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^[1][2] 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 👁 {\displaystyle X}
is an order filter on the power set of 👁 {\displaystyle X}
ordered by inclusion.

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

Given a set 👁 {\displaystyle X}
, a filter 👁 {\displaystyle {\mathcal {F}}}
on 👁 {\displaystyle X}
is a set of subsets of 👁 {\displaystyle X}
such that:^[3][4][5]

• 👁 {\displaystyle {\mathcal {F}}}
  is upwards-closed: If 👁 {\displaystyle A,B\subseteq X}
  are such that 👁 {\displaystyle A\in {\mathcal {F}}}
  and 👁 {\displaystyle A\subseteq B}
  then 👁 {\displaystyle B\in {\mathcal {F}}}
  ,
• 👁 {\displaystyle {\mathcal {F}}}
  is closed under finite intersections: 👁 {\displaystyle X\in {\mathcal {F}}}
  ,^[a], and if 👁 {\displaystyle A\in {\mathcal {F}}}
  and 👁 {\displaystyle B\in {\mathcal {F}}}
  then 👁 {\displaystyle A\cap B\in {\mathcal {F}}}
  .

A proper (or non-degenerate) filter is a filter which is proper as a subset of the powerset 👁 {\displaystyle {\mathcal {P}}(X)}
(i.e., the only improper filter is 👁 {\displaystyle {\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.^[4] Many authors adopt the convention that a filter must be proper by definition.^[6][7][8][9]

When 👁 {\displaystyle {\mathcal {F}}}
and 👁 {\displaystyle {\mathcal {G}}}
are two filters on the same set such that 👁 {\displaystyle {\mathcal {F}}\subseteq {\mathcal {G}}}
holds, 👁 {\displaystyle {\mathcal {F}}}
is said to be coarser^[10] than 👁 {\displaystyle {\mathcal {G}}}
(or a subfilter of 👁 {\displaystyle {\mathcal {G}}}
) while 👁 {\displaystyle {\mathcal {G}}}
is said to be finer^[10] than 👁 {\displaystyle {\mathcal {F}}}
(or subordinate to 👁 {\displaystyle {\mathcal {F}}}
or a superfilter^[11] of 👁 {\displaystyle {\mathcal {F}}}
).

• The singleton set 👁 {\displaystyle {\mathcal {F}}=\{X\}}
  is called the trivial or indiscrete filter on 👁 {\displaystyle X}
  .^[12]
• If 👁 {\displaystyle Y}
  is a subset of 👁 {\displaystyle X}
  , the subsets of 👁 {\displaystyle X}
  which are supersets of 👁 {\displaystyle Y}
  form a principal filter.^[3]
• 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^[13] or vicinity filter^[14] 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^[12][15][5] (or cofinite filter^[13]).
  • Similarly, if 👁 {\displaystyle X}
    is a set, the cocountable subsets of 👁 {\displaystyle X}
    (those whose complement is countable) form a filter, the cocountable filter^[14] which is finer than the Fréchet filter. More generally, for any cardinal 👁 {\displaystyle \kappa }
    , the subsets whose complement has cardinal 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}
    .^[16]
  • 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} }
    .^[17]
• 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.^[18]

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^[3] (or atomic^[13]) 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^[3] (or discrete filter^[19]) 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.^[20] 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}
).^[21] A filter on a set 👁 {\displaystyle X}
is free if and only if it contains the Fréchet filter on 👁 {\displaystyle X}
.^[22]

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}}
.^[23] 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.^[22]

A filter 👁 {\displaystyle {\mathcal {F}}}
is countably deep if the kernel of any countable subset of 👁 {\displaystyle {\mathcal {F}}}
belongs to 👁 {\displaystyle {\mathcal {F}}}
.^[14]

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\leq 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.^[b]

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.^[14][24]

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}
.

