Filter on a set

 ( General Topology ) 

In mathematics, a filter on a set is a family of subsets which is closed under supersets and finite intersections. The concept originates in topology, where the neighborhoods of a point form a filter on the space. Filters were introduced by Henri Cartan in 1937 and, as described in the article dedicated to filters in topology, they were subsequently used by Nicolas Bourbaki in their book Topologie Générale as an alternative to the related notion of a net developed in 1922 by E. H. Moore and Herman L. Smith. They have also found applications in model theory and set theory.

Filters on a set were later generalized to order filters. Specifically, a filter on a set $X$ is a order filter on the power set of $X$ ordered by inclusion.

The notion dual to a filter is an ideal. Ultrafilters are a particularly important subclass of filters.

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Definition Given a set $X$, a filter $\backslash mathcal\{F\}$ on $X$ is a set of subsets of $X$ such that:

• $F$ is upwards-closed: If $A,\; B\; \backslash subseteq\; X$ are such that $A\; \backslash in\; \backslash mathcal\{F\}$ and $A\; \backslash subseteq\; B$ then $B\; \backslash in\; \backslash mathcal\{F\}$,
• $F$ is closed under finite intersections: $X\; \backslash in\; \backslash mathcal\{F\}$,, and if $A\; \backslash in\; \backslash mathcal\{F\}$ and $B\; \backslash in\; \backslash mathcal\{F\}$ then $A\; \backslash cap\; B\; \backslash in\; \backslash mathcal\{F\}$.

A (or non-degenerate) filter is a filter which is proper as a subset of the powerset $\backslash mathcal\{P\}(X)$ (i.e., the only improper filter is $\backslash mathcal\{P\}(X)$, consisting of all possible subsets). By upwards-closure, a filter is proper if and only if it does not contain the empty set. Many authors adopt the convention that a filter must be proper by definition.

When $\backslash mathcal\{F\}$ and $\backslash mathcal\{G\}$ are two filters on the same set such that $\backslash mathcal\{F\}\; \backslash subseteq\; \backslash mathcal\{G\}$ holds, $\backslash mathcal\{F\}$ is said to be coarser than $\backslash mathcal\{G\}$ (or a subfilter of $\backslash mathcal\{G\}$) while $\backslash mathcal\{G\}$ is said to be finer than $\backslash mathcal\{F\}$ (or to $\backslash mathcal\{F\}$ or a superfilter of $\backslash mathcal\{F\}$).

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Examples

• The singleton set $\backslash mathcal\{F\}\; =\; \backslash \{X\backslash \}$ is called the trivial or indiscrete filter on $X$.
• If $Y$ is a subset of $X$, the subsets of $X$ which are supersets of $Y$ form a principal filter.
• If $X$ is a topological space and $x\; \backslash in\; X$, then the set of neighborhoods of $x$ is a filter on $X$, the neighborhood filter or vicinity filter of $x$.
• Many examples arise from various "largeness" conditions:
• If $X$ is a set, the set of all Cofiniteness of $X$ (i.e., those sets whose complement in $X$ is finite) is a filter on $X$, the Fréchet filter (or cofinite filter).
• Similarly, if $X$ is a set, the cocountable subsets of $X$ (those whose complement is countable) form a filter, the cocountable filter which is finer than the Fréchet filter. More generally, for any cardinal number $\backslash kappa$, the subsets whose complement has cardinal at most $\backslash kappa$ form a filter.
• If $X$ is a metric space, e.g., $\backslash R^n$, the co-bounded subsets of $X$ (those whose complement is bounded set) form a filter on $X$.
• If $X$ is a complete measure space (e.g., $\backslash R^n$ with the Lebesgue measure), the conull set subsets of $X$, i.e., the subsets whose complement has measure zero, form a filter on $X$. (For a non-complete measure space, one can take the subsets which, while not necessarily measurable, are contained in a measurable subset of measure zero.)
• Similarly, if $X$ is a measure space, the subsets whose complement is contained in a measurable subset of finite measure form a filter on $X$.
• If $X$ is a topological space, the comeager set subsets of $X$, i.e., those whose complement is meager set, form a filter on $X$.
• The subsets of $\backslash mathbb\{N\}$ which have a natural density of 1 form a filter on $\backslash mathbb\{N\}$.
  Thomas Jech (2025). 9783540440857, Springer Science & Business Media. ISBN 9783540440857
• The club filter of a Regular cardinal uncountable Cardinal number $\backslash kappa$ is the filter of all sets containing a Club set of $\backslash kappa$.
• If $(\backslash mathcal\{F\}\_i)\_\{i\; \backslash in\; I\}$ is a family of filters on $X$ and $\backslash mathcal\{J\}$ is a filter on $I$ then $\backslash bigcup\_\{A\; \backslash in\; \backslash mathcal\{J\}\}\; \backslash bigcap\_\{i\; \backslash in\; A\}\; \backslash mathcal\{F\}\_i$ is a filter on $X$ called Kowalsky's filter.

