if they are equal.{{sfn|Császár|1978|pp=53-65}}

If $\backslash mathcal\{B\}\; \backslash subseteq\; \backslash wp(X)$ then necessarily $\backslash varnothing\; \backslash leq\; \backslash mathcal\{B\}\; \backslash leq\; \backslash wp(X)$ and $\backslash mathcal\{B\}$ is equivalent to $\backslash mathcal\{B\}^\{\backslash uparrow\; X\}.$

Every equivalence class other than $\backslash \{\backslash varnothing\backslash \}$ contains a unique representative (that is, element of the equivalence class) that is upward closed in $X.${{sfn|Császár|1978|pp=53-65}}

Properties preserved between equivalent families

Let $\backslash mathcal\{B\},\; \backslash mathcal\{C\}\; \backslash in\; \backslash wp(\backslash wp(X))$ be arbitrary and let $\backslash mathcal\{F\}$ be any family of sets. If $\backslash mathcal\{B\}\; \backslash text\{\; and\; \}\; \backslash mathcal\{C\}$ are equivalent (which implies that $\backslash ker\; \backslash mathcal\{B\}\; =\; \backslash ker\; \backslash mathcal\{C\}$) then for each of the statements/properties listed below, either it is true of {{em|both}} $\backslash mathcal\{B\}\; \backslash text\{\; and\; \}\; \backslash mathcal\{C\}$ or else it is false of {{em|both}} $\backslash mathcal\{B\}\; \backslash text\{\; and\; \}\; \backslash mathcal\{C\}$:{{sfn|Császár|1978|pp=53-65, 82-91, 102-120}}

1. Not empty
2. Proper (that is, $\backslash varnothing$ is not an element)
   • Moreover, any two degenerate families are necessarily equivalent.

3. Filter subbase
4. Prefilter
   • In which case $\backslash mathcal\{B\}\; \backslash text\{\; and\; \}\; \backslash mathcal\{C\}$ generate the same filter on $X$ (that is, their upward closures in $X$ are equal).

5. Free
6. Principal
7. Ultra
8. Is equal to the trivial filter $\backslash \{X\backslash \}$
   • In words, this means that the only subset of $\backslash wp(X)$ that is equivalent to the trivial filter {{em|is}} the trivial filter. In general, this conclusion of equality does not extend to non−trivial filters (one exception is when both families are filters).

9. Meshes with $\backslash mathcal\{F\}$
10. Is finer than $\backslash mathcal\{F\}$
11. Is coarser than $\backslash mathcal\{F\}$
12. Is equivalent to $\backslash mathcal\{F\}$

Missing from the above list is the word "filter" because this property is {{em|not}} preserved by equivalence.

However, if $\backslash mathcal\{B\}\; \backslash text\{\; and\; \}\; \backslash mathcal\{C\}$ are filters on $X,$ then they are equivalent if and only if they are equal; this characterization does {{em|not}} extend to prefilters.

Equivalence of prefilters and filter subbases

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

1. $\backslash mathcal\{B\}$;
2. the {{pi}}-system generated by $\backslash mathcal\{B\}$;
3. the filter on $X$ generated by $\backslash mathcal\{B\}$;

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.{{sfn|Császár|1978|pp=53-65}}

Every prefilter is equivalent to exactly one filter on $X,$ which is the filter that it generates (that is, the prefilter's upward closure).

Said differently, every equivalence class of prefilters contains exactly one representative that is a filter.

In this way, filters can be considered as just being distinguished elements of these equivalence classes of prefilters.{{sfn|Császár|1978|pp=53-65}}

A filter subbase that is {{em|not}} also a prefilter can{{em|not}} 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.

{{See also|Filter (set theory)}}

If $\backslash mathcal\{B\}$ is a prefilter (resp. filter) on $X\; \backslash text\{\; and\; \}\; S\; \backslash subseteq\; X$ then the trace of $\backslash mathcal\{B\}\; \backslash text\{\; on\; \}\; S,$ which is the family $\backslash mathcal\{B\}\backslash big\backslash vert\_S\; :=\; \backslash mathcal\{B\}\; (\backslash cap)\; \backslash \{S\backslash \},$ is a prefilter (resp. a filter) if and only if $\backslash mathcal\{B\}\; \backslash text\{\; and\; \}\; S$ mesh (that is, $\backslash varnothing\; \backslash not\backslash in\; \backslash mathcal\{B\}\; (\backslash cap)\; \backslash \{S\backslash \}${{sfn|Bourbaki|1989|pp=57–68}}), in which case the trace of $\backslash mathcal\{B\}\; \backslash text\{\; on\; \}\; S$ is said to be {{em|induced by $S$}}.

The trace is always finer than the original family; that is, $\backslash mathcal\{B\}\; \backslash leq\; \backslash mathcal\{B\}\backslash big\backslash vert\_S.$

If $\backslash mathcal\{B\}$ is ultra and if $\backslash mathcal\{B\}\; \backslash text\{\; and\; \}\; S$ mesh then the trace $\backslash mathcal\{B\}\backslash big\backslash vert\_S$ is ultra.

If $\backslash mathcal\{B\}$ is an ultrafilter on $X$ then the trace of $\backslash mathcal\{B\}\; \backslash text\{\; on\; \}\; S$ is a filter on $S$ if and only if $S\; \backslash in\; \backslash mathcal\{B\}.$

For example, suppose that $\backslash mathcal\{B\}$ is a filter on $X\; \backslash text\{\; and\; \}\; S\; \backslash subseteq\; X$ is such that $S\; \backslash neq\; X\; \backslash text\{\; and\; \}\; X\; \backslash setminus\; S\; \backslash not\backslash in\; \backslash mathcal\{B\}.$ Then $\backslash mathcal\{B\}\; \backslash text\{\; and\; \}\; S$ mesh and $\backslash mathcal\{B\}\; \backslash cup\; \backslash \{S\backslash \}$ generates a filter on $X$ that is strictly finer than $\backslash mathcal\{B\}.${{sfn|Bourbaki|1989|pp=57–68}}

When prefilters mesh

Given non-empty families $\backslash mathcal\{B\}\; \backslash text\{\; and\; \}\; \backslash mathcal\{C\},$ the family

$$\backslash mathcal\{B\}\; (\backslash cap)\; \backslash mathcal\{C\}\; :=\; \backslash \{B\; \backslash cap\; C\; ~:~\; B\; \backslash in\; \backslash mathcal\{B\}\; \backslash text\{\; and\; \}\; C\; \backslash in\; \backslash mathcal\{C\}\backslash \}$$

satisfies $\backslash mathcal\{C\}\; \backslash leq\; \backslash mathcal\{B\}\; (\backslash cap)\; \backslash mathcal\{C\}$ and $\backslash mathcal\{B\}\; \backslash leq\; \backslash mathcal\{B\}\; (\backslash cap)\; \backslash mathcal\{C\}.$

If $\backslash mathcal\{B\}\; (\backslash cap)\; \backslash mathcal\{C\}$ is proper (resp. a prefilter, a filter subbase) then this is also true of both $\backslash mathcal\{B\}\; \backslash text\{\; and\; \}\; \backslash mathcal\{C\}.$

In order to make any meaningful deductions about $\backslash mathcal\{B\}\; (\backslash cap)\; \backslash mathcal\{C\}$ from $\backslash mathcal\{B\}\; \backslash text\{\; and\; \}\; \backslash mathcal\{C\},\; \backslash mathcal\{B\}\; (\backslash cap)\; \backslash mathcal\{C\}$ needs to be proper (that is, $\backslash varnothing\; \backslash not\backslash in\; \backslash mathcal\{B\}\; (\backslash cap)\; \backslash mathcal\{C\},$ which is the motivation for the definition of "mesh".

In this case, $\backslash mathcal\{B\}\; (\backslash cap)\; \backslash mathcal\{C\}$ is a prefilter (resp. filter subbase) if and only if this is true of both $\backslash mathcal\{B\}\; \backslash text\{\; and\; \}\; \backslash mathcal\{C\}.$

Said differently, if $\backslash mathcal\{B\}\; \backslash text\{\; and\; \}\; \backslash mathcal\{C\}$ are prefilters then they mesh if and only if $\backslash mathcal\{B\}\; (\backslash cap)\; \backslash mathcal\{C\}$ is a prefilter.

Generalizing gives a well known characterization of "mesh" entirely in terms of subordination (that is, $\backslash ,\backslash leq\backslash ,$):

{{in5}}Two prefilters (resp. filter subbases) $\backslash mathcal\{B\}\; \backslash text\{\; and\; \}\; \backslash mathcal\{C\}$ mesh if and only if there exists a prefilter (resp. filter subbase) $\backslash mathcal\{F\}$ such that $\backslash mathcal\{C\}\; \backslash leq\; \backslash mathcal\{F\}$ and $\backslash mathcal\{B\}\; \backslash leq\; \backslash mathcal\{F\}.$

If the least upper bound of two filters $\backslash mathcal\{B\}\; \backslash text\{\; and\; \}\; \backslash mathcal\{C\}$ exists in $\backslash operatorname\{Filters\}(X)$ then this least upper bound is equal to $\backslash mathcal\{B\}\; (\backslash cap)\; \backslash mathcal\{C\}.${{sfn|Dolecki|Mynard|2016|pp=31–32}}

{{See also|List of set identities and relations|Algebra of sets}}

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

Images of prefilters

Let $\backslash mathcal\{B\}\; \backslash subseteq\; \backslash wp(Y).$ Many of the properties that $\backslash mathcal\{B\}$ 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 $\backslash mathcal\{B\}\; \backslash text\{\; on\; \}\; Y,$ then it will necessarily also be true of $g(\backslash mathcal\{B\})\; \backslash text\{\; on\; \}\; g(Y)$ (although possibly not on the codomain $Z$ unless $g$ is surjective):{{sfn|Bourbaki|1989|pp=57–68}}{{sfn|Dugundji|1966|pp=215–221}}{{sfn|Dolecki|Mynard|2016|pp=37–39}}{{sfn|Arkhangel'skii|Ponomarev|1984|pp=20–22}}{{sfn|Császár|1978|pp=102-120}}{{sfn|Jech|2006|pp=73-89}}

ultra, ultrafilter, filter, prefilter, filter subbase, dual ideal, upward closed, proper/non-degenerate, ideal, closed under finite unions, downward closed, directed upward.

Moreover, if $\backslash mathcal\{B\}\; \backslash subseteq\; \backslash wp(Y)$ is a prefilter then so are both $g(\backslash mathcal\{B\})\; \backslash text\{\; and\; \}\; g^\{-1\}(g(\backslash mathcal\{B\})).${{sfn|Bourbaki|1989|pp=57–68}}

The image under a map $f\; :\; X\; \backslash to\; Y$ of an ultra set $\backslash mathcal\{B\}\; \backslash subseteq\; \backslash wp(X)$ is again ultra and if $\backslash mathcal\{B\}$ is an ultra prefilter then so is $f(\backslash mathcal\{B\}).$

If $\backslash mathcal\{B\}$ is a filter then $g(\backslash mathcal\{B\})$ is a filter on the range $g(Y),$ but it is a filter on the codomain $Z$ if and only if $g$ is surjective.{{sfn|Dolecki|Mynard|2016|pp=37–39}}

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 $g(\backslash mathcal\{B\})\; \backslash text\{\; in\; \}\; Z$ is

$$g(\backslash mathcal\{B\})^\{\backslash uparrow\; Z\}\; =\; \backslash left\backslash \{S\; \backslash subseteq\; Z\; ~:~\; B\; \backslash subseteq\; g^\{-1\}(S)\; \backslash text\{\; for\; some\; \}\; B\; \backslash in\; \backslash mathcal\{B\}\backslash right\backslash \}$$

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

$$g(\backslash mathcal\{B\})^\{\backslash uparrow\; Z\}\; =\; \backslash left\backslash \{S\; \backslash subseteq\; Z\; ~:~\; g^\{-1\}(S)\; \backslash in\; \backslash mathcal\{B\}\backslash right\backslash \}.$$

If $X\; \backslash subseteq\; Y$ then taking $g$ to be the inclusion map $X\; \backslash to\; Y$ shows that any prefilter (resp. ultra prefilter, filter subbase) on $X$ is also a prefilter (resp. ultra prefilter, filter subbase) on $Y.${{sfn|Bourbaki|1989|pp=57–68}}

Preimages of prefilters

Let $\backslash mathcal\{B\}\; \backslash subseteq\; \backslash wp(Y).$

Under the assumption that $f\; :\; X\; \backslash to\; Y$ is surjective:

{{in5}}$f^\{-1\}(\backslash mathcal\{B\})$ is a prefilter (resp. filter subbase, {{pi}}-system, closed under finite unions, proper) if and only if this is true of $\backslash mathcal\{B\}.$

However, if $\backslash mathcal\{B\}$ is an ultrafilter on $Y$ then even if $f$ is surjective (which would make $f^\{-1\}(\backslash mathcal\{B\})$ a prefilter), it is nevertheless still possible for the prefilter $f^\{-1\}(\backslash mathcal\{B\})$ to be neither ultra nor a filter on $X.${{hsp}}{{sfn|Arkhangel'skii|Ponomarev|1984|pp=20–22}}

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

$$\backslash mathcal\{B\}\backslash big\backslash vert\_\{f(X)\}\; =\; f\backslash left(f^\{-1\}(\backslash mathcal\{B\})\backslash right)$$

and consequently also:

$$f^\{-1\}(\backslash mathcal\{B\})\; =\; f^\{-1\}\backslash left(\backslash mathcal\{B\}\backslash big\backslash vert\_\{f(X)\}\backslash right).$$

This last equality and the fact that the trace $\backslash mathcal\{B\}\backslash big\backslash vert\_\{f(X)\}$ is a family of sets over $f(X)$ means that to draw conclusions about $f^\{-1\}(\backslash mathcal\{B\}),$ the trace $\backslash mathcal\{B\}\backslash big\backslash vert\_\{f(X)\}$ can be used in place of $\backslash mathcal\{B\}$ and the {{em|surjection}} $f\; :\; X\; \backslash to\; f(X)$ can be used in place of $f\; :\; X\; \backslash to\; Y.$

For example:{{sfn|Dugundji|1966|pp=215–221}}{{sfn|Bourbaki|1989|pp=57–68}}{{sfn|Császár|1978|pp=102-120}}

{{in5}}$f^\{-1\}(\backslash mathcal\{B\})$ is a prefilter (resp. filter subbase, {{pi}}-system, proper) if and only if this is true of $\backslash mathcal\{B\}\backslash big\backslash vert\_\{f(X)\}.$

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

Even if $\backslash mathcal\{B\}$ is an ultrafilter on $Y,$ if $f$ is not surjective then it is nevertheless possible that $\backslash varnothing\; \backslash in\; \backslash mathcal\{B\}\backslash big\backslash vert\_\{f(X)\},$ which would make $f^\{-1\}(\backslash mathcal\{B\})$ degenerate as well. The next characterization shows that degeneracy is the only obstacle. If $\backslash mathcal\{B\}$ is a prefilter then the following are equivalent:{{sfn|Dugundji|1966|pp=215–221}}{{sfn|Bourbaki|1989|pp=57–68}}{{sfn|Császár|1978|pp=102-120}}

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

and moreover, if $f^\{-1\}(\backslash mathcal\{B\})$ is a prefilter then so is $f\backslash left(f^\{-1\}(\backslash mathcal\{B\})\backslash right).${{sfn|Dugundji|1966|pp=215–221}}{{sfn|Bourbaki|1989|pp=57–68}}

If $S\; \backslash subseteq\; Y$ and if $\backslash operatorname\{In\}\; :\; S\; \backslash to\; Y$ denotes the inclusion map then the trace of $\backslash mathcal\{B\}\; \backslash text\{\; on\; \}\; S$ is equal to $\backslash operatorname\{In\}^\{-1\}(\backslash mathcal\{B\}).${{sfn|Bourbaki|1989|pp=57–68}} This observation allows the results in this subsection to be applied to investigating the trace on a set.

