Ultrafilter (set theory)
In the mathematical field of set theory, an ultrafilter is a maximal proper filter: it is a filter on a given non-empty set which is a certain type of non-empty family of subsets of that is not equal to the power set of (such filters are called proper) and that is also "maximal" in that there does not exist any other proper filter on that contains it as a proper subset. Said differently, a proper filter is called an ultrafilter if there exists exactly one proper filter that contains it as a subset, that proper filter (necessarily) being itself.
More formally, an ultrafilter on is a proper filter that is also a maximal filter on with respect to set inclusion, meaning that there does not exist any proper filter on that contains as a proper subset. Ultrafilters on sets are an important special instance of ultrafilters on partially ordered sets, where the partially ordered set consists of the power set and the partial order is subset inclusion
Given an arbitrary set an ultrafilter on is a non-empty family of subsets of such that:
- Proper or non-degenerate: The empty set is not an element of
- Upward closed in : If and if is any superset of (that is, if ) then
- π−system: If and are elements of then so is their intersection
- If then either[note 1] or its relative complement is an element of
Properties (1), (2), and (3) are the defining properties of a filter on Some authors do not include non-degeneracy (which is property (1) above) in their definition of "filter". However, the definition of "ultrafilter" (and also of "prefilter" and "filter subbase") always includes non-degeneracy as a defining condition. This article requires that all filters be proper although a filter might be described as "proper" for emphasis.
A filter subbase is a non-empty family of sets that has the finite intersection property (i.e. all finite intersections are non-empty). Equivalently, a filter subbase is a non-empty family of sets that is contained in some (proper) filter. The smallest (relative to ) filter containing a given filter subbase is said to be generated by the filter subbase.
The upward closure in of a family of sets is the set
A prefilter or filter base is a non-empty and proper (i.e. ) family of sets that is downward directed, which means that if then there exists some such that Equivalently, a prefilter is any family of sets whose upward closure is a filter, in which case this filter is called the filter generated by and is said to be a filter base for
The dual in  of a family of sets is the set
Generalization to ultra prefiltersEdit
- For every set there exists some set such that or (or equivalently, such that equals or ).
- For every set there exists some set such that equals or
- Here, is defined to be the union of all sets in
- This characterization of " is ultra" does not depend on the set so mentioning the set is optional when using the term "ultra."
- For every set (not necessarily even a subset of ) there exists some set such that equals or
- If satisfies this condition then so does every superset In particular, a set is ultra if and only if and contains as a subset some ultra family of sets.
A filter subbase that is ultra is necessarily a prefilter.[proof 1]
The ultra property can now be used to define both ultrafilters and ultra prefilters:
- An ultra prefilter is a prefilter that is ultra. Equivalently, it is a filter subbase that is ultra.
- An ultrafilter on is a (proper) filter on that is ultra. Equivalently, it is any filter on that is generated by an ultra prefilter.
- Interpretation as large sets
The elements of a proper filter on may be thought of as being "large sets (relative to )" and the complements in of a large sets can be thought of as being "small" sets (the "small sets" are exactly the elements in the ideal ). In general, there may be subsets of that are neither large nor small, or possibly simultaneously large and small. A dual ideal is a filter (i.e. proper) if there is no set that is both large and small, or equivalently, if the is not large. A filter is ultra if and only if every subset of is either large or else small. With this terminology, the defining properties of a filter can be restarted as: (1) any superset of a large set is large set, (2) the intersection of any two (or finitely many) large sets is large, (3) is a large set (i.e. ), (4) the empty set is not large. Different dual ideals give different notions of "large" sets.
Another way of looking at ultrafilters on a power set is as follows: for a given ultrafilter define a function on by setting if is an element of and otherwise. Such a function is called a 2-valued morphism. Then is finitely additive, and hence a content on and every property of elements of is either true almost everywhere or false almost everywhere. However, is usually not countably additive, and hence does not define a measure in the usual sense.
- Ultra prefilters as maximal prefilters
To characterize ultra prefilters in terms of "maximality," the following relation is needed.
- Given two families of sets and the family is said to be coarser than and is finer than and subordinate to written or N ⊢ M, if for every there is some such that The families and are called equivalent if and The families and are comparable if one of these sets is finer than the other.