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Principal and free filters The kernel of a filter $\backslash mathcal\{F\}$ on $X$ is the intersection of all the subsets of $X$ in $F$.

A filter $\backslash mathcal\{F\}$ on $X$ is principal (or atomic) when it has a particularly simple form: it contains exactly the supersets of $Y$, for some fixed subset $Y\; \backslash subseteq\; X$. When $Y\; =\; \backslash varnothing$, this yields the improper filter. When $Y\; =\; \backslash \{y\backslash \}$ is a singleton, this filter (which consists of all subsets that contain $y$) is called the fundamental filter (or discrete filter) associated with $y$.

A filter $\backslash mathcal\{F\}$ is principal if and only if the kernel of $\backslash mathcal\{F\}$ is an element of $\backslash mathcal\{F\}$, and when this is the case, $\backslash mathcal\{F\}$ consists of the supersets of its kernel. On a finite set, every filter is principal (since the intersection defining the kernel is finite).

A filter is said to be free when it has empty kernel, otherwise it is fixed (and if $x$ is an element of the kernel, it is fixed by $x$). A filter on a set $X$ is free if and only if it contains the Fréchet filter on $X$.

Two filters $\backslash mathcal\{F\}\_1$ and $\backslash mathcal\{F\}\_2$ on $X$ mesh when every member of $\backslash mathcal\{F\}\_1$ intersects every member of $\backslash mathcal\{F\}\_2$. For every filter $\backslash mathcal\{F\}$ on $X$, there exists a unique pair of filters $\backslash mathcal\{F\}\_f$ (the free part of $\backslash mathcal\{F\}$) and $\backslash mathcal\{F\}\_p$ (the principal part of $\backslash mathcal\{F\}$) on $X$ such that $\backslash mathcal\{F\}\_f$ is free, $\backslash mathcal\{F\}\_p$ is principal, $\backslash mathcal\{F\}\_f\; \backslash cap\; \backslash mathcal\{F\}\_p\; =\; \backslash mathcal\{F\}$, and $\backslash mathcal\{F\}\_p$ does not mesh with $\backslash mathcal\{F\}\_f$. The principal part $\backslash mathcal\{F\}\_p$ is the principal filter generated by the kernel of $\backslash mathcal\{F\}$, and the free part $\backslash mathcal\{F\}\_f$ consists of elements of $\backslash mathcal\{F\}$ with any number of elements from the kernel possibly removed.

A filter $\backslash mathcal\{F\}$ is countably deep if the kernel of any countable subset of $\backslash mathcal\{F\}$ belongs to $\backslash mathcal\{F\}$.

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Correspondence with order filters The concept of a filter on a set is a special case of the more general concept of a filter on a partially ordered set. By definition, a filter on a partially ordered set $P$ is a subset $F$ of $P$ which is upwards-closed (if $x\; \backslash in\; F$ and $x\; \backslash leq\; y$ then $y\; \backslash in\; F$) and downwards-directed (every finite subset of $F$ has a lower bound in $F$). A filter on a set $X$ is the same as a filter on the powerset $\backslash mathcal\{P\}(X)$ ordered by inclusion.

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Constructions of filters

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Intersection of filters If $(\backslash mathcal\{F\}\_i)\_\{i\; \backslash in\; I\}$ is a family of filters on $X$, its intersection $\backslash bigcap\_\{i\; \backslash in\; I\}\; \backslash mathcal\{F\}\_i$ is a filter on $X$. The intersection is a greatest lower bound operation in the set of filters on $X$ partially ordered by inclusion, which endows the filters on $X$ with a complete lattice structure.

The intersection $\backslash bigcap\_\{i\; \backslash in\; I\}\; \backslash mathcal\{F\}\_i$ consists of the subsets which can be written as $\backslash bigcup\_\{i\; \backslash in\; I\}\; A\_i$ where $A\_i\; \backslash in\; \backslash mathcal\{F\}\_i$ for each $i\; \backslash in\; I$.

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Filter generated by a family of subsets Given a family of subsets $\backslash mathcal\{S\}\; \backslash subseteq\; \backslash mathcal\{P\}(X)$, there exists a minimum filter on $X$ (in the sense of inclusion) which contains $\backslash mathcal\{S\}$. It can be constructed as the intersection (greatest lower bound) of all filters on $X$ containing $\backslash mathcal\{S\}$. This filter $\backslash langle\; \backslash mathcal\{S\}\; \backslash rangle$ is called the filter generated by $\backslash mathcal\{S\}$, and $\backslash mathcal\{S\}$ is said to be a filter subbase of $\backslash langle\; \backslash mathcal\{S\}\; \backslash rangle$.