The relation $\backslash ,\backslash leq\backslash ,$ is preserved under both images and preimages of families of sets.{{sfn|Bourbaki|1989|pp=57–68}}

This means that for {{em|any}} families $\backslash mathcal\{C\}\; \backslash text\{\; and\; \}\; \backslash mathcal\{F\},${{sfn|Császár|1978|pp=102-120}}

$$\backslash mathcal\{C\}\; \backslash leq\; \backslash mathcal\{F\}\; \backslash quad\; \backslash text\{\; implies\; \}\; \backslash quad\; g(\backslash mathcal\{C\})\; \backslash leq\; g(\backslash mathcal\{F\})\; \backslash quad\; \backslash text\{\; and\; \}\; \backslash quad\; f^\{-1\}(\backslash mathcal\{C\})\; \backslash leq\; f^\{-1\}(\backslash mathcal\{F\}).$$

Moreover, the following relations always hold for {{em|any}} family of sets $\backslash mathcal\{C\}$:{{sfn|Császár|1978|pp=102-120}}

$$\backslash mathcal\{C\}\; \backslash leq\; f\backslash left(f^\{-1\}(\backslash mathcal\{C\})\backslash right)$$

where equality will hold if $f$ is surjective.{{sfn|Császár|1978|pp=102-120}}

Furthermore,

$$f^\{-1\}(\backslash mathcal\{C\})\; =\; f^\{-1\}\backslash left(f\backslash left(f^\{-1\}(\backslash mathcal\{C\})\backslash right)\backslash right)\; \backslash quad\; \backslash text\{\; and\; \}\; \backslash quad\; g(\backslash mathcal\{C\})\; =\; g\backslash left(g^\{-1\}(g(\backslash mathcal\{C\}))\backslash right).$$

If $\backslash mathcal\{B\}\; \backslash subseteq\; \backslash wp(X)\; \backslash text\{\; and\; \}\; \backslash mathcal\{C\}\; \backslash subseteq\; \backslash wp(Y)$ then{{sfn|Dolecki|Mynard|2016|pp=27-54}}

$$f(\backslash mathcal\{B\})\; \backslash leq\; \backslash mathcal\{C\}\; \backslash quad\; \backslash text\{\; if\; and\; only\; if\; \}\; \backslash quad\; \backslash mathcal\{B\}\; \backslash leq\; f^\{-1\}(\backslash mathcal\{C\})$$

and $g^\{-1\}(g(\backslash mathcal\{C\}))\; \backslash leq\; \backslash mathcal\{C\}${{sfn|Császár|1978|pp=102-120}} where equality will hold if $g$ is injective.{{sfn|Császár|1978|pp=102-120}}

Suppose $X\_\backslash bull\; =\; \backslash left(X\_i\backslash right)\_\{i\; \backslash in\; I\}$ is a family of one or more non-empty sets, whose product will be denoted by $\{\backslash textstyle\backslash prod\_\{\}\}\; X\_\backslash bull\; :=\; \{\backslash textstyle\backslash prod\backslash limits\_\{i\; \backslash in\; I\}\}\; X\_i,$ and for every index $i\; \backslash in\; I,$ let

$$\backslash Pr\{\}\_\{X\_i\}\; :\; \backslash prod\; X\_\backslash bull\; \backslash to\; X\_i$$

denote the canonical projection.

Let $\backslash mathcal\{B\}\_\backslash bull\; :=\; \backslash left(\backslash mathcal\{B\}\_i\backslash right)\_\{i\; \backslash in\; I\}$ be non−empty families, also indexed by $I,$ such that $\backslash mathcal\{B\}\_i\; \backslash subseteq\; \backslash wp\backslash left(X\_i\backslash right)$ for each $i\; \backslash in\; I.$

The {{em|product}} of the families $\backslash mathcal\{B\}\_\backslash bull${{sfn|Bourbaki|1989|pp=57–68}} is defined identically to how the basic open subsets of the product topology are defined (had all of these $\backslash mathcal\{B\}\_i$ been topologies). That is, both the notations

$$\backslash prod\_\{\}\; \backslash mathcal\{B\}\_\backslash bull\; =\; \backslash prod\_\{i\; \backslash in\; I\}\; \backslash mathcal\{B\}\_i$$

denote the family of all cylinder subsets $\{\backslash textstyle\backslash prod\backslash limits\_\{i\; \backslash in\; I\}\}\; S\_i\; \backslash subseteq\; \{\backslash textstyle\backslash prod\}\; X\_\backslash bull$ such that $S\_i\; =\; X\_i$ for all but finitely many $i\; \backslash in\; I$ and where $S\_i\; \backslash in\; \backslash mathcal\{B\}\_i$ for any one of these finitely many exceptions (that is, for any $i$ such that $S\_i\; \backslash neq\; X\_i,$ necessarily $S\_i\; \backslash in\; \backslash mathcal\{B\}\_i$).

When every $\backslash mathcal\{B\}\_i$ is a filter subbase then the family $\{\backslash textstyle\backslash bigcup\backslash limits\_\{i\; \backslash in\; I\}\}\; \backslash Pr\{\}\_\{X\_i\}^\{-1\}\; \backslash left(\backslash mathcal\{B\}\_i\backslash right)$ is a filter subbase for the filter on $\{\backslash textstyle\backslash prod\}\; X\_\backslash bull$ generated by $\backslash mathcal\{B\}\_\backslash bull.${{sfn|Bourbaki|1989|pp=57–68}}

If $\{\backslash textstyle\backslash prod\}\; \backslash mathcal\{B\}\_\backslash bull$ is a filter subbase then the filter on $\{\backslash textstyle\backslash prod\}\; X\_\backslash bull$ that it generates is called the {{em|filter generated by $\backslash mathcal\{B\}\_\backslash bull$}}.{{sfn|Bourbaki|1989|pp=57–68}}

If every $\backslash mathcal\{B\}\_i$ is a prefilter on $X\_i$ then $\{\backslash textstyle\backslash prod\}\; \backslash mathcal\{B\}\_\backslash bull$ will be a prefilter on $\{\backslash textstyle\backslash prod\}\; X\_\backslash bull$ and moreover, this prefilter is equal to the coarsest prefilter $\backslash mathcal\{F\}\; \backslash text\{\; on\; \}\; \{\backslash textstyle\backslash prod\}\; X\_\backslash bull$ such that

$\backslash Pr\{\}\_\{X\_i\}\; (\backslash mathcal\{F\})\; =\; \backslash mathcal\{B\}\_i$

for every $i\; \backslash in\; I.${{sfn|Bourbaki|1989|pp=57–68}}

However, $\{\backslash textstyle\backslash prod\}\; \backslash mathcal\{B\}\_\backslash bull$ may fail to be a filter on $\{\backslash textstyle\backslash prod\}\; X\_\backslash bull$ even if every $\backslash mathcal\{B\}\_i$ is a filter on $X\_i.${{sfn|Bourbaki|1989|pp=57–68}}

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

Prefilters vs. filters

With respect to maps and subsets, the property of being a prefilter is in general more well behaved and better preserved than the property of being a filter. For instance, the image of a prefilter under some map is again a prefilter; but the image of a filter under a non-surjective map is {{em|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 $S\; \backslash subseteq\; X$ 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 $\backslash ,\backslash leq$), 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 the construction of completions of uniform spaces via Cauchy filters). The many properties that characterize ultrafilters are also often useful. They are used to, for example, construct the Stone–Čech compactification. The use of ultrafilters generally requires that the ultrafilter lemma be assumed. But in the many fields where the axiom of choice (or the Hahn–Banach theorem) is assumed, the ultrafilter lemma necessarily holds and does not require an addition assumption.

A note on intuition

Suppose that $\backslash mathcal\{F\}$ is a non-principal filter on an infinite set $X.$ $\backslash mathcal\{F\}$ has one "upward" property (that of being closed upward) and one "downward" property (that of being directed downward).

Starting with any $F\_0\; \backslash in\; \backslash mathcal\{F\},$ there always exists some $F\_1\; \backslash in\; \backslash mathcal\{F\}$ that is a {{em|proper}} subset of $F\_0$; this may be continued ad infinitum to get a sequence $F\_0\; \backslash supsetneq\; F\_1\; \backslash supsetneq\; \backslash cdots$ of sets in $\backslash mathcal\{F\}$ with each $F\_\{i+1\}$ being a {{em|proper}} subset of $F\_i.$ The same is {{em|not}} true going "upward", for if $F\_0\; =\; X\; \backslash in\; \backslash mathcal\{F\}$ then there is no set in $\backslash mathcal\{F\}$ 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 $\backslash ,\backslash subseteq,$ every filter subbase is contained in a unique smallest filter but there may not exist a unique smallest prefilter containing it.

{{in5}}A family $\backslash mathcal\{B\}$ is said to {{em|{{visible anchor|Convergent filter|Convergent prefilter|Convergent family|text=converge}} in $(X,\; \backslash tau)$}} to a point $x$ of $X${{sfn|Narici|Beckenstein|2011|pp=2–7}} if $\backslash mathcal\{B\}\; \backslash geq\; \backslash mathcal\{N\}(x).$ Explicitly, $\backslash mathcal\{N\}(x)\; \backslash leq\; \backslash mathcal\{B\}$ means that every neighborhood $N\; \backslash text\{\; of\; \}\; x$ contains some $B\; \backslash in\; \backslash mathcal\{B\}$ as a subset (that is, $B\; \backslash subseteq\; N$); thus the following then holds: $\backslash mathcal\{N\}\; \backslash ni\; N\; \backslash supseteq\; B\; \backslash in\; \backslash mathcal\{B\}.$ In words, a family converges to a point or subset $x$ if and only if it is {{em|finer}} than the neighborhood filter at $x.$

A family $\backslash mathcal\{B\}$ converging to a point $x$ may be indicated by writing $\backslash mathcal\{B\}\; \backslash to\; x\; \backslash text\{\; or\; \}\; \backslash lim\; \backslash mathcal\{B\}\; \backslash to\; x\; \backslash text\{\; in\; \}\; X${{sfn|Wilansky|2008|pp=32-35}} and saying that $x$ is a {{em|{{visible anchor|Limit of a filter|Limit of a prefilter|Limit of a family|Limit|Limit point|text=limit}}}} of $\backslash mathcal\{B\}\; \backslash text\{\; in\; \}\; X;$ if this limit $x$ is a point (and not a subset), then $x$ is also called a {{em|{{visible anchor|Limit point of a filter|Limit point of a prefilter|Limit point of a family|text=limit point}}}}.{{sfn|Bourbaki|1989|pp=68-83}}

As usual, $\backslash lim\; \backslash mathcal\{B\}\; =\; x$ is defined to mean that $\backslash mathcal\{B\}\; \backslash to\; x$ and $x\; \backslash in\; X$ is the {{em|only}} limit point of $\backslash mathcal\{B\};$ that is, if also $\backslash mathcal\{B\}\; \backslash to\; z\; \backslash text\{\; then\; \}\; z\; =\; x.${{sfn|Wilansky|2008|pp=32-35}} (If the notation "$\backslash lim\; \backslash mathcal\{B\}\; =\; x$" did not also require that the limit point $x$ be unique then the equals sign {{=}} would no longer be guaranteed to be transitive).

The set of all limit points of $\backslash mathcal\{B\}$ is denoted by $\backslash lim\; \{\}\_X\; \backslash mathcal\{B\}\; \backslash text\{\; or\; \}\; \backslash lim\; \backslash mathcal\{B\}.${{sfn|Narici|Beckenstein|2011|pp=2–7}}

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

Examples

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

1. the prefilter of all open balls centered at the origin, where
2. the prefilter of all closed balls centered at the origin, where $B\_\{\backslash leq\; r\}(z)\; =\; \backslash \{x\; :\; \backslash |x\; -\; z\backslash |\; \backslash leq\; r\backslash \}.$ This prefilter is equivalent to the one above.
3. the prefilter where $R\; =\; S\_1\; \backslash cup\; S\_\{1/2\}\; \backslash cup\; S\_\{1/3\}\; \backslash cup\; \backslash cdots$ is a union of spheres $S\_r\; =\; \backslash \{x\; :\; \backslash |x\backslash |\; =\; r\backslash \}$ centered at the origin having progressively smaller radii. This family consists of the sets $S\_\{1/n\}\; \backslash cup\; S\_\{1/(n+1)\}\; \backslash cup\; S\_\{1/(n+2)\}\; \backslash cup\; \backslash cdots$ as $n$ ranges over the positive integers.
4. any of the families above but with the radius $r$ ranging over $1,\; \backslash ,\; 1/2,\; \backslash ,\; 1/3,\; \backslash ,\; 1/4,\; \backslash ldots$ (or over any other positive decreasing sequence) instead of over all positive reals.
5. * Drawing or imagining any one of these sequences of sets when $X\; =\; \backslash R^2$ has dimension $n\; =\; 2$ suggests that intuitively, these sets "should" converge to the origin (and indeed they do). This is the intuition that the above definition of a "convergent prefilter" make rigorous.

