Filter on a set

Examples

• The singleton set ${\displaystyle \mathcal{F}=\{X\}}$ is called the trivial or indiscrete filter on ${\displaystyle X}$.
• If ${\displaystyle Y}$ is a subset of ${\displaystyle X}$, the subsets of ${\displaystyle X}$ which are supersets of ${\displaystyle Y}$ form a principal filter.
• If ${\displaystyle X}$ is a topological space and ${\displaystyle x\in X}$, then the set of neighborhoods of ${\displaystyle x}$ is a filter on ${\displaystyle X}$, the neighborhood filter or vicinity filter of ${\displaystyle x}$.
• Many examples arise from various "largeness" conditions:
  • If ${\displaystyle X}$ is a set, the set of all cofinite subsets of ${\displaystyle X}$ (i.e., those sets whose complement in ${\displaystyle X}$ is finite) is a filter on ${\displaystyle X}$, the Fréchet filter (or cofinite filter).
  • Similarly, if ${\displaystyle X}$ is a set, the cocountable subsets of ${\displaystyle X}$ (those whose complement is countable) form a filter, the cocountable filter which is finer than the Fréchet filter. More generally, for any infinite cardinal ${\displaystyle \kappa }$, the subsets whose complement has cardinality at most ${\displaystyle \kappa }$ form a filter.
  • If ${\displaystyle X}$ is a metric space, e.g., ${\displaystyle {\mathbb{R}}^n}$, the co-bounded subsets of ${\displaystyle X}$ (those whose complement is bounded set) form a filter on ${\displaystyle X}$.
  • If ${\displaystyle X}$ is a complete measure space (e.g., ${\displaystyle {\mathbb{R}}^n}$ with the Lebesgue measure), the conull subsets of ${\displaystyle X}$, i.e., the subsets whose complement has measure zero, form a filter on ${\displaystyle X}$. (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 ${\displaystyle X}$ is a measure space, the subsets whose complement is contained in a measurable subset of finite measure form a filter on ${\displaystyle X}$.
  • If ${\displaystyle X}$ is a topological space, the comeager subsets of ${\displaystyle X}$, i.e., those whose complement is meager, form a filter on ${\displaystyle X}$.
  • The subsets of ${\displaystyle \mathbb{N}}$ which have a natural density of 1 form a filter on ${\displaystyle \mathbb{N}}$.
• The club filter of a regular uncountable cardinal ${\displaystyle \kappa }$ is the filter of all sets containing a club subset of ${\displaystyle \kappa }$.
• If ${\displaystyle ({\mathcal{F}}_i{)}_{i\in I}}$ is a family of filters on ${\displaystyle X}$ and ${\displaystyle \mathcal{J}}$ is a filter on ${\displaystyle I}$ then ${\displaystyle \bigcup _{A\in \mathcal{J}}\bigcap _{i\in A}{\mathcal{F}}_i}$ is a filter on ${\displaystyle X}$ called Kowalsky's filter.

Principal and free filters

The kernel of a filter ${\displaystyle \mathcal{F}}$ on ${\displaystyle X}$ is the intersection of all the subsets of ${\displaystyle X}$ in ${\displaystyle \mathcal{F}}$.

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

A filter ${\displaystyle \mathcal{F}}$ is principal if and only if the kernel of ${\displaystyle \mathcal{F}}$ is an element of ${\displaystyle \mathcal{F}}$, and when this is the case, ${\displaystyle \mathcal{F}}$ 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 ${\displaystyle x}$ is an element of the kernel, it is fixed by ${\displaystyle x}$). A filter on a set ${\displaystyle X}$ is free if and only if it contains the Fréchet filter on ${\displaystyle X}$.

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

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

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 ${\displaystyle P}$ is a subset ${\displaystyle \mathcal{F}}$ of ${\displaystyle P}$ which is upwards-closed (if ${\displaystyle x\in \mathcal{F}}$ and ${\displaystyle x\le y}$ then ${\displaystyle y\in \mathcal{F}}$) and downwards-directed (every finite subset of ${\displaystyle \mathcal{F}}$ has a lower bound in ${\displaystyle \mathcal{F}}$). A filter on a set ${\displaystyle X}$ is the same as a filter on the powerset ${\displaystyle \mathcal{P}(X)}$ ordered by inclusion.