The subordination relationship, i.e. is a preorder so the above definition of "equivalent" does form an equivalence relation. If then but the converse does not hold in general. However, if is upward closed, such as a filter, then if and only if Every prefilter is equivalent to the filter that it generates. This shows that it is possible for filters to be equivalent to sets that are not filters.
If two families of sets and are equivalent then either both and are ultra (resp. prefilters, filter subbases) or otherwise neither one of them is ultra (resp. a prefilter, a filter subbase). In particular, if a filter subbase is not also a prefilter, then it is not equivalent to the filter or prefilter that it generates. If and are both filters on then and are equivalent if and only if If a proper filter (resp. ultrafilter) is equivalent to a family of sets then is necessarily a prefilter (resp. ultra prefilter). Using the following characterization, it is possible to define prefilters (resp. ultra prefilters) using only the concept of filters (resp. ultrafilters) and subordination:
- An arbitrary family of sets is a prefilter if and only it is equivalent to a (proper) filter.
- An arbitrary family of sets is an ultra prefilter if and only it is equivalent to an ultrafilter.
- A maximal prefilter on  is a prefilter that satisfies any of the following equivalent conditions:
There are no ultrafilters on the empty set, so it is henceforth assumed that is nonempty.
- for any either or
- is a maximal filter subbase on meaning that if is any filter subbase on then implies 
A (proper) filter on is an ultrafilter on if and only if any of the following equivalent conditions hold:
- is ultra;
- is generated by an ultra prefilter;
- For any subset or 
- So an ultrafilter decides for every whether is "large" (i.e. ) or "small" (i.e. ).
- For each subset either[note 1] is in or ( ) is.
- This condition can be restated as: is partitioned by and its dual
- The sets and are disjoint for all prefilters on
- is an ideal on 
- For any finite family of subsets of (where ), if then for some index
- In words, a "large" set cannot be a finite union of sets that aren't large.
- For any if then or
- For any if then or (a filter with this property is called a prime filter).
- For any if and then either or
- is a maximal filter; that is, if is a filter on such that then Equivalently, is a maximal filter if there is no filter on that contains as a proper subset (that is, no filter is strictly finer than ).
Grills and Filter-GrillsEdit
If then its grill on is the family
where may be written if is clear from context. For example, and if then If then and moreover, if is a filter subbase then  The grill is upward closed in if and only if which will henceforth be assumed. Moreover, so that is upward closed in if and only if
The grill of a filter on is called a filter-grill on  For any is a filter-grill on if and only if (1) is upward closed in and (2) for all sets and if then or The grill operation induces a bijection
whose inverse is also given by  If then is a filter-grill on if and only if  or equivalently, if and only if is an ultrafilter on  That is, a filter on is a filter-grill if and only if it is ultra. For any non-empty is both a filter on and a filter-grill on if and only if (1) and (2) for all the following equivalences hold:
- if and only if if and only if 
Free or principalEdit
If is any non-empty family of sets then the Kernel of is the intersection of all sets in :
A non-empty family of sets is called:
- free if and fixed otherwise (that is, if ),
- principal if
- principal at a point if and is a singleton set; in this case, if then is said to be principal at
If a family of sets is fixed then is ultra if and only if some element of is a singleton set, in which case will necessarily be a prefilter. Every principal prefilter is fixed, so a principal prefilter is ultra if and only if is a singleton set. A singleton set is ultra if and only if its sole element is also a singleton set.
Every filter on that is principal at a single point is an ultrafilter, and if in addition is finite, then there are no ultrafilters on other than these. If there exists a free ultrafilter (or even filter subbase) on a set then must be infinite.
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.
Proposition — If is an ultrafilter on then the following are equivalent:
- is fixed, or equivalently, not free.
- is principal.
- Some element of is a finite set.
- Some element of is a singleton set.
- is principal at some point of which means for some
- does not contain the Fréchet filter on as a subset.
- is sequential.
Examples, properties, and sufficient conditionsEdit
If and are families of sets such that is ultra, and then is necessarily ultra. A filter subbase that is not a prefilter cannot be ultra; but it is nevertheless still possible for the prefilter and filter generated by to be ultra.
Suppose is ultra and is a set. The trace is ultra if and only if it does not contain the empty set. Furthermore, at least one of the sets and will be ultra (this result extends to any finite partition of ). If are filters on is an ultrafilter on and then there is some that satisfies  This result is not necessarily true for an infinite family of filters.