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

The one and only limit point in $X\; :=\; \backslash R$ of the free prefilter $\backslash \{(0,\; r)\; :\; r\; >\; 0\backslash \}$ is $0$ since every open ball around the origin contains some open interval of this form.

The fixed prefilter $\backslash mathcal\{B\}\; :=\; \backslash \{[0,\; 1\; +\; r)\; :\; r\; >\; 0\backslash \}$ does not converges in $\backslash R$ to any {{em|point}} and so $\backslash lim\; \backslash mathcal\{B\}\; =\; \backslash varnothing,$ although $\backslash mathcal\{B\}$ does converge to the {{em|set}} $\backslash ker\; \backslash mathcal\{B\}\; =\; [0,\; 1]$ since $\backslash mathcal\{N\}([0,\; 1])\; \backslash leq\; \backslash mathcal\{B\}.$

However, not every fixed prefilter converges to its kernel. For instance, the fixed prefilter $\backslash \{[0,\; 1\; +\; r)\; \backslash cup\; (1\; +\; 1/r,\; \backslash infty)\; :\; r\; >\; 0\backslash \}$ also has kernel $[0,\; 1]$ but does not converges (in $\backslash R$) to it.

The free prefilter $(\backslash R,\; \backslash infty)\; :=\; \backslash \{(r,\; \backslash infty)\; :\; r\; \backslash in\; \backslash R\backslash \}$ of intervals does not converge (in $\backslash R$) to any point.

The same is also true of the prefilter $[\backslash R,\; \backslash infty)\; :=\; \backslash \{[r,\; \backslash infty)\; :\; r\; \backslash in\; \backslash R\backslash \}$ because it is equivalent to $(\backslash R,\; \backslash infty)$ and equivalent families have the same limits.

In fact, if $\backslash mathcal\{B\}$ is any prefilter in any topological space $X$ then for every $S\; \backslash in\; \backslash mathcal\{B\}^\{\backslash uparrow\; X\},$ $\backslash mathcal\{B\}\; \backslash to\; S.$

More generally, because the only neighborhood of $X$ is itself (that is, $\backslash mathcal\{N\}(X)\; =\; \backslash \{X\backslash \}$), every non-empty family (including every filter subbase) converges to $X.$

For any point $x,$ its neighborhood filter $\backslash mathcal\{N\}(x)\; \backslash to\; x$ always converges to $x.$ More generally, any neighborhood basis at $x$ converges to $x.$

A point $x$ is always a limit point of the principle ultra prefilter $\backslash \{\backslash \{x\backslash \}\backslash \}$ and of the ultrafilter that it generates.

The empty family $\backslash mathcal\{B\}\; =\; \backslash varnothing$ does not converge to any point.

Basic properties

If $\backslash mathcal\{B\}$ converges to a point then the same is true of any family finer than $\backslash mathcal\{B\}.$

This has many important consequences.

One consequence is that the limit points of a family $\backslash mathcal\{B\}$ are the same as the limit points of its upward closure: $$\backslash operatorname\{lim\}\_X\; \backslash mathcal\{B\}\; ~=~\; \backslash operatorname\{lim\}\_X\; \backslash left(\backslash mathcal\{B\}^\{\backslash uparrow\; X\}\backslash right).$$

In particular, the limit points of a prefilter are the same as the limit points of the filter that it generates.

Another consequence is that if a family converges to a point then the same is true of the family's trace/restriction to any given subset of $X.$

If $\backslash mathcal\{B\}$ is a prefilter and $B\; \backslash in\; \backslash mathcal\{B\}$ then $\backslash mathcal\{B\}$ converges to a point of $X$ if and only if this is true of the trace $\backslash mathcal\{B\}\backslash big\backslash vert\_B.${{sfn|Dixmier|1984|pp=13–18}}

If a filter subbase converges to a point then so do the filter and the {{pi}}-system that it generates, although the converse is not guaranteed. For example, the filter subbase $\backslash \{(-\backslash infty,\; 0],\; [0,\; \backslash infty)\backslash \}$ does not converge to $0$ in $X\; :=\; \backslash R$ although the filter that it generates—which is equal to the principal filter generated by $\backslash \{0\backslash \}$—does.

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

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

Because subordination is transitive, if $\backslash mathcal\{B\}\; \backslash leq\; \backslash mathcal\{C\}\; \backslash text\{\; then\; \}\; \backslash lim\; \{\}\_\{X\}\; \backslash mathcal\{B\}\; \backslash subseteq\; \backslash lim\; \{\}\_\{X\}\; \backslash mathcal\{C\}$ and moreover, for every $x\; \backslash in\; X,$ both $\backslash \{x\backslash \}$ and the maximal/ultrafilter $\backslash \{x\backslash \}^\{\backslash uparrow\; X\}$ converge to $x.$ Thus every topological space $(X,\; \backslash tau)$ induces a canonical convergence $\backslash xi\; \backslash subseteq\; X\; \backslash times\; \backslash operatorname\{Filters\}(X)$ defined by $(x,\; \backslash mathcal\{B\})\; \backslash in\; \backslash xi\; \backslash text\{\; if\; and\; only\; if\; \}\; x\; \backslash in\; \backslash lim\; \{\}\_\{(X,\; \backslash tau)\}\; \backslash mathcal\{B\}.$

At the other extreme, the neighborhood filter $\backslash mathcal\{N\}(x)$ is the smallest (that is, coarsest) filter on $X$ that converges to $x;$ that is, any filter converging to $x$ must contain $\backslash mathcal\{N\}(x)$ as a subset. Said differently, the family of filters that converge to $x$ consists exactly of those filter on $X$ that contain $\backslash mathcal\{N\}(x)$ as a subset.

Consequently, the finer the topology on $X$ then the {{em|fewer}} prefilters exist that have any limit points in $X.$

A family $\backslash mathcal\{B\}$ is said to {{em|cluster at}} a point $x$ of $X$ if it meshes with the neighborhood filter of $x;$ that is, if $\backslash mathcal\{B\}\; \backslash \#\; \backslash mathcal\{N\}(x).$ Explicitly, this means that $B\; \backslash cap\; N\; \backslash neq\; \backslash varnothing\; \backslash text\{\; for\; every\; \}\; B\; \backslash in\; \backslash mathcal\{B\}$ and every neighborhood $N$ of $x.$

In particular, a point $x\; \backslash in\; X$ is a {{em|{{visible anchor|Cluster point of a filter|Cluster point of a prefilter|Cluster point of a family|Cluster point|text=cluster point}}}} or an {{em|{{visible anchor|Accumulation point of a filter|Accumulation point of a prefilter|Accumulation point of a family|Accumulation point|text=accumulation point}}}} of a family $\backslash mathcal\{B\}${{sfn|Narici|Beckenstein|2011|pp=2–7}} if $\backslash mathcal\{B\}$ meshes with the neighborhood filter at $x:\; \backslash \; \backslash mathcal\{B\}\; \backslash \#\; \backslash mathcal\{N\}(x).$ The set of all cluster points of $\backslash mathcal\{B\}$ is denoted by $\backslash operatorname\{cl\}\_X\; \backslash mathcal\{B\},$ where the subscript may be dropped if not needed.

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

When $\backslash mathcal\{B\}$ is a prefilter then the definition of "$\backslash mathcal\{B\}\; \backslash text\{\; and\; \}\; \backslash mathcal\{N\}$ mesh" can be characterized entirely in terms of the subordination preorder $\backslash ,\backslash leq\backslash ,.$

Two equivalent families of sets have the exact same limit points and also the same cluster points. No matter the topology, for every $x\; \backslash in\; X,$ both $\backslash \{x\backslash \}$ and the principal ultrafilter $\backslash \{x\backslash \}^\{\backslash uparrow\; X\}$ cluster at $x.$

If $\backslash mathcal\{B\}$ clusters to a point then the same is true of any family coarser than $\backslash mathcal\{B\}.$ Consequently, the cluster points of a family $\backslash mathcal\{B\}$ are the same as the cluster points of its upward closure: $$\backslash operatorname\{cl\}\_X\; \backslash mathcal\{B\}\; ~=~\; \backslash operatorname\{cl\}\_X\; \backslash left(\backslash mathcal\{B\}^\{\backslash uparrow\; X\}\backslash right).$$

In particular, the cluster points of a prefilter are the same as the cluster points of the filter that it generates.

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

1. $\backslash mathcal\{B\}$ clusters at $x.$
2. The family $\backslash mathcal\{B\}^\{\backslash uparrow\; X\}$ generated by $\backslash mathcal\{B\}$ clusters at $x.$
3. There exists a family equivalent to $\backslash mathcal\{B\}$ that clusters at $x.$
4. $x\; \backslash in\; \{\backslash textstyle\backslash bigcap\backslash limits\_\{F\; \backslash in\; \backslash mathcal\{B\}\}\}\; \backslash operatorname\{cl\}\_X\; F.${{sfn|Bourbaki|1989|pp=69}}
5. $X\; \backslash setminus\; N\; \backslash not\backslash in\; \backslash mathcal\{B\}^\{\backslash uparrow\; X\}$ for every neighborhood $N$ of $x.$
   • If $\backslash mathcal\{B\}$ is a filter on $X$ then $x\; \backslash in\; \backslash operatorname\{cl\}\_X\; \backslash mathcal\{B\}\; \backslash text\{\; if\; and\; only\; if\; \}\; X\; \backslash setminus\; N\; \backslash not\backslash in\; \backslash mathcal\{B\}$ for every neighborhood $N\; \backslash text\{\; of\; \}\; x.$

6. There exists a prefilter $\backslash mathcal\{F\}$ subordinate to $\backslash mathcal\{B\}$ (that is, $\backslash mathcal\{F\}\; \backslash geq\; \backslash mathcal\{B\}$) that converges to $x.$
   • This is the filter equivalent of "$x$ is a cluster point of a sequence if and only if there exists a subsequence converging to $x.$
   • In particular, if $x$ is a cluster point of a prefilter $\backslash mathcal\{B\}$ then $\backslash mathcal\{B\}\; (\backslash cap)\; \backslash mathcal\{N\}(x)$ is a prefilter subordinate to $\backslash mathcal\{B\}$ that converges to $x.$

The set $\backslash operatorname\{cl\}\_X\; \backslash mathcal\{B\}$ of all cluster points of a prefilter $\backslash mathcal\{B\}$ satisfies

$$\backslash operatorname\{cl\}\_X\; \backslash mathcal\{B\}\; =\; \backslash bigcap\_\{B\; \backslash in\; \backslash mathcal\{B\}\}\; \backslash operatorname\{cl\}\_X\; B.$$

Consequently, the set $\backslash operatorname\{cl\}\_X\; \backslash mathcal\{B\}$ of all cluster points of {{em|any}} prefilter $\backslash mathcal\{B\}$ is a closed subset of $X.${{sfn|Bourbaki|1989|pp=68–74}}{{sfn|Narici|Beckenstein|2011|pp=2–7}} This also justifies the notation $\backslash operatorname\{cl\}\_X\; \backslash mathcal\{B\}$ for the set of cluster points.{{sfn|Narici|Beckenstein|2011|pp=2–7}}

In particular, if $K\; \backslash subseteq\; X$ is non-empty (so that $\backslash mathcal\{B\}\; :=\; \backslash \{K\backslash \}$ is a prefilter) then $\backslash operatorname\{cl\}\_X\; \backslash \{K\backslash \}\; =\; \backslash operatorname\{cl\}\_X\; K$ since both sides are equal to $\{\backslash textstyle\backslash bigcap\backslash limits\_\{B\; \backslash in\; \backslash mathcal\{B\}\}\}\; \backslash operatorname\{cl\}\_X\; B.$

Just like sequences and nets, it is possible for a prefilter on a topological space of infinite cardinality to not have {{em|any}} cluster points or limit points.{{sfn|Bourbaki|1989|pp=68–74}}

If $x$ is a limit point of $\backslash mathcal\{B\}$ then $x$ is necessarily a limit point of any family $\backslash mathcal\{C\}$ {{em|finer}} than $\backslash mathcal\{B\}$ (that is, if $\backslash mathcal\{N\}(x)\; \backslash leq\; \backslash mathcal\{B\}\; \backslash text\{\; and\; \}\; \backslash mathcal\{B\}\; \backslash leq\; \backslash mathcal\{C\}$ then $\backslash mathcal\{N\}(x)\; \backslash leq\; \backslash mathcal\{C\}$).{{sfn|Bourbaki|1989|pp=68–74}}

In contrast, if $x$ is a cluster point of $\backslash mathcal\{B\}$ then $x$ is necessarily a cluster point of any family $\backslash mathcal\{C\}$ {{em|coarser}} than $\backslash mathcal\{B\}$ (that is, if $\backslash mathcal\{N\}(x)\; \backslash text\{\; and\; \}\; \backslash mathcal\{B\}$ mesh and $\backslash mathcal\{C\}\; \backslash leq\; \backslash mathcal\{B\}$ then $\backslash mathcal\{N\}(x)\; \backslash text\{\; and\; \}\; \backslash mathcal\{C\}$ mesh).

Equivalent families and subordination

Any two equivalent families $\backslash mathcal\{B\}\; \backslash text\{\; and\; \}\; \backslash mathcal\{C\}$ can be used {{em|interchangeably}} in the definitions of "limit of" and "cluster at" because their equivalency guarantees that $\backslash mathcal\{N\}\; \backslash leq\; \backslash mathcal\{B\}$ if and only if $\backslash mathcal\{N\}\; \backslash leq\; \backslash mathcal\{C\},$ and also that $\backslash mathcal\{N\}\; \backslash \#\; \backslash mathcal\{B\}$ if and only if $\backslash mathcal\{N\}\; \backslash \#\; \backslash mathcal\{C\}.$

In essence, the preorder $\backslash ,\backslash leq\backslash ,$ is incapable of distinguishing between equivalent families.

Given two prefilters, whether or not they mesh can be characterized entirely in terms of subordination.

Thus the two most fundamental concepts related to (pre)filters to Topology (that is, limit and cluster points) can both be defined {{em|entirely}} in terms of the subordination relation. This is why the preorder $\backslash ,\backslash leq\backslash ,$ is of such great importance in applying (pre)filters to Topology.