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 \langle {\mathcal {S}}\rangle }
is called the filter generated by 👁 {\displaystyle {\mathcal {S}}}
, and 👁 {\displaystyle {\mathcal {S}}}
is said to be a filter subbase of 👁 {\displaystyle \langle {\mathcal {S}}\rangle }
. ^[25]

The generated filter can also be described more explicitly: 👁 {\displaystyle \langle {\mathcal {S}}\rangle }
is obtained by closing 👁 {\displaystyle {\mathcal {S}}}
under finite intersections, then upwards, i.e., 👁 {\displaystyle \langle {\mathcal {S}}\rangle }
consists of the subsets 👁 {\displaystyle Y\subseteq X}
such that 👁 {\displaystyle A_{0}\cap \dots \cap A_{n-1}\subseteq Y}
for some 👁 {\displaystyle A_{0},\dots ,A_{n-1}\in {\mathcal {B}}}
.^[11]

Since these operations preserve the kernel, it follows that 👁 {\displaystyle \langle {\mathcal {S}}\rangle }
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.^[16]

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}}
.^[20]

Two filters 👁 {\displaystyle {\mathcal {F}}_{1}}
and 👁 {\displaystyle {\mathcal {F}}_{2}}
on 👁 {\displaystyle X}
mesh if and only if 👁 {\displaystyle \langle {\mathcal {F}}_{1}\cup {\mathcal {F}}_{2}\rangle }
is proper.^[23]

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}}}
.^[6]

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}
.^[6][13] 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 \langle {\mathcal {B}}\rangle }
.^[16] Hence, being a filter base of 👁 {\displaystyle {\mathcal {F}}}
is a stronger property than being a filter subbase of 👁 {\displaystyle {\mathcal {F}}}
.

• 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.^{[citation needed]}
• 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\geq 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^[26] or elementary filter^[9] is a filter which is the eventuality filter of some net. This example is fundamental in the application of filters in topology.^[13][27]
• Every π-system is a filter base.

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.^[15]

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 \langle f({\mathcal {F}})\rangle }
.^[28] 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 \langle f({\mathcal {F}})\rangle }
is its upwards closure.^[29] Furthermore, if 👁 {\displaystyle {\mathcal {B}}}
is a filter base of 👁 {\displaystyle {\mathcal {F}}}
then 👁 {\displaystyle f({\mathcal {B}})}
is a filter base of 👁 {\displaystyle \langle f({\mathcal {F}})\rangle }
.

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

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.^[12][30] 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}
.^[12][31]

• Axiomatic foundations of topological spaces, for a definition of topological spaces in terms of filters
• Filters in topology – Use of filters to describe and characterize all basic topological notions and results
• Convergence space, a generalization of topological spaces using filters
• Filter quantifier
• Ultrafilter – Maximal proper filter
• Generic filter, a kind of filter used in set-theoretic forcing

