Filter (mathematics)

Last updated December 03, 2024

In mathematics, a filter or order filter is a special subset of a partially ordered set (poset), describing "large" or "eventual" elements. Filters appear in order and lattice theory, but also topology, whence they originate. The notion dual to a filter is an order ideal.

Filters on sets were introduced by Henri Cartan in 1937. Nicolas Bourbaki, in their book Topologie Générale , popularized filters as an alternative to E. H. Moore and Herman L. Smith's 1922 notion of a net; order filters generalize this notion from the specific case of a power set under inclusion to arbitrary partially ordered sets. Nevertheless, the theory of power-set filters retains interest in its own right, in part for substantial applications in topology.

Motivation

Fix a partially ordered set (poset) $P$ . Intuitively, a filter $F$ is a subset of $P$ whose members are elements large enough to satisfy some criterion.^[1] For instance, if $x \in P$ , then the set of elements above $x$ is a filter, called the principal filter at $x$ . (If $x$ and $y$ are incomparable elements of $P$ , then neither the principal filter at $x$ nor $y$ is contained in the other.)

Similarly, a filter on a set $S$ contains those subsets that are sufficiently large to contain some given thing. For example, if $S$ is the real line and $x \in S$ , then the family of sets including $x$ in their interior is a filter, called the neighborhood filter at $x$ . The thing in this case is slightly larger than $x$ , but it still does not contain any other specific point of the line.

The above considerations motivate the upward closure requirement in the definition below: "large enough" objects can always be made larger.

To understand the other two conditions, reverse the roles and instead consider $F$ as a "locating scheme" to find $x$ . In this interpretation, one searches in some space $X$ , and expects $F$ to describe those subsets of $X$ that contain the goal. The goal must be located somewhere; thus the empty set $\emptyset$ can never be in $F$ . And if two subsets both contain the goal, then should "zoom in" to their common region.

An ultrafilter describes a "perfect locating scheme" where each scheme component gives new information (either "look here" or "look elsewhere"). Compactness is the property that "every search is fruitful," or, to put it another way, "every locating scheme ends in a search result."

A common use for a filter is to define properties that are satisfied by "generic" elements of some topological space.^[2] This application generalizes the "locating scheme" to find points that might be hard to write down explicitly.

Definition

A subset $F$ of a partially ordered set $(P, \leq)$ is a filter or dual ideal if the following are satisfied:^[3]

Nontriviality: The set $F$ is non-empty.
Downward directed: For every $x, y \in F$ , there is some $z \in F$ such that $z \leq x$ and $z \leq y$ .
Upward closure: For every $x \in F$ and $p \in P$ , the condition $x \leq p$ implies $p \in F$ .

If, additionally, $F \neq P$ , then $F$ is said to be a proper filter. Authors in set theory and mathematical logic often require all filters to be proper;^[4] this article will eschew that convention. An ultrafilter is a filter contained in no other proper filter.

Filter bases

A subset $S$ of $F$ is a base or basis for $F$ if the upper set generated by $S$ (i.e., the smallest upwards-closed set containing $S$ ) is equal to $F$ . Since every filter is upwards-closed, every filter is a base for itself.

Moreover, if $B \subseteq P$ is nonempty and downward directed, then $B$ generates an upper set $F$ that is a filter (for which $B$ is a base). Such sets are called prefilters, as well as the aforementioned filter base/basis, and $F$ is said to be generated or spanned by $B$ . A prefilter is proper if and only if it generates a proper filter.

Given $p \in P$ , the set ${x : p \leq x}$ is the smallest filter containing $p$ , and sometimes written $↑ p$ . Such a filter is called a principal filter; $p$ is said to be the principal element of $F$ , or generate $F$ .

Suppose $B$ and $C$ are two prefilters on $P$ , and, for each $c \in C$ , there is a $b \in B$ , such that $b \leq c$ . Then we say that $B$ is finer than (or refines) $C$ ; likewise, $C$ is coarser than (or coarsens) $B$ . Refinement is a preorder on the set of prefilters. In fact, if $C$ also refines $B$ , then $B$ and $C$ are called equivalent, for they generate the same filter. Thus passage from prefilter to filter is an instance of passing from a preordering to associated partial ordering.