Limit and cluster point relationships and sufficient conditions

Every limit point of a non-degenerate family $\backslash mathcal\{B\}$ is also a cluster point; in symbols:

$$\backslash operatorname\{lim\}\_X\; \backslash mathcal\{B\}\; ~\backslash subseteq~\; \backslash operatorname\{cl\}\_X\; \backslash mathcal\{B\}.$$

This is because if $x$ is a limit point of $\backslash mathcal\{B\}$ then $\backslash mathcal\{N\}(x)\; \backslash text\{\; and\; \}\; \backslash mathcal\{B\}$ mesh,{{sfn|Dugundji|1966|pp=211–213}}{{sfn|Bourbaki|1989|pp=68–74}} which makes $x$ a cluster point of $\backslash mathcal\{B\}.${{sfn|Narici|Beckenstein|2011|pp=2–7}} But in general, a cluster point need not be a limit point. For instance, every point in any given non-empty subset $K\; \backslash subseteq\; X$ is a cluster point of the principle prefilter $\backslash mathcal\{B\}\; :=\; \backslash \{K\backslash \}$ (no matter what topology is on $X$) but if $X$ is Hausdorff and $K$ has more than one point then this prefilter has no limit points; the same is true of the filter $\backslash \{K\backslash \}^\{\backslash uparrow\; X\}$ that this prefilter generates.

However, every cluster point of an {{em|ultra}} prefilter is a limit point. Consequently, the limit points of an {{em|ultra}} prefilter $\backslash mathcal\{B\}$ are the same as its cluster points: $\backslash operatorname\{lim\}\_X\; \backslash mathcal\{B\}\; =\; \backslash operatorname\{cl\}\_X\; \backslash mathcal\{B\};$ that is to say, a given point is a cluster point of an ultra prefilter $\backslash mathcal\{B\}$ if and only if $\backslash mathcal\{B\}$ converges to that point.{{sfn|Dugundji|1966|pp=219–221}}{{sfn|Bourbaki|1989|p=70}}

Although a cluster point of a filter need not be a limit point, there will always exist a finer filter that does converge to it; in particular, if $\backslash mathcal\{B\}$ clusters at $x$ then $\backslash mathcal\{B\}\; \backslash ,(\backslash cap)\backslash ,\; \backslash mathcal\{N\}(x)\; =\; \backslash \{B\; \backslash cap\; N\; :\; B\; \backslash in\; \backslash mathcal\{B\},\; N\; \backslash in\; \backslash mathcal\{N\}(x)\backslash \}$ is a filter subbase whose generated filter converges to $x.$

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

If $x$ is a cluster point of a prefilter $\backslash mathcal\{B\}$ then $\backslash mathcal\{B\}\; (\backslash cap)\; \backslash mathcal\{N\}(x)$ is a prefilter subordinate to $\backslash mathcal\{B\}$ that converges to $x\; \backslash text\{\; in\; \}\; X.$

If $S\; \backslash subseteq\; X$ and if $\backslash mathcal\{B\}$ is a prefilter on $S$ then every cluster point of $\backslash mathcal\{B\}\; \backslash text\{\; in\; \}\; X$ belongs to $\backslash operatorname\{cl\}\_X\; S$ and any point in $\backslash operatorname\{cl\}\_X\; S$ is a limit point of a filter on $S.${{sfn|Bourbaki|1989|pp=68–74}}

Primitive sets

A subset $P\; \backslash subseteq\; X$ is called {{em|{{visible anchor|Primitive set|text=primitive}}}}{{sfn|Bourbaki|1989|pp=132–133}} if it is the set of limit points of some ultrafilter (or equivalently, some ultra prefilter). That is, if there exists an ultrafilter $\backslash mathcal\{B\}\; \backslash text\{\; on\; \}\; X$ such that $P$ is equal to $\backslash operatorname\{lim\}\_X\; \backslash mathcal\{B\},$ which recall denotes the set of limit points of $\backslash mathcal\{B\}\; \backslash text\{\; in\; \}\; X.$ Since limit points are the same as cluster points for ultra prefilters, a subset is primitive if and only if it is equal to the set $\backslash operatorname\{cl\}\_X\; \backslash mathcal\{B\}$ of cluster points of some ultra prefilter $\backslash mathcal\{B\}.$

For example, every closed singleton subset is primitive.{{sfn|Bourbaki|1989|pp=132–133}} The image of a primitive subset of $X$ under a continuous map $f\; :\; X\; \backslash to\; Y$ is contained in a primitive subset of $Y.${{sfn|Bourbaki|1989|pp=132–133}}

Assume that $P,\; Q\; \backslash subseteq\; X$ are two primitive subset of $X.$

If $U$ is an open subset of $X$ that intersects $P$ then $U\; \backslash in\; \backslash mathcal\{B\}$ for any ultrafilter $\backslash mathcal\{B\}\; \backslash text\{\; on\; \}\; X$ such that $P\; =\; \backslash operatorname\{lim\}\_X\; \backslash mathcal\{B\}.${{sfn|Bourbaki|1989|pp=132–133}}

In addition, if $P\; \backslash text\{\; and\; \}\; Q$ are distinct then there exists some $S\; \backslash subseteq\; X$ and some ultrafilters $\backslash mathcal\{B\}\_P\; \backslash text\{\; and\; \}\; \backslash mathcal\{B\}\_Q\; \backslash text\{\; on\; \}\; X$ such that $P\; =\; \backslash operatorname\{lim\}\_X\; \backslash mathcal\{B\}\_P,\; Q\; =\; \backslash operatorname\{lim\}\_X\; \backslash mathcal\{B\}\_Q,\; S\; \backslash in\; \backslash mathcal\{B\}\_P,$ and $X\; \backslash setminus\; S\; \backslash in\; \backslash mathcal\{B\}\_Q.${{sfn|Bourbaki|1989|pp=132–133}}

Other results

{{See also|Limit superior and limit inferior#Definition for filter bases}}

If $X$ is a complete lattice then:{{citation needed|date=November 2020}}

• 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.

{{anchor|Limits of functions|Limit points of functions}}

{{anchor|Limit of a function|Limit point of a function}}

{{anchor|Limit of a function with respect to a filter|Limit point of a function with respect to a filter}}

{{anchor|Cluster point of a function|Cluster points of functions}}

{{See also|Limit of a function|One-sided limit}}

Suppose $f\; :\; X\; \backslash to\; Y$ is a map from a set into a topological space $Y,$ $\backslash mathcal\{B\}\; \backslash subseteq\; \backslash wp(X),$ and $y\; \backslash in\; Y.$ If $y$ is a limit point (respectively, a cluster point) of $f(\backslash mathcal\{B\})\; \backslash text\{\; in\; \}\; Y$ then $y$ is called a {{em|limit point}} or {{em|limit}} (respectively, a {{em|cluster point}}) {{em|of $f$ with respect to $\backslash mathcal\{B\}.$}}{{sfn|Bourbaki|1989|pp=68–74}}

Explicitly, $y$ is a limit of $f$ with respect to $\backslash mathcal\{B\}$ if and only if $\backslash mathcal\{N\}(y)\; \backslash leq\; f(\backslash mathcal\{B\}),$ which can be written as $f(\backslash mathcal\{B\})\; \backslash to\; y\; \backslash text\{\; or\; \}\; \backslash lim\; f(\backslash mathcal\{B\})\; \backslash to\; y\; \backslash text\{\; in\; \}\; Y$ (by definition of this notation) and stated as $f$ {{em|tend to $y$ along $\backslash mathcal\{B\}.$}}{{sfn|Dixmier|1984|pp=14–17}} If the limit $y$ is unique then the arrow $\backslash to$ may be replaced with an equals sign $=.${{sfn|Wilansky|2008|pp=32-35}} The neighborhood filter $\backslash mathcal\{N\}(y)$ can be replaced with any family equivalent to it and the same is true of $\backslash mathcal\{B\}.$

The definition of a convergent net is a special case of the above definition of a limit of a function.

Specifically, if $x\; \backslash in\; X\; \backslash text\{\; and\; \}\; \backslash chi\; :\; (I,\; \backslash leq)\; \backslash to\; X$ is a net then

$$\backslash chi\; \backslash to\; x\; \backslash text\{\; in\; \}\; X\; \backslash quad\; \backslash text\{\; if\; and\; only\; if\; \}\; \backslash quad\; \backslash chi(\backslash operatorname\{Tails\}(I,\; \backslash leq))\; \backslash to\; x\; \backslash text\{\; in\; \}\; X,$$

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

The table below shows how various types of limits encountered in analysis and topology can be defined in terms of the convergence of images (under $f$) of particular prefilters on the domain $X.$

This shows that prefilters provide a general framework into which many of the various definitions of limits fit.{{sfn|Dixmier|1984|pp=13–18}}

The limits in the left-most column are defined in their usual way with their obvious definitions.

Throughout, let $f\; :\; X\; \backslash to\; Y$ be a map between topological spaces, $x\_0\; \backslash in\; X,\; \backslash text\{\; and\; \}\; y\; \backslash in\; Y.$

If $Y$ is Hausdorff then all arrows {{nowrap|"$\backslash to\; y$"}} in the table may be replaced with equal signs {{nowrap|"$=\; y$"}} and {{nowrap|1="$\backslash lim\; f(\backslash mathcal\{B\})\; \backslash to\; y$"}} may be replaced with {{nowrap|1="$\backslash lim\; f(\backslash mathcal\{B\})\; =\; y$".}}{{sfn|Wilansky|2008|pp=32-35}}

By defining different prefilters, many other notions of limits can be defined; for example, $\backslash lim\_\{\backslash stackrel$

Divergence to infinity

Divergence of a real-valued function to infinity can be defined/characterized by using the prefilters

$$(\backslash R,\; \backslash infty)\; :=\; \backslash \{(r,\; \backslash infty)\; :\; r\; \backslash in\; \backslash R\backslash \}\; ~~\; \backslash text\{\; and\; \}\; ~~\; (-\backslash infty,\; \backslash R)\; :=\; \backslash \{(-\backslash infty,\; r)\; :\; r\; \backslash in\; \backslash R\backslash \},$$

where $f\; \backslash to\; \backslash infty$ along $\backslash mathcal\{B\}$ if and only if $(\backslash R,\; \backslash infty)\; \backslash leq\; f(\backslash mathcal\{B\})$ and similarly, $f\; \backslash to\; -\backslash infty$ along $\backslash mathcal\{B\}$ if and only if $(-\backslash infty,\; \backslash R)\; \backslash leq\; f(\backslash mathcal\{B\}).$ The family $(\backslash R,\; \backslash infty)$ can be replaced by any family equivalent to it, such as $[\backslash R,\; \backslash infty)\; :=\; \backslash \{[r,\; \backslash infty)\; :\; r\; \backslash in\; \backslash R\backslash \}$ for instance (in real analysis, this would correspond to replacing the strict inequality {{nowrap|"$f(x)\; >\; r$"}} in the definition with {{nowrap|"$f(x)\; \backslash geq\; r$"),}} and the same is true of $\backslash mathcal\{B\}$ and $(-\backslash infty,\; \backslash R).$

So for example, if $\backslash mathcal\{B\}\backslash ,:=\backslash ,\backslash mathcal\{N\}\backslash left(x\_0\backslash right)$ then $\backslash lim\_\{x\; \backslash to\; x\_0\}\; f(x)\; \backslash to\; \backslash infty$ if and only if $(\backslash R,\; \backslash infty)\; \backslash leq\; f(\backslash mathcal\{B\})$ holds. Similarly, $\backslash lim\_\{x\; \backslash to\; x\_0\}\; f(x)\; \backslash to\; -\; \backslash infty$ if and only if $(-\backslash infty,\; \backslash R)\; \backslash leq\; f\backslash left(\backslash mathcal\{N\}\backslash left(x\_0\backslash right)\backslash right),$ or equivalently, if and only if $(-\backslash infty,\; \backslash R]\; \backslash leq\; f\backslash left(\backslash mathcal\{N\}\backslash left(x\_0\backslash right)\backslash right).$

More generally, if $f$ is valued in $Y\; =\; \backslash R^n\; \backslash text\{\; or\; \}\; Y\; =\; \backslash Complex^n$ (or some other seminormed vector space) and if then $\backslash lim\_\{x\; \backslash to\; x\_0\}\; |f(x)|\; \backslash to\; \backslash infty$ if and only if $B\_\{\backslash geq\; \backslash R\}\; \backslash leq\; f\backslash left(\backslash mathcal\{N\}\backslash left(x\_0\backslash right)\backslash right)$ holds, where $B\_\{\backslash geq\; \backslash R\}\; :=\; \backslash left\backslash \{B\_\{\backslash geq\; r\}\; :\; r\; \backslash in\; \backslash R\backslash right\backslash \}.$

{{See also|Filter (set theory)#Filters and nets}}

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

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

{{Quote frame|1=A net $x\_\backslash bull\; =\; \backslash left(x\_i\backslash right)\_\{i\; \backslash in\; I\}$ is said to {{anchor|Convergent net}}{{em|converge in}} $(X,\; \backslash tau)$ to a point $x\; \backslash in\; X,$ written $x\_\backslash bull\; \backslash to\; x\; \backslash text\{\; in\; \}\; X,$ and $x$ is called a {{em|limit}} or {{em|limit point}} of $x\_\backslash bull,${{sfn|Kelley|1975|pp=65–72}} if any of the following equivalent conditions hold:

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

As usual, $\backslash lim\; x\_\backslash bull\; =\; x$ is defined to mean that $x\_\backslash bull\; \backslash to\; x$ and $x$ is the {{em|only}} limit point of $x\_\backslash bull;$ that is, if also $x\_\backslash bull\; \backslash to\; z\; \backslash text\{\; then\; \}\; z\; =\; x.${{sfn|Kelley|1975|pp=65–72}}

}}

{{Quote frame|1=A point $x\; \backslash in\; X$ is called a {{anchor|Cluster point of a net}}{{em|cluster}} or {{em|accumulation point}} of a net $x\_\backslash bull\; =\; \backslash left(x\_i\backslash right)\_\{i\; \backslash in\; I\}\; \backslash text\{\; in\; \}\; (X,\; \backslash tau)$ if any of the following equivalent conditions hold:

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

}}

If $f\; :\; X\; \backslash to\; Y$ is a map and $x\_\backslash bull$ is a net in $X$ then $\backslash operatorname\{Tails\}\backslash left(f\backslash left(x\_\backslash bull\backslash right)\backslash right)\; =\; f\backslash left(\backslash operatorname\{Tails\}\backslash left(x\_\backslash bull\backslash right)\backslash right).${{sfn|Schechter|1996|pp=155-171}}

A {{em|pointed set}} is a pair $(S,\; s)$ consisting of a non-empty set $S$ and an element $s\; \backslash in\; S.$

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

$$\backslash operatorname\{PointedSets\}(\backslash mathcal\{B\})\; :=\; \backslash \{(B,\; b)\; ~:~\; B\; \backslash in\; \backslash mathcal\{B\}\; \backslash text\{\; and\; \}\; b\; \backslash in\; B\backslash \}.$$

Define a canonical preorder $\backslash ,\backslash leq\backslash ,$ on pointed sets by declaring

$$(R,\; r)\; \backslash leq\; (S,\; s)\; \backslash quad\; \backslash text\{\; if\; and\; only\; if\; \}\; \backslash quad\; R\; \backslash supseteq\; S.$$

There is a canonical map $\backslash operatorname\{Point\}\_\{\backslash mathcal\{B\}\}\; ~:~\; \backslash operatorname\{PointedSets\}(\backslash mathcal\{B\})\; \backslash to\; X$ defined by $(B,\; b)\; \backslash mapsto\; b.$

If $i\_0\; =\; \backslash left(B\_0,\; b\_0\backslash right)\; \backslash in\; \backslash operatorname\{PointedSets\}(\backslash mathcal\{B\})$ then the tail of the assignment $\backslash operatorname\{Point\}\_\{\backslash mathcal\{B\}\}$ starting at $i\_0$ is $\backslash left\backslash \{c\; ~:~\; (C,\; c)\; \backslash in\; \backslash operatorname\{PointedSets\}(\backslash mathcal\{B\})\; \backslash text\{\; and\; \}\; \backslash left(B\_0,\; b\_0\backslash right)\; \backslash leq\; (C,\; c)\backslash right\backslash \}\; =\; B\_0.$

Although $(\backslash operatorname\{PointedSets\}(\backslash mathcal\{B\}),\; \backslash leq)$ is not, in general, a partially ordered set, it is a directed set if (and only if) $\backslash mathcal\{B\}$ is a prefilter.