Families 👁 {\displaystyle {\mathcal {F}}} of sets over 👁 {\displaystyle \Omega } v t e
Is necessarily true of 👁 {\displaystyle {\mathcal {F}}\colon } or, is 👁 {\displaystyle {\mathcal {F}}} closed under:	Directed by 👁 {\displaystyle \,\supseteq }	👁 {\displaystyle A\cap B}	👁 {\displaystyle A\cup B}	👁 {\displaystyle B\setminus A}	👁 {\displaystyle \Omega \setminus A}	👁 {\displaystyle A_{1}\cap A_{2}\cap \cdots }	👁 {\displaystyle A_{1}\cup A_{2}\cup \cdots }	👁 {\displaystyle \Omega \in {\mathcal {F}}}	👁 {\displaystyle \varnothing \in {\mathcal {F}}}	F.I.P.
π-system	👁 Yes	👁 Yes	👁 No	👁 No	👁 No	👁 No	👁 No	👁 No	👁 No	👁 No
Semiring	👁 Yes	👁 Yes	👁 No	👁 No	👁 No	👁 No	👁 No	👁 No	👁 Yes	Never
Semialgebra (semifield)	👁 Yes	👁 Yes	👁 No	👁 No	👁 No	👁 No	👁 No	👁 No	👁 Yes	Never
Monotone class	👁 No	👁 No	👁 No	👁 No	👁 No	only if 👁 {\displaystyle A_{i}\searrow }	only if 👁 {\displaystyle A_{i}\nearrow }	👁 No	👁 No	👁 No
𝜆-system (Dynkin system)	👁 Yes	👁 No	👁 No	only if 👁 {\displaystyle A\subseteq B}	👁 Yes	👁 No	only if 👁 {\displaystyle A_{i}\nearrow } or they are disjoint	👁 Yes	👁 Yes	Never
Ring (order theory)	👁 Yes	👁 Yes	👁 Yes	👁 No	👁 No	👁 No	👁 No	👁 No	👁 No	👁 No
Ring (measure theory)	👁 Yes	👁 Yes	👁 Yes	👁 Yes	👁 No	👁 No	👁 No	👁 No	👁 Yes	Never
δ-ring	👁 Yes	👁 Yes	👁 Yes	👁 Yes	👁 No	👁 Yes	👁 No	👁 No	👁 Yes	Never
𝜎-ring	👁 Yes	👁 Yes	👁 Yes	👁 Yes	👁 No	👁 Yes	👁 Yes	👁 No	👁 Yes	Never
Algebra (field)	👁 Yes	👁 Yes	👁 Yes	👁 Yes	👁 Yes	👁 No	👁 No	👁 Yes	👁 Yes	Never
𝜎-algebra (𝜎-field)	👁 Yes	👁 Yes	👁 Yes	👁 Yes	👁 Yes	👁 Yes	👁 Yes	👁 Yes	👁 Yes	Never
	👁 Yes	👁 Yes	👁 Yes	👁 No	👁 No	👁 No	👁 Yes	👁 Yes	👁 No	👁 No
Proper filter	👁 Yes	👁 Yes	👁 Yes	Never	Never	👁 No	👁 Yes	👁 Yes	Never	👁 Yes
Prefilter (filter base)	👁 Yes	👁 No	👁 No	👁 No	👁 No	👁 No	👁 No	👁 No	👁 No	👁 Yes
Filter subbase	👁 No	👁 No	👁 No	👁 No	👁 No	👁 No	👁 No	👁 No	👁 No	👁 Yes
Open topology	👁 Yes	👁 Yes	👁 Yes	👁 No	👁 No	👁 No	👁 Image (even arbitrary 👁 {\displaystyle \cup } )	👁 Yes	👁 Yes	Never
Closed topology	👁 Yes	👁 Yes	👁 Yes	👁 No	👁 No	👁 Image (even arbitrary 👁 {\displaystyle \cap } )	👁 No	👁 Yes	👁 Yes	Never
Is necessarily true of 👁 {\displaystyle {\mathcal {F}}\colon } or, is 👁 {\displaystyle {\mathcal {F}}} closed under:	directed downward	finite intersections	finite unions	relative complements	complements in 👁 {\displaystyle \Omega }	countable intersections	countable unions	contains 👁 {\displaystyle \Omega }	contains 👁 {\displaystyle \varnothing }	Finite intersection property
Additionally, a semiring is a π-system where every complement 👁 {\displaystyle B\setminus A} is equal to a finite disjoint union of sets in 👁 {\displaystyle {\mathcal {F}}.} A semialgebra is a semiring where every complement 👁 {\displaystyle \Omega \setminus A} is equal to a finite disjoint union of sets in 👁 {\displaystyle {\mathcal {F}}.} 👁 {\displaystyle A,B,A_{1},A_{2},\ldots } are arbitrary elements of 👁 {\displaystyle {\mathcal {F}}} and it is assumed that 👁 {\displaystyle {\mathcal {F}}\neq \varnothing .}

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