Special cases

Historically, filters generalized to order-theoretic lattices before arbitrary partial orders. In the case of lattices, downward direction can be written as closure under finite meets: for all $x, y \in F$ , one has $x \land y \in F$ .^[3]

Linear filters

A linear (ultra)filter is an (ultra)filter on the lattice of vector subspaces of a given vector space, ordered by inclusion. Explicitly, a linear filter on a vector space $X$ is a family $B$ of vector subspaces of $X$ such that if $A, B \in B$ and $C$ is a vector subspace of $X$ that contains $A$ , then $A \cap B \in B$ and $C \in B$ .^[5]

A linear filter is proper if it does not contain ${0}$ .^[5]

Filters on a set; subbases

Families ${\mathcal {F}}$ of sets over $\Omega$ v t e
Is necessarily true of ${\mathcal {F}}\colon$ or, is ${\mathcal {F}}$ closed under:	Directed by $\,\supseteq$	$A\cap B$	$A\cup B$	$B\setminus A$	$\Omega \setminus A$	$A_{1}\cap A_{2}\cap \cdots$	$A_{1}\cup A_{2}\cup \cdots$	$\Omega \in {\mathcal {F}}$	$\varnothing \in {\mathcal {F}}$	F.I.P.
$π$ -system
Semiring										Never
Semialgebra(Semifield)										Never
Monotone class						only if $A_{i}\searrow$	only if $A_{i}\nearrow$
𝜆-system(Dynkin System)				only if $A\subseteq B$			only if $A_{i}\nearrow$ or they are disjoint			Never
Ring (Order theory)
Ring (Measure theory)										Never
δ-Ring										Never
𝜎-Ring										Never
Algebra (Field)										Never
𝜎-Algebra(𝜎-Field)										Never
Dual ideal
Filter				Never	Never				$\varnothing \not \in {\mathcal {F}}$
Prefilter(Filter base)				Never	Never				$\varnothing \not \in {\mathcal {F}}$
Filter subbase				Never	Never				$\varnothing \not \in {\mathcal {F}}$
Open Topology							(even arbitrary $\cup$ )			Never
Closed Topology						(even arbitrary $\cap$ )				Never
Is necessarily true of ${\mathcal {F}}\colon$ or, is ${\mathcal {F}}$ closed under:	directed downward	finite intersections	finite unions	relative complements	complements in $\Omega$	countable intersections	countable unions	contains $\Omega$	contains $\varnothing$	Finite Intersection Property
Additionally, a *semiring* is a $π$ -system where every complement $B\setminus A$ is equal to a finite disjoint union of sets in ${\mathcal {F}}.$ A *semialgebra* is a semiring where every complement $\Omega \setminus A$ is equal to a finite disjoint union of sets in ${\mathcal {F}}.$ $A,B,A_{1},A_{2},\ldots$ are arbitrary elements of ${\mathcal {F}}$ and it is assumed that ${\mathcal {F}}\neq \varnothing .$

Given a set $S$ , the power set $P (S)$ is partially ordered by set inclusion; filters on this poset are often just called "filters on $S$ ," in an abuse of terminology. For such posets, downward direction and upward closure reduce to:^[4]

Closure under finite intersections: If $A, B \in F$ , then so too is $A \cap B \in F$ .
Isotony: ^[6] If $A \in F$ and $A \subseteq B \subseteq S$ , then $B \in F$ .

A proper^[7]/non-degenerate^[8] filter is one that does not contain $\emptyset$ , and these three conditions (including non-degeneracy) are Henri Cartan's original definition of a filter.^[9]^[10] It is common — though not universal — to require filters on sets to be proper (whatever one's stance on poset filters); we shall again eschew this convention.

Prefilters on a set are proper if and only if they do not contain $\emptyset$ either.

For every subset $T$ of $P (S)$ , there is a smallest filter $F$ containing $T$ . As with prefilters, $T$ is said to generate or span $F$ ; a base for $F$ is the set $U$ of all finite intersections of $T$ . The set $T$ is said to be a filter subbase when $F$ (and thus $U$ ) is proper.

Proper filters on sets have the finite intersection property.

If $S = \emptyset$ , then $S$ admits only the improper filter ${\emptyset}$ .

Free filters

A filter is said to be a free if the intersection of its members is empty. A proper principal filter is not free.