So the most immediate choice for the definition of "the net in $X$ induced by a prefilter $\backslash mathcal\{B\}$" is the assignment $(B,\; b)\; \backslash mapsto\; b$ from $\backslash operatorname\{PointedSets\}(\backslash mathcal\{B\})$ into $X.$

{{Quote frame|1=If $\backslash mathcal\{B\}$ is a prefilter on $X$ then the {{em|net associated with $\backslash mathcal\{B\}$}} is the map

$$\backslash begin\{alignat\}\{4\}$$

\operatorname{Net}_{\mathcal{B}} :\;&& (\operatorname{PointedSets}(\mathcal{B}), \leq) &&\,\to \;& X \\

&& (B, b) &&\,\mapsto\;& b \\

\end{alignat}

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

}}

If $\backslash mathcal\{B\}$ is a prefilter on $X\; \backslash text\{\; then\; \}\; \backslash operatorname\{Net\}\_\{\backslash mathcal\{B\}\}$ is a net in $X$ and the prefilter associated with $\backslash operatorname\{Net\}\_\{\backslash mathcal\{B\}\}$ is $\backslash mathcal\{B\}$; that is:The set equality $\backslash operatorname\{Tails\}\backslash left(\backslash operatorname\{Net\}\_\{\backslash mathcal\{B\}\}\backslash right)\; =\; \backslash mathcal\{B\}$ holds more generally: if the family of sets $\backslash mathcal\{B\}\; \backslash neq\; \backslash varnothing\; \backslash text\{\; satisfies\; \}\; \backslash varnothing\; \backslash not\backslash in\; \backslash mathcal\{B\}$ then the family of tails of the map $\backslash operatorname\{PointedSets\}(\backslash mathcal\{B\})\; \backslash to\; X$ (defined by $(B,\; b)\; \backslash mapsto\; b$) is equal to $\backslash mathcal\{B\}.$

$$\backslash operatorname\{Tails\}\backslash left(\backslash operatorname\{Net\}\_\{\backslash mathcal\{B\}\}\backslash right)\; =\; \backslash mathcal\{B\}.$$

This would not necessarily be true had $\backslash operatorname\{Net\}\_\{\backslash mathcal\{B\}\}$ been defined on a proper subset of $\backslash operatorname\{PointedSets\}(\backslash mathcal\{B\}).$

If $x\_\backslash bull$ is a net in $X$ then it is {{em|not}} in general true that $\backslash operatorname\{Net\}\_\{\backslash operatorname\{Tails\}\backslash left(x\_\backslash bull\backslash right)\}$ is equal to $x\_\backslash bull$ because, for example, the domain of $x\_\backslash bull$ may be of a completely different cardinality than that of $\backslash operatorname\{Net\}\_\{\backslash operatorname\{Tails\}\backslash left(x\_\backslash bull\backslash right)\}$ (since unlike the domain of $\backslash operatorname\{Net\}\_\{\backslash operatorname\{Tails\}\backslash left(x\_\backslash bull\backslash right)\},$ the domain of an arbitrary net in $X$ could have {{em|any}} cardinality).

{{Math theorem|name=Proposition|math_statement=If $\backslash mathcal\{B\}$ is a prefilter on $X$ and $x\; \backslash in\; X$ then

1. $\backslash mathcal\{B\}\; \backslash to\; x\; \backslash text\{\; if\; and\; only\; if\; \}\; \backslash operatorname\{Net\}\_\{\backslash mathcal\{B\}\}\; \backslash to\; x.$
2. $x$ is a cluster point of $\backslash mathcal\{B\}$ if and only if $x$ is a cluster point of $\backslash operatorname\{Net\}\_\{\backslash mathcal\{B\}\}.$

}}

{{Math proof|drop=hidden|proof=

Recall that $\backslash mathcal\{B\}\; =\; \backslash operatorname\{Tails\}\backslash left(\backslash operatorname\{Net\}\_\{\backslash mathcal\{B\}\}\backslash right)$ and that if $x\_\backslash bull$ is a net in $X$ then (1) $x\_\backslash bull\; \backslash to\; x\; \backslash text\{\; if\; and\; only\; if\; \}\; \backslash operatorname\{Tails\}\backslash left(x\_\backslash bull\backslash right)\; \backslash to\; x,$ and (2) $x$ is a cluster point of $x\_\backslash bull$ if and only if $x$ is a cluster point of $\backslash operatorname\{Tails\}\backslash left(x\_\backslash bull\backslash right).$

By using $x\_\backslash bull\; :=\; \backslash operatorname\{Net\}\_\{\backslash mathcal\{B\}\}\; \backslash text\{\; and\; \}\; \backslash mathcal\{B\}\; =\; \backslash operatorname\{Tails\}\backslash left(\backslash operatorname\{Net\}\_\{\backslash mathcal\{B\}\}\backslash right),$ it follows that

$$\backslash mathcal\{B\}\; \backslash to\; x\; \backslash quad\; \backslash text\{\; if\; and\; only\; if\; \}\; \backslash quad\; \backslash operatorname\{Tails\}\backslash left(\backslash operatorname\{Net\}\_\{\backslash mathcal\{B\}\}\backslash right)\; \backslash to\; x\; \backslash quad\; \backslash text\{\; if\; and\; only\; if\; \}\; \backslash quad\; \backslash operatorname\{Net\}\_\{\backslash mathcal\{B\}\}\; \backslash to\; x.$$

It also follows that $x$ is a cluster point of $\backslash mathcal\{B\}$ if and only if $x$ is a cluster point of $\backslash operatorname\{Tails\}\backslash left(\backslash operatorname\{Net\}\_\{\backslash mathcal\{B\}\}\backslash right)$ if and only if $x$ is a cluster point of $\backslash operatorname\{Net\}\_\{\backslash mathcal\{B\}\}.$

}}

Partially ordered net

The domain of the canonical net $\backslash operatorname\{Net\}\_\{\backslash mathcal\{B\}\}$ is in general not partially ordered. However, in 1955 Bruns and Schmidt discoveredBruns G., Schmidt J.,Zur Aquivalenz von Moore-Smith-Folgen und Filtern, Math. Nachr. 13 (1955), 169-186. a construction (detailed here: Filter (set theory)#Partially ordered net) that allows for the canonical net to have a domain that is both partially ordered and directed; this was independently rediscovered by Albert Wilansky in 1970.{{sfn|Schechter|1996|pp=155-171}}

Because the tails of this partially ordered net are identical to the tails of $\backslash operatorname\{Net\}\_\{\backslash mathcal\{B\}\}$ (since both are equal to the prefilter $\backslash mathcal\{B\}$), there is typically nothing lost by assuming that the domain of the net associated with a prefilter is both directed {{em|and}} partially ordered.{{sfn|Schechter|1996|pp=155-171}} If can further be assumed that the partially ordered domain is also a dense order.

{{anchor|Subnets}}

The notion of "$\backslash mathcal\{B\}$ is subordinate to $\backslash mathcal\{C\}$" (written $\backslash mathcal\{B\}\; \backslash vdash\; \backslash mathcal\{C\}$) is for filters and prefilters what "$x\_\{n\_\backslash bull\}\; =\; \backslash left(x\_\{n\_i\}\backslash right)\_\{i=1\}^\{\backslash infty\}$ is a subsequence of $x\_\backslash bull\; =\; \backslash left(x\_i\backslash right)\_\{i=1\}^\{\backslash infty\}$" is for sequences.{{sfn|Dugundji|1966|p=212}}

For example, if $\backslash operatorname\{Tails\}\backslash left(x\_\backslash bull\backslash right)\; =\; \backslash left\backslash \{x\_\{\backslash geq\; i\}\; :\; i\; \backslash in\; \backslash N\backslash right\backslash \}$ denotes the set of tails of $x\_\backslash bull$ and if $\backslash operatorname\{Tails\}\backslash left(x\_\{n\_\backslash bull\}\backslash right)\; =\; \backslash left\backslash \{x\_\{n\_\{\backslash geq\; i\}\}\; :\; i\; \backslash in\; \backslash N\backslash right\backslash \}$ denotes the set of tails of the subsequence $x\_\{n\_\backslash bull\}$ (where $x\_\{n\_\{\backslash geq\; i\}\}\; :=\; \backslash left\backslash \{x\_\{n\_j\}\; ~:~\; j\; \backslash geq\; i\; \backslash text\{\; and\; \}\; j\; \backslash in\; \backslash N\backslash right\backslash \}$) then $\backslash operatorname\{Tails\}\backslash left(x\_\{n\_\backslash bull\}\backslash right)\; ~\backslash vdash~\; \backslash operatorname\{Tails\}\backslash left(x\_\backslash bull\backslash right)$ (which by definition means $\backslash operatorname\{Tails\}\backslash left(x\_\backslash bull\backslash right)\; \backslash leq\; \backslash operatorname\{Tails\}\backslash left(x\_\{n\_\backslash bull\}\backslash right)$) is true but $\backslash operatorname\{Tails\}\backslash left(x\_\backslash bull\backslash right)\; ~\backslash vdash~\; \backslash operatorname\{Tails\}\backslash left(x\_\{n\_\backslash bull\}\backslash right)$ is in general false.

If $x\_\backslash bull\; =\; \backslash left(x\_i\backslash right)\_\{i\; \backslash in\; I\}$ is a net in a topological space $X$ and if $\backslash mathcal\{N\}(x)$ is the neighborhood filter at a point $x\; \backslash in\; X,$ then $x\_\backslash bull\; \backslash to\; x\; \backslash text\{\; if\; and\; only\; if\; \}\; \backslash mathcal\{N\}(x)\; \backslash leq\; \backslash operatorname\{Tails\}\backslash left(x\_\backslash bull\backslash right).$

If $f\; :\; X\; \backslash to\; Y$ is an surjective open map, $x\; \backslash in\; X,$ and $\backslash mathcal\{C\}$ is a prefilter on $Y$ that converges to $f(x),$ then there exist a prefilter $\backslash mathcal\{B\}$ on $X$ such that $\backslash mathcal\{B\}\; \backslash to\; x$ and $f(\backslash mathcal\{B\})$ is equivalent to $\backslash mathcal\{C\}$ (that is, $\backslash mathcal\{C\}\; \backslash leq\; f(\backslash mathcal\{B\})\; \backslash leq\; \backslash mathcal\{C\}$).{{sfn|Dugundji|1966|p=220–221}}

{{See also|Sequential space|Fréchet–Urysohn space}}

The following results are the prefilter analogs of statements involving subsequences.{{sfn|Dugundji|1966|pp=211–221}} The condition "$\backslash mathcal\{C\}\; \backslash geq\; \backslash mathcal\{B\},$" which is also written $\backslash mathcal\{C\}\; \backslash vdash\; \backslash mathcal\{B\},$ is the analog of "$\backslash mathcal\{C\}$ is a subsequence of $\backslash mathcal\{B\}.$" 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."

{{Math theorem|name=Proposition{{sfn|Dugundji|1966|pp=211–221}}{{sfn|Bourbaki|1989|pp=68–74}}|math_statement=

Let $\backslash mathcal\{B\}$ be a prefilter on $X$ and let $x\; \backslash in\; X.$

1. Suppose $\backslash mathcal\{C\}$ is a prefilter such that $\backslash mathcal\{C\}\; \backslash geq\; \backslash mathcal\{B\}.$

   1. If $\backslash mathcal\{B\}\; \backslash to\; x$ then $\backslash mathcal\{C\}\; \backslash to\; x.$By definition, $\backslash mathcal\{B\}\; \backslash to\; x$ if and only if $\backslash mathcal\{B\}\; \backslash geq\; \backslash mathcal\{N\}(x).$ Since $\backslash mathcal\{C\}\; \backslash geq\; \backslash mathcal\{B\}$ and $\backslash mathcal\{B\}\; \backslash geq\; \backslash mathcal\{N\}(x),$ transitivity implies $\backslash mathcal\{C\}\; \backslash geq\; \backslash mathcal\{N\}(x).\; \backslash blacksquare$
      • This is the analog of "if a sequence converges to $x$ then so does every subsequence."

   2. If $x$ is a cluster point of $\backslash mathcal\{C\}$ then $x$ is a cluster point of $\backslash mathcal\{B\}.$
      • This is the analog of "if $x$ is a cluster point of some subsequence, then $x$ is a cluster point of the original sequence."