The generated filter can also be described more explicitly: $\backslash langle\; \backslash mathcal\{S\}\; \backslash rangle$ is obtained by closing $\backslash mathcal\{S\}$ under finite intersections, then upwards, i.e., $\backslash langle\; \backslash mathcal\{S\}\; \backslash rangle$ consists of the subsets $Y\; \backslash subseteq\; X$ such that $A\_0\; \backslash cap\; \backslash dots\; \backslash cap\; A\_\{n-1\}\; \backslash subseteq\; Y$ for some $A\_0,\; \backslash dots,\; A\_\{n-1\}\; \backslash in\; \backslash mathcal\{B\}$.

Since these operations preserve the kernel, it follows that $\backslash langle\; \backslash mathcal\{S\}\; \backslash rangle$ is a proper filter if and only if $\backslash mathcal\{S\}$ has the finite intersection property: the intersection of a finite subfamily of $\backslash mathcal\{S\}$ is non-empty.

In the complete lattice of filters on $X$ ordered by inclusion, the least upper bound of a family of filters $(\backslash mathcal\{F\}\_i)\_\{i\; \backslash in\; I\}$ is the filter generated by $\backslash bigcup\_\{i\; \backslash in\; I\}\; \backslash mathcal\{F\}\_i$.

Two filters $\backslash mathcal\{F\}\_1$ and $\backslash mathcal\{F\}\_2$ on $X$ mesh if and only if $\backslash langle\; \backslash mathcal\{F\}\_1\; \backslash cup\; \backslash mathcal\{F\}\_2\; \backslash rangle$ is proper.

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Filter bases Let $\backslash mathcal\{F\}$ be a filter on $X$. A filter base of $\backslash mathcal\{F\}$ is a family of subsets $\backslash mathcal\{B\}\; \backslash subseteq\; \backslash mathcal\{P\}(X)$ such that $\backslash mathcal\{F\}$ is the upwards closure of $\backslash mathcal\{B\}$, i.e., $\backslash mathcal\{F\}$ consists of those subsets $Y\; \backslash subseteq\; X$ for which $A\; \backslash subseteq\; Y$ for some $A\; \backslash in\; \backslash mathcal\{B\}$.

This upwards closure is a filter if and only if $\backslash mathcal\{B\}$ is downwards-directed, i.e., $\backslash mathcal\{B\}$ is non-empty and for all $A,\; B\; \backslash in\; \backslash mathcal\{B\}$ there exists $C\; \backslash in\; \backslash mathcal\{B\}$ such that $C\; \backslash subseteq\; A\; \backslash cap\; B$. When this is the case, $\backslash mathcal\{B\}$ is also called a prefilter, and the upwards closure is also equal to the generated filter $\backslash langle\; \backslash mathcal\{B\}\; \backslash rangle$. Hence, being a filter base of $\backslash mathcal\{F\}$ is a stronger property than being a filter subbase of $\backslash mathcal\{F\}$.

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Examples

• When $X$ is a topological space and $x\; \backslash in\; X$, a filter base of the neighborhood filter of $x$ is known as a neighborhood base for $x$, and similarly, a filter subbase of the neighborhood filter of $x$ is known as a neighborhood subbase for $x$. The open set neighborhoods of $x$ always form a neighborhood base for $x$, by definition of the neighborhood filter. In $X\; =\; \backslash R^n$, the of positive radius around $x$ also form a neighborhood base for $x$.
• Let $X$ be an infinite set and let $\backslash mathcal\{F\}$ consist of the subsets of $X$ which contain all points but one. Then $\backslash mathcal\{F\}$ is a filter subbase of the Fréchet filter on $X$, which consists of the . Its closure under finite intersections is the entire Fréchet filter, but there are smaller bases of the Fréchet filter which contain the subbase $\backslash mathcal\{F\}$, such as the one formed by the subsets of $X$ which contain all points but a finite odd number. In fact, for every base of the Fréchet filter, removing any subset yields another base of the Fréchet filter.
• If $X$ is a topological space, the dense set of $X$ form a filter base on $X$, because they are closed under finite intersection. The filter they generate consists of the complements of nowhere dense subsets. On $X\; =\; \backslash R^n$, restricting to the null set dense open subsets yields another filter base for the same filter.
• Similarly, if $X$ is a topological space, the countable intersections of dense open subsets form a filter base which generates the filter of comeager subsets.
• Let $X$ be a set and let $(x\_i)\_\{i\; \backslash in\; I\}$ be a net with values in $X$, i.e., a family whose domain $I$ is a directed set. The filter base of tails of $(x\_i)$ consists of the sets $\backslash \{x\_j,\; j\; \backslash geq\; i\backslash \}$ for $i\; \backslash in\; I$; it is downwards-closed by directedness of $I$. The generated filter is called the eventuality filter or filter of tails of $(x\_n)$. A sequential filter or is a filter which is the eventuality filter of some net. This example is fundamental in the application of filters in topology.
• Every π-system is a filter base.