Since the intersection of any finite number of members of a filter is also a member, no proper filter on a finite set is free, and indeed is the principal filter generated by the common intersection of all of its members. But a nonprincipal filter on an infinite set is not necessarily free: a filter is free if and only if it includes the Fréchet filter (see § Examples).

Examples

See the image at the top of this article for a simple example of filters on the finite poset $P ({1, 2, 3, 4})$ .

Partially order $ℝ \to ℝ$ , the space of real-valued functions on $ℝ$ , by pointwise comparison. Then the set of functions "large at infinity," $\left\{f:\lim _{x\to \pm \infty }{f(x)}=\infty \right\}{\text{,}}$ is a filter on $ℝ \to ℝ$ . One can generalize this construction quite far by compactifying the domain and completing the codomain: if $X$ is a set with distinguished subset $S$ and $Y$ is a poset with distinguished element $m$ , then ${f : f | S \geq m}$ is a filter in $X \to Y$ .

The set ${{k : k \geq N} : N \in ℕ}$ is a filter in $P (ℕ)$ . More generally, if $D$ is any directed set, then $\{\{k:k\geq N\}:N\in D\}$ is a filter in $P (D)$ , called the tail filter. Likewise any net ${x α} α \in Α$ generates the eventuality filter ${{x β : α \leq β} : α \in Α}$ . A tail filter is the eventuality filter for $x α = α$ .

The Fréchet filter on an infinite set $X$ is $\{A:X\setminus A{\text{ finite}}\}{\text{.}}$ If $(X, μ)$ is a measure space, then the collection ${A : μ (X ∖ A) = 0}$ is a filter. If $μ (X) = \infty$ , then ${A : μ (X ∖ A) < \infty}$ is also a filter; the Fréchet filter is the case where $μ$ is counting measure.

Given an ordinal $a$ , a subset of $a$ is called a club if it is closed in the order topology of $a$ but has net-theoretic limit $a$ . The clubs of $a$ form a filter: the club filter, $♣(a)$ .

The previous construction generalizes as follows: any club $C$ is also a collection of dense subsets (in the ordinal topology) of $a$ , and $♣(a)$ meets each element of $C$ . Replacing $C$ with an arbitrary collection $C ̃$ of dense sets, there "typically" exists a filter meeting each element of $C ̃$ , called a generic filter. For countable $C ̃$ , the Rasiowa–Sikorski lemma implies that such a filter must exist; for "small" uncountable $C ̃$ , the existence of such a filter can be forced through Martin's axiom.

Let $P$ denote the set of partial orders of limited cardinality, modulo isomorphism. Partially order $P$ by:

A \leq B

if there exists a strictly increasing

f : A \to B

.

Then the subset of non-atomic partial orders forms a filter. Likewise, if $I$ is the set of injective modules over some given commutative ring, of limited cardinality, modulo isomorphism, then a partial order on $I$ is:

A \leq B

if there exists an injective linear map

f : A \to B

.^[11]

Given any infinite cardinal $κ$ , the modules in $I$ that cannot be generated by fewer than $κ$ elements form a filter.

Every uniform structure on a set $X$ is a filter on $X \times X$ .

Relationship to ideals

The dual notion to a filter — that is, the concept obtained by reversing all $\leq$ and exchanging $\land$ with $\lor$ — is an order ideal. Because of this duality, any question of filters can be mechanically translated to a question about ideals and vice-versa; in particular, a prime or maximal filter is a filter whose corresponding ideal is (respectively) prime or maximal.

A filter is an ultrafilter if and only if the corresponding ideal is minimal.

In model theory

For every filter $F$ on a set $S$ , the set function defined by $m(A)={\begin{cases}1&{\text{if }}A\in F\\0&{\text{if }}S\smallsetminus A\in F\\{\text{is undefined}}&{\text{otherwise}}\end{cases}}$ is finitely additive — a "measure," if that term is construed rather loosely. Moreover, the measures so constructed are defined everywhere if $F$ is an ultrafilter. Therefore, the statement $\left\{\,x\in S:\varphi (x)\,\right\}\in F$ can be considered somewhat analogous to the statement that $φ$ holds "almost everywhere." That interpretation of membership in a filter is used (for motivation, not actual proofs) in the theory of ultraproducts in model theory, a branch of mathematical logic.