2. $\backslash mathcal\{B\}\; \backslash to\; x$ if and only if for any finer prefilter $\backslash mathcal\{C\}\; \backslash geq\; \backslash mathcal\{B\}$ there exists some even more fine prefilter $\backslash mathcal\{F\}\; \backslash geq\; \backslash mathcal\{C\}$ such that $\backslash mathcal\{F\}\; \backslash to\; x.${{sfn|Bourbaki|1989|pp=68–74}}
   • This is the analog of "a sequence converges to $x$ if and only if every subsequence has a sub-subsequence that converges to $x.$"

3. $x$ is a cluster point of $\backslash mathcal\{B\}$ if and only if there exists some finer prefilter $\backslash mathcal\{C\}\; \backslash geq\; \backslash mathcal\{B\}$ such that $\backslash mathcal\{C\}\; \backslash to\; x.$
   • This is the analog of the following {{em|false}} statement: "$x$ is a cluster point of a sequence if and only if it has a subsequence that converges to $x$" (that is, if and only if $x$ is a subsequential limit).
   • The analog for sequences is false since there is a Hausdorff topology on $X\; :=\; [\backslash N\; \backslash times\; \backslash N]\; \backslash cup\; \backslash \{(0,\; 0)\backslash \}$ and a sequence in this space (both defined hereThe topology on $X\; :=\; [\backslash N\; \backslash times\; \backslash N]\; \backslash cup\; \backslash \{(0,\; 0)\backslash \}$ is defined as follows: Every subset of $\backslash N\; \backslash times\; \backslash N$ is open in this topology and the neighborhoods of $(0,\; 0)$ are all those subsets $U\; \backslash subseteq\; X$ containing $(0,\; 0)$ for which there exists some positive integer $N\; >\; 0$ such that for every integer $n\; \backslash geq\; N,$ $U$ contains all but at most finitely many points of $\backslash \{n\backslash \}\; \backslash times\; \backslash N.$ For example, the set $W\; :=\; [\backslash \{2,\; 3,\; \backslash ldots\backslash \}\; \backslash times\; \backslash N]\; \backslash cup\; \backslash \{(0,\; 0)\backslash \}$ is a neighborhood of $(0,\; 0).$ Any diagonal enumeration of $\backslash N\; \backslash times\; \backslash N$ furnishes a sequence that clusters at $(0,\; 0)$ but possess not convergent subsequence. An explicit example is the inverse of the bijective Hopcroft and Ullman pairing function $\backslash N\; \backslash times\; \backslash N\; \backslash to\; \backslash N,$ which is defined by $(p,\; q)\; \backslash mapsto\; p\; +\; \backslash tfrac\{1\}\{2\}(p\; +\; q\; -\; 1)(p\; +\; q\; -\; 2).${{sfn|Dugundji|1966|p=60}}) that clusters at $(0,\; 0)$ but that also does not have any subsequence that converges to $(0,\; 0).${{sfn|Dugundji|1966|pp=215–216}}

}}

{{anchor|Subnets versus subordinate filters}}

{{See also|Net (mathematics)|Subnet (mathematics)|Filter (set theory)#Subnets versus subordinate filters}}

Subnets in the sense of Willard and subnets in the sense of Kelley are the most commonly used definitions of "subnet."{{sfn|Schechter|1996|pp=157–168}}

The first definition of a subnet ("Kelley-subnet") was introduced by John L. Kelley in 1955.{{sfn|Schechter|1996|pp=157–168}}

Stephen Willard introduced in 1970 his own variant ("Willard-subnet") of Kelley's definition of subnet.{{sfn|Schechter|1996|pp=157–168}}

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.{{sfn|Schechter|1996|pp=157–168}}

A subset $R\; \backslash subseteq\; I$ of a preordered space $(I,\; \backslash leq)$ is {{em|{{visible anchor|frequent}}}} or {{em|cofinal}} in $I$ if for every $i\; \backslash in\; I$ there exists some $r\; \backslash in\; R$ such that $i\; \backslash leq\; r.$ If $R\; \backslash subseteq\; I$ contains a tail of $I$ then $R$ is said to be {{em|{{visible anchor|eventual|eventually}}}} in $I$; explicitly, this means that there exists some $i\; \backslash in\; I$ such that $I\_\{\backslash geq\; i\}\; \backslash subseteq\; R$ (that is, $j\; \backslash in\; R$ for all $j\; \backslash in\; I$ satisfying $i\; \backslash leq\; j$). A subset is eventual if and only if its complement is not frequent (which is termed {{em|{{visible anchor|infrequent}}}}).{{sfn|Schechter|1996|pp=157–168}}

A map $h\; :\; A\; \backslash to\; I$ between two preordered sets is {{em|Order-preserving function}} if whenever $a,\; b\; \backslash in\; A$ satisfy $a\; \backslash leq\; b,$ then $h(a)\; \backslash leq\; h(b).$

{{Quote frame|1=Definitions: Let $S\; =\; S\_\backslash bull\; ~:~\; (A,\; \backslash leq)\; \backslash to\; X\; \backslash text\{\; and\; \}\; N\; =\; N\_\backslash bull\; ~:~\; (I,\; \backslash leq)\; \backslash to\; X$ be nets. Then{{sfn|Schechter|1996|pp=157–168}}

1. $S\_\backslash bull$ is a {{em|{{visible anchor|Willard-subnet|Willard-subnet}}}} of $N\_\backslash bull$ or a {{em|subnet in the sense of Willard}} if there exists an order-preserving map $h\; :\; A\; \backslash to\; I$ such that $S\; =\; N\; \backslash circ\; h\; \backslash text\{\; and\; \}\; h(A)$ is cofinal in $I.$
2. $S\_\backslash bull$ is a {{em|{{visible anchor|Kelley-subnet|Kelley-subnet}}}} of $N\_\backslash bull$ or a {{em|subnet in the sense of Kelley}} if there exists a map $h\; ~:~\; A\; \backslash to\; I$ such that $S\; =\; N\; \backslash circ\; h$ and whenever $E\; \backslash subseteq\; I$ is eventual in $I$ then $h^\{-1\}(E)$ is eventual in $A.$
3. $S\_\backslash bull$ is an {{em|{{visible anchor|AA-subnet|AA-subnet}}}} of $N\_\backslash bull$ or a {{em|subnet in the sense of Aarnes and Andenaes}} if any of the following equivalent conditions are satisfied:

   1. $\backslash operatorname\{Tails\}\backslash left(N\_\backslash bull\backslash right)\; \backslash leq\; \backslash operatorname\{Tails\}\backslash left(S\_\backslash bull\backslash right).$
   2. $\backslash operatorname\{TailsFilter\}\backslash left(N\_\backslash bull\backslash right)\; \backslash subseteq\; \backslash operatorname\{TailsFilter\}\backslash left(S\_\backslash bull\backslash right).$
   3. If $J$ is eventual in $I\; \backslash text\{\; then\; \}\; S^\{-1\}(N(J))$ is eventual in $A.$
   4. For any subset $R\; \backslash subseteq\; X,\; \backslash text\{\; if\; \}\; \backslash operatorname\{Tails\}\backslash left(S\_\backslash bull\backslash right)\; \backslash text\{\; and\; \}\; \backslash \{R\backslash \}$ mesh, then so do $\backslash operatorname\{Tails\}\backslash left(N\_\backslash bull\backslash right)\; \backslash text\{\; and\; \}\; \backslash \{R\backslash \}.$
   5. For any subset $R\; \backslash subseteq\; X,\; \backslash text\{\; if\; \}\; \backslash operatorname\{Tails\}\backslash left(S\_\backslash bull\backslash right)\; \backslash leq\; \backslash \{R\backslash \}\; \backslash text\{\; then\; \}\; \backslash operatorname\{Tails\}\backslash left(N\_\backslash bull\backslash right)\; \backslash leq\; \backslash \{R\backslash \}.$

}}

Kelley did not require the map $h$ to be order preserving while the definition of an AA-subnet does away entirely with any map between the two nets' domains and instead focuses entirely on $X$ − the nets' common codomain.

Every Willard-subnet is a Kelley-subnet and both are AA-subnets.{{sfn|Schechter|1996|pp=157–168}}

In particular, if $y\_\backslash bull\; =\; \backslash left(y\_a\backslash right)\_\{a\; \backslash in\; A\}$ is a Willard-subnet or a Kelley-subnet of $x\_\backslash bull\; =\; \backslash left(x\_i\backslash right)\_\{i\; \backslash in\; I\}$ then $\backslash operatorname\{Tails\}\backslash left(x\_\backslash bull\backslash right)\; \backslash leq\; \backslash operatorname\{Tails\}\backslash left(y\_\backslash bull\backslash right).$

:Example: If $I\; =\; \backslash N$ and $x\_\backslash bull\; =\; \backslash left(x\_i\backslash right)\_\{i\; \backslash in\; I\}$ is a constant sequence and if $A\; =\; \backslash \{1\backslash \}$ and $s\_1\; :=\; x\_1$ then $\backslash left(s\_a\backslash right)\_\{a\; \backslash in\; A\}$ is an AA-subnet of $x\_\backslash bull$ but it is neither a Willard-subnet nor a Kelley-subnet of $x\_\backslash bull.$

AA-subnets have a defining characterization that immediately shows that they are fully interchangeable with sub(ordinate)filters.{{sfn|Schechter|1996|pp=157–168}}{{cite web|last=Clark|first=Pete L.|title=Convergence| website=math.uga.edu/|url=http://math.uga.edu/~pete/convergence.pdf|date=18 October 2016|access-date=18 August 2020}}

Explicitly, what is meant is that the following statement is true for AA-subnets:

{{in5}}If $\backslash mathcal\{B\}\; \backslash text\{\; and\; \}\; \backslash mathcal\{F\}$ are prefilters then $\backslash mathcal\{B\}\; \backslash leq\; \backslash mathcal\{F\}$ if and only if $\backslash operatorname\{Net\}\_\{\backslash mathcal\{F\}\}$ is an AA-subnet of $\backslash operatorname\{Net\}\_\{\backslash mathcal\{B\}\}.$

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

{{in5}}{{em|False}} statement: If $\backslash mathcal\{B\}\; \backslash text\{\; and\; \}\; \backslash mathcal\{F\}$ are prefilters such that $\backslash mathcal\{B\}\; \backslash leq\; \backslash mathcal\{F\}\; \backslash text\{\; then\; \}\; \backslash operatorname\{Net\}\_\{\backslash mathcal\{F\}\}$ is a Kelley-subnet of $\backslash operatorname\{Net\}\_\{\backslash mathcal\{B\}\}.$

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

If "subnet" is defined to mean Willard-subnet or Kelley-subnet then nets and filters are not completely interchangeable because there exists a filter–sub(ordinate)filter relationships that cannot be expressed in terms of a net–subnet relationship between the two induced nets. In particular, the problem is that Kelley-subnets and Willard-subnets are {{em|not}} fully interchangeable with subordinate filters. If the notion of "subnet" is not used or if "subnet" is defined to mean AA-subnet, then this ceases to be a problem and so it becomes correct to say that nets and filters are interchangeable. Despite the fact that AA-subnets do not have the problem that Willard and Kelley subnets have, they are not widely used or known about.{{sfn|Schechter|1996|pp=157–168}}

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

Bases and prefilters

Let $\backslash mathcal\{B\}\; \backslash neq\; \backslash varnothing$ be a family of sets that covers $X$ and define $\backslash mathcal\{B\}\_x\; =\; \backslash \{B\; \backslash in\; \backslash mathcal\{B\}\; ~:~\; x\; \backslash in\; B\backslash \}$ for every $x\; \backslash in\; X.$ The definition of a base for some topology can be immediately reworded as: $\backslash mathcal\{B\}$ is a base for some topology on $X$ if and only if $\backslash mathcal\{B\}\_x$ is a filter base for every $x\; \backslash in\; X.$

If $\backslash tau$ is a topology on $X$ and $\backslash mathcal\{B\}\; \backslash subseteq\; \backslash tau$ then the definitions of $\backslash mathcal\{B\}$ is a basis (resp. subbase) for $\backslash tau$ can be reworded as:

{{in5}}$\backslash mathcal\{B\}$ is a base (resp. subbase) for $\backslash tau$ if and only if for every $x\; \backslash in\; X,\; \backslash mathcal\{B\}\_x$ is a filter base (resp. filter subbase) that generates the neighborhood filter of $(X,\; \backslash tau)$ at $x.$

Neighborhood filters

The archetypical example of a filter is the set of all neighborhoods of a point in a topological space.

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

Neighborhood bases at points are examples of prefilters that are fixed but may or may not be principal.

If $X\; =\; \backslash R$ has its usual topology and if $x\; \backslash in\; X,$ then any neighborhood filter base $\backslash mathcal\{B\}$ of $x$ is fixed by $x$ (in fact, it is even true that $\backslash ker\; \backslash mathcal\{B\}\; =\; \backslash \{x\backslash \}$) but $\backslash mathcal\{B\}$ is {{em|not}} principal since $\backslash \{x\backslash \}\; \backslash not\backslash in\; \backslash mathcal\{B\}.$

In contrast, a topological space has the discrete topology if and only if the neighborhood filter of every point is a principal filter generated by exactly one point.

This shows that a non-principal filter on an infinite set is not necessarily free.

The neighborhood filter of every point $x$ in topological space $X$ is fixed since its kernel contains $x$ (and possibly other points if, for instance, $X$ is not a T₁ space). This is also true of any neighborhood basis at $x.$

For any point $x$ in a T₁ space (for example, a Hausdorff space), the kernel of the neighborhood filter of $x$ is equal to the singleton set $\backslash \{x\backslash \}.$

However, it is possible for a neighborhood filter at a point to be principal but {{em|not}} discrete (that is, not principal at a {{em|single}} point).

A neighborhood basis $\backslash mathcal\{B\}$ of a point $x$ in a topological space is principal if and only if the kernel of $\backslash mathcal\{B\}$ is an open set. If in addition the space is T₁ then $\backslash ker\; \backslash mathcal\{B\}\; =\; \backslash \{x\backslash \}$ so that this basis $\backslash mathcal\{B\}$ is principal if and only if $\backslash \{x\backslash \}$ is an open set.