Constructions of filters

Intersection of filters

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

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

Filter generated by a family of subsets

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

The generated filter can also be described more explicitly: ${\displaystyle ⟨\mathcal{S}⟩}$ is obtained by closing ${\displaystyle \mathcal{S}}$ under finite intersections, then upwards, i.e., ${\displaystyle ⟨\mathcal{S}⟩}$ consists of the subsets ${\displaystyle Y\subseteq X}$ such that ${\displaystyle A_0\cap \cdots \cap A_{n-1}\subseteq Y}$ for some ${\displaystyle A_0,\dots ,A_{n-1}\in \mathcal{B}}$.

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

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

Two filters ${\displaystyle {\mathcal{F}}_1}$ and ${\displaystyle {\mathcal{F}}_2}$ on ${\displaystyle X}$ mesh if and only if ${\displaystyle ⟨{\mathcal{F}}_1\cup {\mathcal{F}}_2⟩}$ is proper.

Filter bases

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

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

Examples

• When ${\displaystyle X}$ is a topological space and ${\displaystyle x\in X}$, a filter base of the neighborhood filter of ${\displaystyle x}$ is known as a neighborhood base for ${\displaystyle x}$, and similarly, a filter subbase of the neighborhood filter of ${\displaystyle x}$ is known as a neighborhood subbase for ${\displaystyle x}$. The open neighborhoods of ${\displaystyle x}$ always form a neighborhood base for ${\displaystyle x}$, by definition of the neighborhood filter. In ${\displaystyle X={\mathbb{R}}^n}$, the closed balls of positive radius around ${\displaystyle x}$ also form a neighborhood base for ${\displaystyle x}$.
• Let ${\displaystyle X}$ be an infinite set and let ${\displaystyle \mathcal{F}}$ consist of the subsets of ${\displaystyle X}$ which contain all points but one. Then ${\displaystyle \mathcal{F}}$ is a filter subbase of the Fréchet filter on ${\displaystyle X}$, which consists of the cofinite subsets. Its closure under finite intersections is the entire Fréchet filter, but there are smaller bases of the Fréchet filter which contain the subbase ${\displaystyle \mathcal{F}}$, such as the one formed by the subsets of ${\displaystyle X}$ 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 ${\displaystyle X}$ is a topological space, the dense open subsets of ${\displaystyle X}$ form a filter base on ${\displaystyle X}$, because they are closed under finite intersection. The filter they generate consists of the complements of nowhere dense subsets. On ${\displaystyle X={\mathbb{R}}^n}$, restricting to the null dense open subsets yields another filter base for the same filter.
• Similarly, if ${\displaystyle X}$ is a topological space, the countable intersections of dense open subsets form a filter base which generates the filter of comeager subsets.
• Let ${\displaystyle X}$ be a set and let ${\displaystyle (x_i{)}_{i\in I}}$ be a net with values in ${\displaystyle X}$, i.e., a family whose domain ${\displaystyle I}$ is a directed set. The filter base of tails of ${\displaystyle (x_i)}$ consists of the sets ${\displaystyle \{x_j,j\ge i\}}$ for ${\displaystyle i\in I}$; it is downwards-closed by directedness of ${\displaystyle I}$. The generated filter is called the eventuality filter or filter of tails of ${\displaystyle (x_n)}$. A sequential filter or elementary filter 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.

Trace of a filter on a subset

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

Image of a filter by a function

Let ${\displaystyle f:X\to Y}$ be a function.

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

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

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

The kernels of ${\displaystyle \mathcal{F}}$ and ${\displaystyle ⟨f(\mathcal{F})⟩}$ are linked by ${\displaystyle f\left(\bigcap \mathcal{F}\right)\subseteq \bigcap ⟨f(\mathcal{F})⟩}$.

Product of filters

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

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