In topology

In general topology and analysis, filters are used to define convergence in a manner similar to the role of sequences in a metric space. They unify the concept of a limit across the wide variety of arbitrary topological spaces.

To understand the need for filters, begin with the equivalent concept of a net. A sequence is usually indexed by the natural numbers $ℕ$ , which are a totally ordered set. Nets generalize the notion of a sequence by replacing $ℕ$ with an arbitrary directed set. In certain categories of topological spaces, such as first-countable spaces, sequences characterize most topological properties, but this is not true in general. However, nets — as well as filters — always do characterize those topological properties.

Filters do not involve any set external to the topological space $X$ , whereas sequences and nets rely on other directed sets. For this reason, the collection of all filters on $X$ is always a set, whereas the collection of all $X$ -valued nets is a proper class.

Neighborhood bases

Any point $x$ in the topological space $X$ defines a neighborhood filter or system $N x$ : namely, the family of all sets containing $x$ in their interior. A set $N$ of neighborhoods of $x$ is a neighborhood base at $x$ if $N$ generates $N x$ . Equivalently, $S \subseteq X$ is a neighborhood of $x$ if and only if there exists $N \in N$ such that $N \subseteq S$ .

Convergent filters and cluster points

A prefilter $B$ converges to a point $x$ , written $B \to x$ , if and only if $B$ generates a filter $F$ that contains the neighborhood filter $N x$ — explicitly, for every neighborhood $U$ of $x$ , there is some $V \in B$ such that $V \subseteq U$ . Less explicitly, $B \to x$ if and only if $B$ refines $N x$ , and any neighborhood base at $x$ can replace $N x$ in this condition. Clearly, every neighborhood base at $x$ converges to $x$ .

A filter $F$ (which generates itself) converges to $x$ if $N x \subseteq F$ . The above can also be reversed to characterize the neighborhood filter $N x$ : $N x$ is the finest filter coarser than each filter converging to $x$ .

If $B \to x$ , then $x$ is called a limit (point) of $B$ . The prefilter $B$ is said to cluster at $x$ (or have $x$ as a cluster point) if and only if each element of $B$ has non-empty intersection with each neighborhood of $x$ . Every limit point is a cluster point but the converse is not true in general. However, every cluster point of an ultrafilter is a limit point.

Notes

↑ Koutras et al. 2021.
↑ Igarashi, Ayumi; Zwicker, William S. (16 February 2021). "Fair division of graphs and of tangled cakes". arXiv: 2102.08560 [math.CO].
1 2 Davey, B. A.; Priestley, H. A. (2002) [1990]. Introduction to Lattices and Order (2nd ed.). Cambridge University Press. p. 44. ISBN 9780521784511.
1 2 Dugundji 1966, pp. 211–213.
1 2 Bergman & Hrushovski 1998.
↑ Dolecki & Mynard 2016, pp. 27–29.
↑ Goldblatt, R. Lectures on the Hyperreals: an Introduction to Nonstandard Analysis. p. 32.
↑ Narici & Beckenstein 2011, pp. 2–7.
↑ Cartan 1937a.
↑ Cartan 1937b.
↑ Bumby, R. T. (1965-12-01). "Modules which are isomorphic to submodules of each other". Archiv der Mathematik. 16 (1): 184–185. doi:10.1007/BF01220018. ISSN 1420-8938.

Related Research Articles

In mathematics, a directed set is a nonempty set $together with a reflexive and transitive binary relation, with the additional property that every pair of elements has an upper bound. In other words, for any and in there must exist in with and A directed set's preorder is called a direction .$

This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric topology. For a list of terms specific to algebraic topology, see Glossary of algebraic topology.

In the mathematical field of order theory, an ultrafilter on a given partially ordered set $is a certain subset of namely a maximal filter on that is, a proper filter on that cannot be enlarged to a bigger proper filter on$

In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures together and consequently they are not independent from each other.

In mathematics, a topological vector space is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is also a topological space with the property that the vector space operations are also continuous functions. Such a topology is called a vector topology and every topological vector space has a uniform topological structure, allowing a notion of uniform convergence and completeness. Some authors also require that the space is a Hausdorff space. One of the most widely studied categories of TVSs are locally convex topological vector spaces. This article focuses on TVSs that are not necessarily locally convex. Other well-known examples of TVSs include Banach spaces, Hilbert spaces and Sobolev spaces.