Generating topologies from filters and prefilters

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

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

Neighborhoods and topologies

The neighborhood filter of a nonempty subset $S\; \backslash subseteq\; X$ in a topological space $X$ is equal to the intersection of all neighborhood filters of all points in $S.${{sfn|Bourbaki|1989|p=129}}

A subset $S\; \backslash subseteq\; X$ is open in $X$ if and only if whenever $\backslash mathcal\{F\}$ is a filter on $X$ and $s\; \backslash in\; S,$ then $\backslash mathcal\{F\}\; \backslash to\; s\; \backslash text\{\; in\; \}\; X\; \backslash text\{\; implies\; \}\; S\; \backslash in\; \backslash mathcal\{F\}.$

Suppose $\backslash sigma\; \backslash text\{\; and\; \}\; \backslash tau$ are topologies on $X.$

Then $\backslash tau$ is finer than $\backslash sigma$ (that is, $\backslash sigma\; \backslash subseteq\; \backslash tau$) if and only if whenever $x\; \backslash in\; X\; \backslash text\{\; and\; \}\; \backslash mathcal\{B\}$ is a filter on $X,$ if $\backslash mathcal\{B\}\; \backslash to\; x\; \backslash text\{\; in\; \}\; (X,\; \backslash tau)$ then $\backslash mathcal\{B\}\; \backslash to\; x\; \backslash text\{\; in\; \}\; (X,\; \backslash sigma).${{sfn|Bourbaki|1989|pp=132–133}} Consequently, $\backslash sigma\; =\; \backslash tau$ if and only if for every filter $\backslash mathcal\{B\}\; \backslash text\{\; on\; \}\; X$ and every $x\; \backslash in\; X,\; \backslash mathcal\{B\}\; \backslash to\; x\; \backslash text\{\; in\; \}\; (X,\; \backslash sigma)$ if and only if $\backslash mathcal\{B\}\; \backslash to\; x\; \backslash text\{\; in\; \}\; (X,\; \backslash tau).${{sfn|Wilansky|2008|pp=32-35}}

However, it is possible that $\backslash sigma\; \backslash neq\; \backslash tau$ while also for every filter $\backslash mathcal\{B\}\; \backslash text\{\; on\; \}\; X,\; \backslash mathcal\{B\}$ converges to {{em|some}} point of $X\; \backslash text\{\; in\; \}\; (X,\; \backslash sigma)$ if and only if $\backslash mathcal\{B\}$ converges to {{em|some}} point of $X\; \backslash text\{\; in\; \}\; (X,\; \backslash tau).${{sfn|Wilansky|2008|pp=32-35}}

Closure

If $\backslash mathcal\{B\}$ is a prefilter on a subset $S\; \backslash subseteq\; X$ then every cluster point of $\backslash mathcal\{B\}\; \backslash text\{\; in\; \}\; X$ belongs to $\backslash operatorname\{cl\}\_X\; S.${{sfn|Bourbaki|1989|p=70}}

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

1. $x\; \backslash in\; \backslash operatorname\{cl\}\_X\; S$
2. $x$ is a limit point of a prefilter on $S.$ Explicitly: there exists a prefilter $\backslash mathcal\{F\}\; \backslash subseteq\; \backslash wp(S)\; \backslash text\{\; on\; \}\; S$ such that $\backslash mathcal\{F\}\; \backslash to\; x\; \backslash text\{\; in\; \}\; X.${{sfn|Dugundji|1966|pp=211–221}}
3. $x$ is a limit point of a filter on $S.${{sfn|Bourbaki|1989|p=70}}
4. There exists a prefilter $\backslash mathcal\{F\}\; \backslash text\{\; on\; \}\; X$ such that $S\; \backslash in\; \backslash mathcal\{F\}\; \backslash text\{\; and\; \}\; \backslash mathcal\{F\}\; \backslash to\; x\; \backslash text\{\; in\; \}\; X.$
5. The prefilter $\backslash \{S\backslash \}$ meshes with the neighborhood filter $\backslash mathcal\{N\}(x).$ Said differently, $x$ is a cluster point of the prefilter $\backslash \{S\backslash \}.$
6. The prefilter $\backslash \{S\backslash \}$ meshes with some (or equivalently, with every) filter base for $\backslash mathcal\{N\}(x)$ (that is, with every neighborhood basis at $x$).

The following are equivalent:

1. $x$ is a limit points of $S\; \backslash text\{\; in\; \}\; X.$
2. There exists a prefilter $\backslash mathcal\{F\}\; \backslash subseteq\; \backslash wp(S)\; \backslash text\{\; on\; \}\; \backslash \{S\backslash \}\; \backslash setminus\; \backslash \{x\backslash \}$ such that $\backslash mathcal\{F\}\; \backslash to\; x\; \backslash text\{\; in\; \}\; X.${{sfn|Dugundji|1966|pp=211–221}}

Closed sets

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

1. $S$ is a closed subset of $X.$
2. If $x\; \backslash in\; X\; \backslash text\{\; and\; \}\; \backslash mathcal\{F\}\; \backslash subseteq\; \backslash wp(S)$ is a prefilter on $S$ such that $\backslash mathcal\{F\}\; \backslash to\; x\; \backslash text\{\; in\; \}\; X,$ then $x\; \backslash in\; S.$
3. If $x\; \backslash in\; X\; \backslash text\{\; and\; \}\; \backslash mathcal\{F\}\; \backslash subseteq\; \backslash wp(S)$ is a prefilter on $S$ such that $x$ is an accumulation points of $\backslash mathcal\{F\}\; \backslash text\{\; in\; \}\; X,$ then $x\; \backslash in\; S.${{sfn|Dugundji|1966|pp=211–221}}
4. If $x\; \backslash in\; X$ is such that the neighborhood filter $\backslash mathcal\{N\}(x)$ meshes with $\backslash \{S\backslash \}$ then $x\; \backslash in\; S.$

Hausdorffness

The following are equivalent:

1. $X$ is a Hausdorff space.
2. Every prefilter on $X$ converges to at most one point in $X.${{sfn|Narici|Beckenstein|2011|pp=2–7}}
3. The above statement but with the word "prefilter" replaced by any one of the following: filter, ultra prefilter, ultrafilter.{{sfn|Narici|Beckenstein|2011|pp=2–7}}

Compactness

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

The following are equivalent:

1. $(X,\; \backslash tau)$ is a compact space.
2. Every ultrafilter on $X$ converges to at least one point in $X.${{sfn|Bourbaki|1989|p=83}}
   • That this condition implies compactness can be proven by using only the ultrafilter lemma. That compactness implies this condition can be proven without the ultrafilter lemma (or even the axiom of choice).

3. The above statement but with the word "ultrafilter" replaced by "ultra prefilter".{{sfn|Narici|Beckenstein|2011|pp=2–7}}
4. For every filter $\backslash mathcal\{C\}\; \backslash text\{\; on\; \}\; X$ there exists a filter $\backslash mathcal\{F\}\; \backslash text\{\; on\; \}\; X$ such that $\backslash mathcal\{C\}\; \backslash leq\; \backslash mathcal\{F\}$ and $\backslash mathcal\{F\}$ converges to some point of $X.$
5. The above statement but with each instance of the word "filter" replaced by: prefilter.
6. Every filter on $X$ has at least one cluster point in $X.${{sfn|Bourbaki|1989|p=83}}
   • That this condition is equivalent to compactness can be proven by using only the ultrafilter lemma.

7. The above statement but with the word "filter" replaced by "prefilter".{{sfn|Narici|Beckenstein|2011|pp=2–7}}
8. Alexander subbase theorem: There exists a subbase $\backslash mathcal\{S\}\; \backslash text\{\; for\; \}\; \backslash tau$ such that every cover of $X$ by sets in $\backslash mathcal\{S\}$ has a finite subcover.
   • That this condition is equivalent to compactness can be proven by using only the ultrafilter lemma.

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

{{Math theorem|name=Theorem{{sfn|Bourbaki|1989|pp=83–84}}|math_statement=

If $\backslash mathcal\{B\}$ is a filter on a compact space and $C$ is the set of cluster points of $\backslash mathcal\{B\},$ then every neighborhood of $C$ belongs to $\backslash mathcal\{B\}.$

Thus a filter on a compact Hausdorff space converges if and only if it has a single cluster point.

}}

Continuity

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

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

1. $f\; :\; X\; \backslash to\; Y$ is continuous at $x.$
2. Definition: For every neighborhood $V$ of $f(x)\; \backslash text\{\; in\; \}\; Y$ there exists some neighborhood $N$ of $x\; \backslash text\{\; in\; \}\; X$ such that $f(N)\; \backslash subseteq\; V.$
3. $f(\backslash mathcal\{N\}(x))\; \backslash to\; f(x)\; \backslash text\{\; in\; \}\; Y.${{sfn|Dugundji|1966|pp=215–216}}
4. If $\backslash mathcal\{B\}$ is a filter on $X$ such that $\backslash mathcal\{B\}\; \backslash to\; x\; \backslash text\{\; in\; \}\; X$ then $f(\backslash mathcal\{B\})\; \backslash to\; f(x)\; \backslash text\{\; in\; \}\; Y.$
5. The above statement but with the word "filter" replaced by "prefilter".

The following are equivalent:

1. $f\; :\; X\; \backslash to\; Y$ is continuous.
2. If $x\; \backslash in\; X\; \backslash text\{\; and\; \}\; \backslash mathcal\{B\}$ is a prefilter on $X$ such that $\backslash mathcal\{B\}\; \backslash to\; x\; \backslash text\{\; in\; \}\; X$ then $f(\backslash mathcal\{B\})\; \backslash to\; f(x)\; \backslash text\{\; in\; \}\; Y.${{sfn|Dugundji|1966|pp=215–216}}
3. If $x\; \backslash in\; X$ is a limit point of a prefilter $\backslash mathcal\{B\}\; \backslash text\{\; on\; \}\; X$ then $f(x)$ is a limit point of $f(\backslash mathcal\{B\})\; \backslash text\{\; in\; \}\; Y.$
4. Any one of the above two statements but with the word "prefilter" replaced by "filter".

If $\backslash mathcal\{B\}$ is a prefilter on $X,\; x\; \backslash in\; X$ is a cluster point of $\backslash mathcal\{B\},\; \backslash text\{\; and\; \}\; f\; :\; X\; \backslash to\; Y$ is continuous, then $f(x)$ is a cluster point in $Y$ of the prefilter $f(\backslash mathcal\{B\}).${{sfn|Bourbaki|1989|pp=132–133}}

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

Suppose $f\; :\; D\; \backslash to\; Y$ is a continuous map into a Hausdorff regular space $Y$ and that $D$ is a dense subset of a topological space $X.$ Then $f$ has a continuous extension $F\; :\; X\; \backslash to\; Y$ if and only if for every $x\; \backslash in\; X,$ the prefilter $f\backslash left(\backslash mathcal\{N\}\_X(x)\backslash big\backslash vert\_D\backslash right)$ converges to some point in $Y.$ Furthermore, this continuous extension will be unique whenever it exists.{{sfn|Dugundji|1966|pp=216}}

Products

Suppose $X\_\backslash bull\; :=\; \backslash left(X\_i\backslash right)\_\{i\; \backslash in\; I\}$ is a non-empty family of non-empty topological spaces and that is a family of prefilters where each $\backslash mathcal\{B\}\_i$ is a prefilter on $X\_i.$

Then the product $\backslash mathcal\{B\}\_\backslash bull$ of these prefilters (defined above) is a prefilter on the product space $\{\backslash textstyle\backslash prod\}\; X\_\backslash bull,$ which as usual, is endowed with the product topology.

If $x\_\backslash bull\; :=\; \backslash left(x\_i\backslash right)\_\{i\; \backslash in\; I\}\; \backslash in\; \{\backslash textstyle\backslash prod\}\; X\_\backslash bull,$ then $\backslash mathcal\{B\}\_\backslash bull\; \backslash to\; x\_\backslash bull\; \backslash text\{\; in\; \}\; \{\backslash textstyle\backslash prod\}\; X\_\backslash bull$ if and only if $\backslash mathcal\{B\}\_i\; \backslash to\; x\_i\; \backslash text\{\; in\; \}\; X\_i\; \backslash text\{\; for\; every\; \}\; i\; \backslash in\; I.$

Suppose $X\; \backslash text\{\; and\; \}\; Y$ are topological spaces, $\backslash mathcal\{B\}$ is a prefilter on $X$ having $x\; \backslash in\; X$ as a cluster point, and $\backslash mathcal\{C\}$ is a prefilter on $Y$ having $y\; \backslash in\; Y$ as a cluster point.

Then $(x,\; y)$ is a cluster point of $\backslash mathcal\{B\}\; \backslash times\; \backslash mathcal\{C\}$ in the product space $X\; \backslash times\; Y.${{sfn|Bourbaki|1989|pp=132–133}}

However, if $X\; =\; Y\; =\; \backslash Q$ then there exist sequences $\backslash left(x\_i\backslash right)\_\{i=1\}^\backslash infty\; \backslash subseteq\; X\; \backslash text\{\; and\; \}\; \backslash left(y\_i\backslash right)\_\{i=1\}^\backslash infty\; \backslash subseteq\; Y$ such that both of these sequences have a cluster point in $\backslash Q$ but the sequence $\backslash left(x\_i,\; y\_i\backslash right)\_\{i=1\}^\backslash infty\; \backslash subseteq\; X\; \backslash times\; Y$ does {{em|not}} have a cluster point in $X\; \backslash times\; Y.${{sfn|Bourbaki|1989|pp=132–133}}

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

{{collapse top|title=Proof|left=true}}

Let $X\_\backslash bull\; :=\; \backslash left(X\_i\backslash right)\_\{i\; \backslash in\; I\}$ be compact Hausdorff space topological spaces.

Assume that the ultrafilter lemma holds (because of the Hausdorff assumption, this proof does {{em|not}} need the full strength of the axiom of choice; the ultrafilter lemma suffices).

Let $X\; :=\; \{\backslash textstyle\backslash prod\}\; X\_\backslash bull$ be given the product topology (which makes $X$ a Hausdorff space) and for every $i,$ let $\backslash Pr\{\}\_i\; :\; X\; \backslash to\; X\_i$ denote this product's projections.

If $X\; =\; \backslash varnothing$ then $X$ is compact and the proof is complete so assume $X\; \backslash neq\; \backslash varnothing.$

Despite the fact that $X\; \backslash neq\; \backslash varnothing,$ because the axiom of choice is not assumed, the projection maps $\backslash Pr\{\}\_i\; :\; X\; \backslash to\; X\_i$ are not guaranteed to be surjective.

Let $\backslash mathcal\{B\}$ be an ultrafilter on $X$ and for every $i,$ let $\backslash mathcal\{B\}\_i$ denote the ultrafilter on $X\_i$ generated by the ultra prefilter $\backslash Pr\{\}\_i(\backslash mathcal\{B\}).$

Because $X\_i$ is compact and Hausdorff, the ultrafilter $\backslash mathcal\{B\}\_i$ converges to a unique limit point $x\_i\; \backslash in\; X\_i$ (because of $x\_i$'s uniqueness, this definition does not require the axiom of choice).