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Trace of a filter on a subset If $\backslash mathcal\{F\}$ is a filter on $X$ and $Y\; \backslash subseteq\; X$, the trace of $\backslash mathcal\{F\}$ on $Y$ is $\backslash \{A\; \backslash cap\; Y,\; A\; \backslash in\; \backslash mathcal\{F\}\backslash \}$, which is a filter.

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Image of a filter by a function Let $f\; :\; X\; \backslash to\; Y$ be a function.

When $\backslash mathcal\{F\}$ is a family of subsets of $X$, its image by $f$ is defined as

$$f(\backslash mathcal\{F\})\; =\; \backslash \{\backslash \{f(x),\; x\; \backslash in\; A\backslash \},\; A\; \backslash in\; \backslash mathcal\{F\}\backslash \}$$

The image filter by $f$ of a filter $\backslash mathcal\{F\}$ on $X$ is defined as the generated filter $\backslash langle\; f(\backslash mathcal\{F\})\; \backslash rangle$. If $f$ is surjective, then $f(\backslash mathcal\{F\})$ is already a filter. In the general case, $f(\backslash mathcal\{F\})$ is a filter base and hence $\backslash langle\; f(\backslash mathcal\{F\})\; \backslash rangle$ is its upwards closure. Furthermore, if $\backslash mathcal\{B\}$ is a filter base of $\backslash mathcal\{F\}$ then $f(\backslash mathcal\{B\})$ is a filter base of $\backslash langle\; f(\backslash mathcal\{F\})\; \backslash rangle$.

The kernels of $\backslash mathcal\{F\}$ and $\backslash langle\; f(\backslash mathcal\{F\})\; \backslash rangle$ are linked by $f\backslash left(\backslash bigcap\; \backslash mathcal\{F\}\backslash right)\; \backslash subseteq\; \backslash bigcap\; \backslash langle\; f(\backslash mathcal\{F\})\; \backslash rangle$.

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Product of filters Given a family of sets $(X\_i)\_\{i\; \backslash in\; I\}$ and a filter $\backslash mathcal\{F\}\_i$ on each $X\_i$, the product filter $\backslash prod\_\{i\; \backslash in\; I\}\; \backslash mathcal\{F\}\_i$ on the product set $\backslash prod\_\{i\; \backslash in\; I\}\; X\_i$ is defined as the filter generated by the sets $\backslash pi\_i^\{-1\}(A)$ for $i\; \backslash in\; I$ and $A\; \backslash in\; \backslash mathcal\{F\}\_i$, where $\backslash pi\_i\; :\; \backslash left(\backslash prod\_\{j\; \backslash in\; I\}\; X\_j\backslash right)\; \backslash to\; X\_i$ is the projection from the product set onto the $i$-th component. This construction is similar to the product topology.

If each $\backslash mathcal\{B\}\_i$ is a filter base on $\backslash mathcal\{F\}\_i$, a filter base of $\backslash prod\_\{i\; \backslash in\; I\}\; \backslash mathcal\{F\}\_i$ is given by the sets $\backslash prod\_\{i\; \backslash in\; I\}\; A\_i$ where $(A\_i)$ is a family such that $A\_i\; \backslash in\; \backslash mathcal\{F\}\_i$ for all $i\; \backslash in\; I$ and $A\_i\; =\; X\_i$ for all but finitely many $i\; \backslash in\; I$.

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See also

• Axiomatic foundations of topological spaces, for a definition of topological spaces in terms of filters
• Convergence space, a generalization of topological spaces using filters
• Generic filter, a kind of filter used in set-theoretic forcing

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Notes

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Citations

• (2025). 9780988055209, Springer-Verlag. . ISBN 9780988055209
• (1978). 9780852742754, Adam Hilger Ltd. ISBN 9780852742754
• (2025). 9789814571524, World Scientific Publishing Company. ISBN 9789814571524
• K. D. Joshi (1983). 9780852264447, John Wiley and Sons Ltd. ISBN 9780852264447
• Gottfried Köthe (1983). 9783642649882, Springer Science & Business Media. ISBN 9783642649882
• (Provides an introductory review of filters in topology and in metric spaces.)
• (2025). 9781584888666, CRC Press. ISBN 9781584888666
• (1996). 9780126227604, Academic Press. ISBN 9780126227604
• Horst Schubert (1968). 9780356020778, Macdonald & Co. ISBN 9780356020778
• Albert Wilansky (2025). 9780486493534, Dover Publications, Inc. ISBN 9780486493534
• Stephen Willard (2025). 9780486434797, Dover Publications. . ISBN 9780486434797

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