In the mathematical discipline of general topology, Stone–Čech compactification is a technique for constructing a universal map from a topological space X to a compact Hausdorff space βX. The Stone–Čech compactification βX of a topological space X is the largest, most general compact Hausdorff space "generated" by X, in the sense that any continuous map from X to a compact Hausdorff space factors through βX. If X is a Tychonoff space then the map from X to its image in βX is a homeomorphism, so X can be thought of as a (dense) subspace of βX; every other compact Hausdorff space that densely contains X is a quotient of βX. For general topological spaces X, the map from X to βX need not be injective.

In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact with respect to the product topology. The theorem is named after Andrey Nikolayevich Tikhonov, who proved it first in 1930 for powers of the closed unit interval and in 1935 stated the full theorem along with the remark that its proof was the same as for the special case. The earliest known published proof is contained in a 1935 article by Tychonoff, "Über einen Funktionenraum".

In mathematics, the Boolean prime ideal theorem states that ideals in a Boolean algebra can be extended to prime ideals. A variation of this statement for filters on sets is known as the ultrafilter lemma. Other theorems are obtained by considering different mathematical structures with appropriate notions of ideals, for example, rings and prime ideals, or distributive lattices and maximal ideals. This article focuses on prime ideal theorems from order theory.

In general topology, a branch of mathematics, a non-empty family A of subsets of a set $is said to have the finite intersection property (FIP) if the intersection over any finite subcollection of is non-empty. It has the strong finite intersection property (SFIP) if the intersection over any finite subcollection of is infinite. Sets with the finite intersection property are also called centered systems and filter subbases .$

In mathematics, a closure operator on a set S is a function $:{\mathcal {P}}(S)\rightarrow {\mathcal {P}}(S)}$ from the power set of S to itself that satisfies the following conditions for all sets

In mathematical order theory, an ideal is a special subset of a partially ordered set (poset). Although this term historically was derived from the notion of a ring ideal of abstract algebra, it has subsequently been generalized to a different notion. Ideals are of great importance for many constructions in order and lattice theory.

In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered by finitely many subsets of every fixed “size”.

In the mathematical field of set theory, Martin's axiom, introduced by Donald A. Martin and Robert M. Solovay, is a statement that is independent of the usual axioms of ZFC set theory. It is implied by the continuum hypothesis, but it is consistent with ZFC and the negation of the continuum hypothesis. Informally, it says that all cardinals less than the cardinality of the continuum, 𝔠, behave roughly like ℵ₀. The intuition behind this can be understood by studying the proof of the Rasiowa–Sikorski lemma. It is a principle that is used to control certain forcing arguments.

In mathematics, the Fréchet filter, also called the cofinite filter, on a set $is a certain collection of subsets of . A subset of belongs to the Fréchet filter if and only if the complement of in is finite. Any such set is said to be cofinite in, which is why it is alternatively called the cofinite filter on .$

In mathematics, especially functional analysis, a bornology on a set X is a collection of subsets of X satisfying axioms that generalize the notion of boundedness. One of the key motivations behind bornologies and bornological analysis is the fact that bornological spaces provide a convenient setting for homological algebra in functional analysis. This is because^{pg 9} the category of bornological spaces is additive, complete, cocomplete, and has a tensor product adjoint to an internal hom, all necessary components for homological algebra.

In mathematics, a filter on a set $is a family of subsets such that:$

$and$
if $and, then$
If $and, then$

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

In functional analysis and related areas of mathematics, a complete topological vector space is a topological vector space (TVS) with the property that whenever points get progressively closer to each other, then there exists some point $towards which they all get closer. The notion of "points that get progressively closer" is made rigorous by Cauchy nets or Cauchy filters, which are generalizations of Cauchy sequences, while "point towards which they all get closer" means that this Cauchy net or filter converges to The notion of completeness for TVSs uses the theory of uniform spaces as a framework to generalize the notion of completeness for metric spaces. But unlike metric-completeness, TVS-completeness does not depend on any metric and is defined for all TVSs, including those that are not metrizable or Hausdorff.$

In mathematics, a convergence space, also called a generalized convergence, is a set together with a relation called a convergence that satisfies certain properties relating elements of X with the family of filters on X. Convergence spaces generalize the notions of convergence that are found in point-set topology, including metric convergence and uniform convergence. Every topological space gives rise to a canonical convergence but there are convergences, known as non-topological convergences, that do not arise from any topological space. An example of convergence that is in general non-topological is almost everywhere convergence. Many topological properties have generalizations to convergence spaces.