Let $x\; :=\; \backslash left(x\_i\backslash right)\_\{i\; \backslash in\; I\}$ where $x$ satisfies $\backslash Pr\{\}\_i(x)\; =\; x\_i$ for every $i.$

The characterization of convergence in the product topology that was given above implies that $\backslash mathcal\{B\}\; \backslash to\; x\; \backslash text\{\; in\; \}\; X.$

Thus every ultrafilter on $X$ converges to some point of $X,$ which implies that $X$ is compact (recall that this implication's proof only required the ultrafilter lemma). $\backslash blacksquare$

{{collapse bottom}}

{{Main|Uniform space|Complete uniform space|Complete metric space}}

{{See also|Topological group|Complete topological vector space}}

A uniform space is a set $X$ equipped with a filter on $X\; \backslash times\; X$ that has certain properties. A {{em|base}} or {{em|fundamental system of entourages}} is a prefilter on $X\; \backslash times\; X$ whose upward closure is a uniform space.

A prefilter $\backslash mathcal\{B\}$ on a uniform space $X$ with uniformity $\backslash mathcal\{F\}$ is called a {{em|Cauchy prefilter}} if for every entourage $N\; \backslash in\; \backslash mathcal\{F\},$ there exists some $B\; \backslash in\; \backslash mathcal\{B\}$ that is {{em|$N$-small}}, which means that $B\; \backslash times\; B\; \backslash subseteq\; N.$

A {{em|minimal Cauchy filter}} is a minimal element (with respect to $\backslash ,\backslash leq\backslash ,$ or equivalently, to $\backslash ,\backslash subseteq$) of the set of all Cauchy filters on $X.$

Examples of minimal Cauchy filters include the neighborhood filter $\backslash mathcal\{N\}\_X(x)$ of any point $x\; \backslash in\; X.$

Every convergent filter on a uniform space is Cauchy. Moreover, every cluster point of a Cauchy filter is a limit point.

A uniform space $(X,\; \backslash mathcal\{F\})$ is called {{em|complete}} (resp. {{em|sequentially complete}}) if every Cauchy prefilter (resp. every elementary Cauchy prefilter) on $X$ converges to at least one point of $X$ (replacing all instance of the word "prefilter" with "filter" results in equivalent statement).

Every compact uniform space is complete because any Cauchy filter has a cluster point (by compactness), which is necessarily also a limit point (since the filter is Cauchy).

Uniform spaces were the result of attempts to generalize notions such as "uniform continuity" and "uniform convergence" that are present in metric spaces. Every topological vector space, and more generally, every topological group can be made into a uniform space in a canonical way. Every uniformity also generates a canonical induced topology. Filters and prefilters play an important role in the theory of uniform spaces. For example, the completion of a Hausdorff uniform space (even if it is not metrizable) is typically constructed by using minimal Cauchy filters.

Nets are less ideal for this construction because their domains are extremely varied (for example, the class of all Cauchy nets is not a set); sequences cannot be used in the general case because the topology might not be metrizable, first-countable, or even sequential.

The set of all {{em|minimal Cauchy filters}} on a Hausdorff topological vector space (TVS) $X$ can made into a vector space and topologized in such a way that it becomes a completion of $X$ (with the assignment $x\; \backslash mapsto\; \backslash mathcal\{N\}\_X(x)$ becoming a linear topological embedding that identifies $X$ as a dense vector subspace of this completion).

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

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

The set of all Cauchy filters on a uniform space forms a Cauchy space. Every Cauchy space is also a convergence space.

A map $f\; :\; X\; \backslash to\; Y$ between two Cauchy spaces is called {{em|Cauchy continuous}} if the image of every Cauchy filter in $X$ is a Cauchy filter in $Y.$

Unlike the category of topological spaces, the category of Cauchy spaces and Cauchy continuous maps is Cartesian closed, and contains the category of proximity spaces.

{{See also|Stone space}}

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

$$\backslash mathbb\{P\}\; :=\; \backslash operatorname\{Prefilters\}(X)$$

of all filter bases on $X$ with the {{em|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 $x\; \backslash in\; X.$
• Upper case letters for subsets $S\; \backslash subseteq\; X.$
• Upper case calligraphy letters for subsets $\backslash mathcal\{B\}\; \backslash subseteq\; \backslash wp(X)$ (or equivalently, for elements $\backslash mathcal\{B\}\; \backslash in\; \backslash wp(\backslash wp(X)),$ such as prefilters).
• Upper case double-struck letters for subsets $\backslash mathbb\{P\}\; \backslash subseteq\; \backslash wp(\backslash wp(X)).$

For every $S\; \backslash subseteq\; X,$ let

$$\backslash mathbb\{O\}(S)\; :=\; \backslash left\backslash \{\backslash mathcal\{B\}\; \backslash in\; \backslash mathbb\{P\}\; ~:~\; S\; \backslash in\; \backslash mathcal\{B\}^\{\backslash uparrow\; X\}\backslash right\backslash \}$$

where $\backslash mathbb\{O\}(X)\; =\; \backslash mathbb\{P\}\; \backslash text\{\; and\; \}\; \backslash mathbb\{O\}(\backslash varnothing)\; =\; \backslash varnothing.$As a side note, had the definitions of "filter" and "prefilter" not required propriety then the degenerate dual ideal $\backslash wp(X)$ would have been a prefilter on $X$ so that in particular, $\backslash mathbb\{O\}(\backslash varnothing)\; =\; \backslash \{\backslash wp(X)\backslash \}\; \backslash neq\; \backslash varnothing$ with $\backslash wp(X)\; \backslash in\; \backslash mathbb\{O\}(S)\; \backslash text\{\; for\; every\; \}\; S\; \backslash subseteq\; X.$ These sets will be the basic open subsets of the Stone topology.

If $R\; \backslash subseteq\; S\; \backslash subseteq\; X$ then

$$\backslash left\backslash \{\backslash mathcal\{B\}\; \backslash in\; \backslash wp(\backslash wp(X))\; ~:~\; R\; \backslash in\; \backslash mathcal\{B\}^\{\backslash uparrow\; X\}\backslash right\backslash \}\; ~\backslash subseteq~\; \backslash left\backslash \{\backslash mathcal\{B\}\; \backslash in\; \backslash wp(\backslash wp(X))\; ~:~\; S\; \backslash in\; \backslash mathcal\{B\}^\{\backslash uparrow\; X\}\backslash right\backslash \}.$$

From this inclusion, it is possible to deduce all of the subset inclusions displayed below with the exception of $\backslash mathbb\{O\}(R\; \backslash cap\; S)\; ~\backslash supseteq~\; \backslash mathbb\{O\}(R)\; \backslash cap\; \backslash mathbb\{O\}(S).$This is because the inclusion $\backslash mathbb\{O\}(R\; \backslash cap\; S)\; ~\backslash supseteq~\; \backslash mathbb\{O\}(R)\; \backslash cap\; \backslash mathbb\{O\}(S)$ is the only one in the sequence below whose proof uses the defining assumption that $\backslash mathbb\{O\}(S)\; \backslash subseteq\; \backslash mathbb\{P\}.$

For all $R\; \backslash subseteq\; S\; \backslash subseteq\; X,$

$$\backslash mathbb\{O\}(R\; \backslash cap\; S)\; ~=~\; \backslash mathbb\{O\}(R)\; \backslash cap\; \backslash mathbb\{O\}(S)\; ~\backslash subseteq~\; \backslash mathbb\{O\}(R)\; \backslash cup\; \backslash mathbb\{O\}(S)\; ~\backslash subseteq~\; \backslash mathbb\{O\}(R\; \backslash cup\; S)$$

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

In contrast to most other general constructions of topologies (for example, the product, quotient, subspace topologies, etc.), this topology on $\backslash mathbb\{P\}$ was defined with{{em|out}} using anything other than the set $X;$ there were {{em|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 $\backslash mathbb\{B\}\; \backslash subseteq\; \backslash mathbb\{P\}\; \backslash text\{\; and\; \}\; \backslash mathcal\{F\}\; \backslash in\; \backslash mathbb\{P\}$ then:

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

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

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

Every $\backslash tau\; \backslash in\; \backslash operatorname\{Top\}(X)$ induces a canonical map $\backslash mathcal\{N\}\_\backslash tau\; :\; X\; \backslash to\; \backslash operatorname\{Filters\}(X)$ defined by $x\; \backslash mapsto\; \backslash mathcal\{N\}\_\backslash tau(x),$ which sends $x\; \backslash in\; X$ to the neighborhood filter of $x\; \backslash text\{\; in\; \}\; (X,\; \backslash tau).$

If $\backslash tau,\; \backslash sigma\; \backslash in\; \backslash operatorname\{Top\}(X)$ then $\backslash tau\; =\; \backslash sigma$ if and only if $\backslash mathcal\{N\}\_\backslash tau\; =\; \backslash mathcal\{N\}\_\backslash sigma.$

Thus every topology $\backslash tau\; \backslash in\; \backslash operatorname\{Top\}(X)$ can be identified with the canonical map $\backslash mathcal\{N\}\_\backslash tau\; \backslash in\; \backslash operatorname\{Func\}(X;\; \backslash mathbb\{P\}),$ which allows $\backslash operatorname\{Top\}(X)$ to be canonically identified as a subset of $\backslash operatorname\{Func\}(X;\; \backslash mathbb\{P\})$ (as a side note, it is now possible to place on $\backslash operatorname\{Func\}(X;\; \backslash mathbb\{P\}),$ and thus also on $\backslash operatorname\{Top\}(X),$ the topology of pointwise convergence on $X$ so that it now makes sense to talk about things such as sequences of topologies on $X$ converging pointwise).

For every $\backslash tau\; \backslash in\; \backslash operatorname\{Top\}(X),$ the surjection $\backslash mathcal\{N\}\_\backslash tau\; :\; (X,\; \backslash tau)\; \backslash to\; \backslash operatorname\{image\}\; \backslash mathcal\{N\}\_\backslash tau$ is always continuous, closed, and open, but it is injective if and only if $\backslash tau\; \backslash text\{\; is\; \}\; T\_0$ (that is, a Kolmogorov space).

In particular, for every $T\_0$ topology $\backslash tau\; \backslash text\{\; on\; \}\; X,$ the map $\backslash mathcal\{N\}\_\backslash tau\; :\; (X,\; \backslash tau)\; \backslash to\; \backslash mathbb\{P\}$ is a topological embedding (said differently, every Kolmogorov space is a topological subspace of the space of prefilters).

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

• {{annotated link|Characterizations of the category of topological spaces}}
• {{annotated link|Convergence space}}
• {{annotated link|Filtration (mathematics)}}
• {{annotated link|Filtration (probability theory)}}
• {{annotated link|Filtration (abstract algebra)}}
• {{annotated link|Fréchet filter}}
• {{annotated link|Generic filter}}
• {{annotated link|Ideal (set theory)}}
• {{annotated link|Stone–Čech compactification#Construction using ultrafilters}}
• {{annotated link|The fundamental theorem of ultraproducts}}

{{reflist|group=note}}

Proofs

{{reflist|group=proof}}

{{reflist}}

• {{Adams Franzosa Introduction to Topology Pure and Applied}}
• {{Arkhangel'skii Ponomarev Fundamentals of General Topology Problems and Exercises|edition=2}}
• {{Berberian Lectures in Functional Analysis and Operator Theory}}
• {{Bourbaki General Topology Part I Chapters 1-4}}
• {{Bourbaki General Topology Part II Chapters 5-10}}
• {{Bourbaki Topological Vector Spaces Part 1 Chapters 1–5}}
• {{cite book|last1=Burris|first1=Stanley|author-link1=Stanley Burris|last2=Sankappanavar|first2=Hanamantagouda P.|year=2012|title=A Course in Universal Algebra|publisher=Springer-Verlag|isbn=978-0-9880552-0-9|url=http://www.thoralf.uwaterloo.ca/htdocs/ualg.html|archive-url=https://web.archive.org/web/20220401154440/https://www.math.uwaterloo.ca/~snburris/htdocs/UALG/univ-algebra2012.pdf|archive-date=1 April 2022}}
• {{cite journal|last=Cartan|first=Henri|author-link=Henri Cartan|title=Théorie des filtres|title-link=|journal=Comptes rendus hebdomadaires des séances de l'Académie des sciences|volume=205|year=1937a|pages=595–598|url=http://gallica.bnf.fr/ark:/12148/bpt6k3157c/f594.image}}
• {{cite journal|last=Cartan|first=Henri|author-link=Henri Cartan|title=Filtres et ultrafiltres|title-link=|journal=Comptes rendus hebdomadaires des séances de l'Académie des sciences|volume=205|year=1937b|pages=777–779|url=http://gallica.bnf.fr/ark:/12148/bpt6k3157c/f776.image}}
• {{Comfort Negrepontis The Theory of Ultrafilters 1974}}
• {{Császár General Topology}}
• {{Dixmier General Topology}}
• {{Dolecki Mynard Convergence Foundations Of Topology}}
• {{Dugundji Topology}}
• {{Dunford Schwartz Linear Operators Part 1 General Theory}}
• {{Edwards Functional Analysis Theory and Applications}}
• {{Howes Modern Analysis and Topology 1995}}
• {{Jarchow Locally Convex Spaces}}
• {{cite book|last=Jech|first=Thomas|author-link=Thomas Jech|title=Set Theory: The Third Millennium Edition, Revised and Expanded|publisher=Springer Science & Business Media|publication-place=Berlin New York|year=2006|isbn=978-3-540-44085-7|oclc=50422939}}
• {{Joshi Introduction to General Topology}}
• {{Kelley General Topology}}
• {{Köthe Topological Vector Spaces I}}
• {{cite web|last=MacIver R.|first=David|title=Filters in Analysis and Topology|date=1 July 2004|url=http://www.efnet-math.org/~david/mathematics/filters.pdf |archive-url=https://web.archive.org/web/20071009170540/http://www.efnet-math.org/~david/mathematics/filters.pdf |archive-date=2007-10-09 }} (Provides an introductory review of filters in topology and in metric spaces.)
• {{Narici Beckenstein Topological Vector Spaces|edition=2}}
• {{Robertson Topological Vector Spaces}}
• {{Schaefer Wolff Topological Vector Spaces|edition=2}}
• {{Schechter Handbook of Analysis and Its Foundations}}
• {{Schubert Topology}}
• {{Trèves François Topological vector spaces, distributions and kernels}}
• {{Wilansky Modern Methods in Topological Vector Spaces|edition=1}}
• {{Wilansky Topology for Analysis 2008}}
• {{Willard General Topology}}

{{Topology}}

Category:Filters

Category:General topology