In the mathematical field of set theory, an ultrafilter on a set $is a maximal filter on the set In other words, it is a collection of subsets of that satisfies the definition of a filter on and that is maximal with respect to inclusion, in the sense that there does not exist a strictly larger collection of subsets of that is also a filter. Equivalently, an ultrafilter on the set can also be characterized as a filter on with the property that for every subset of either or its complement belongs to the ultrafilter.$

References

Nicolas Bourbaki, General Topology (Topologie Générale), ISBN 0-387-19374-X (Ch. 1–4): Provides a good reference for filters in general topology (Chapter I) and for Cauchy filters in uniform spaces (Chapter II)
Bourbaki, Nicolas (1987) [1981]. Topological Vector Spaces: Chapters 1–5. Éléments de mathématique. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag. ISBN 3-540-13627-4. OCLC 17499190.
Burris, Stanley; Sankappanavar, Hanamantagouda P. (2012). A Course in Universal Algebra (PDF). Springer-Verlag. ISBN 978-0-9880552-0-9. Archived from the original on 1 April 2022.
Cartan, Henri (1937a). "Théorie des filtres". Comptes rendus hebdomadaires des séances de l'Académie des sciences . 205: 595–598.
Cartan, Henri (1937b). "Filtres et ultrafiltres". Comptes rendus hebdomadaires des séances de l'Académie des sciences . 205: 777–779.
Dolecki, Szymon; Mynard, Frédéric (2016). Convergence Foundations Of Topology. New Jersey: World Scientific Publishing Company. ISBN 978-981-4571-52-4. OCLC 945169917.
Dugundji, James (1966). Topology. Boston: Allyn and Bacon. ISBN 978-0-697-06889-7. OCLC 395340485.
Koutras, Costas D.; Moyzes, Christos; Nomikos, Christos; Tsaprounis, Konstantinos; Zikos, Yorgos (20 October 2021). "On Weak Filters and Ultrafilters: Set Theory From (and for) Knowledge Representation". Logic Journal of the IGPL . doi:10.1093/jigpal/jzab030.
MacIver R., David (1 July 2004). "Filters in Analysis and Topology" (PDF). Archived from the original (PDF) on 2007-10-09. (Provides an introductory review of filters in topology and in metric spaces.)
Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
Willard, Stephen (2004) [1970]. General Topology. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240.
Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.

v t e Order theory
Topics Glossary Category
Key concepts	Binary relation Boolean algebra Cyclic order Lattice Partial order Preorder Total order Weak ordering
Results	Boolean prime ideal theorem Cantor–Bernstein theorem Cantor's isomorphism theorem Dilworth's theorem Dushnik–Miller theorem Hausdorff maximal principle Knaster–Tarski theorem Kruskal's tree theorem Laver's theorem Mirsky's theorem Szpilrajn extension theorem Zorn's lemma
Properties & Types ( list )	Antisymmetric Asymmetric Boolean algebra topics Completeness Connected Covering Dense Directed (Partial) Equivalence Foundational Heyting algebra Homogeneous Idempotent Lattice Bounded Complemented Complete Distributive Join and meet Reflexive Partial order Chain-complete Graded Eulerian Strict Prefix order Preorder Total Semilattice Semiorder Symmetric Total Tolerance Transitive Well-founded Well-quasi-ordering (Better) (Pre) Well-order
Constructions	Composition Converse/Transpose Lexicographic order Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure
Topology & Orders	Alexandrov topology & Specialization preorder Ordered topological vector space Normal cone Order topology Order topology Topological vector lattice Banach Fréchet Locally convex Normed
Related	Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet Order morphism Embedding Isomorphism Order type Ordered field Positive cone of an ordered field Ordered vector space Partially ordered Positive cone of an ordered vector space Riesz space Partially ordered group Positive cone of a partially ordered group Upper set Young's lattice