The image under a map of an ultra set is again ultra and if is an ultra prefilter then so is The property of being ultra is preserved under bijections. However, the preimage of an ultrafilter is not necessarily ultra, not even if the map is surjective. For example, if has more than one point and if the range of consists of a single point then is an ultra prefilter on but its preimage is not ultra. Alternatively, if is a principal filter generated by a point in then the preimage of contains the empty set and so is not ultra.
The elementary filter induced by an infinite sequence, all of whose points are distinct, is not an ultrafilter. If then denotes the set consisting all subsets of having cardinality and if contains at least ( ) distinct points, then is ultra but it is not contained in any prefilter. This example generalizes to any integer and also to if contains more than one element. Ultra sets that are not also prefilters are rarely used.
For every and every let If is an ultrafilter on then the set of all such that is an ultrafilter on 
sends any element to the principal ultrafilter given by
The ultrafilter lemmaEdit
The ultrafilter lemma/principle/theorem — Every proper filter on a set is contained in some ultrafilter on
The ultrafilter lemma is equivalent to each of the following statements:
- For every prefilter on a set there exists a maximal prefilter on subordinate to it.
- Every proper filter subbase on a set is contained in some ultrafilter on
The following results can be proven using the ultrafilter lemma. A free ultrafilter exists on a set if and only if is infinite. Every proper filter is equal to the intersection of all ultrafilters containing it. Since there are filters that are not ultra, this shows that the intersection of a family of ultrafilters need not be ultra. A family of sets can be extended to a free ultrafilter if and only if the intersection of any finite family of elements of is infinite.
Relationships to other statements under ZFEdit
Throughout this section, ZF refers to Zermelo–Fraenkel set theory and ZFC refers to ZF with the Axiom of Choice (AC). The ultrafilter lemma is independent of ZF. That is, there exist models in which the axioms of ZF hold but the ultrafilter lemma does not. There also exist models of ZF in which every ultrafilter is necessarily principal.
Every filter that contains a singleton set is necessarily an ultrafilter and given the definition of the discrete ultrafilter does not require more than ZF. If is finite then every ultrafilter is a discrete filter at a point; consequently, free ultrafilters can only exist on infinite sets. In particular, if is finite then the ultrafilter lemma can be proven from the axioms ZF. The existence of free ultrafilter on infinite sets can be proven if the axiom of choice is assumed. More generally, the ultrafilter lemma can be proven by using the axiom of choice, which in brief states that any Cartesian product of non-empty sets is non-empty. Under ZF, the axiom of choice is, in particular, equivalent to (a) Zorn's lemma, (b) Tychonoff's theorem, (c) the weak form of the vector basis theorem (which states that every vector space has a basis), (d) the strong form of the vector basis theorem, and other statements. However, the ultrafilter lemma is strictly weaker than the axiom of choice. While free ultrafilters can be proven to exist, it is not possible to construct an explicit example of a free ultrafilter; that is, free ultrafilters are intangible.Alfred Tarski proved that under ZFC, the cardinality of the set of all free ultrafilters on an infinite set is equal to the cardinality of where denotes the power set of 
Under ZF, the axiom of choice can be used to prove both the ultrafilter lemma and the Krein–Milman theorem; conversely, under ZF, the ultrafilter lemma together with the Krein–Milman theorem can prove the axiom of choice.
Statements that cannot be deducedEdit
The ultrafilter lemma is a relatively weak axiom. For example, each of the statements in the following list can not be deduced from ZF together with only the ultrafilter lemma:
- A countable union of countable sets is a countable set.
- The axiom of countable choice (ACC).
- The axiom of dependent choice (ADC).
Under ZF, the ultrafilter lemma is equivalent to each of the following statements:
- The Boolean prime ideal theorem (BPIT).
- This equivalence is provable in ZF set theory without the Axiom of Choice (AC).
- Stone's representation theorem for Boolean algebras.
- Any product of Boolean spaces is a Boolean space.
- Boolean Prime Ideal Existence Theorem: Every nondegenerate Boolean algebra has a prime ideal.
- Tychonoff's theorem for Hausdorff spaces: Any product of compact Hausdorff spaces is compact.
- If is endowed with the discrete topology then for any set the product space is compact.
- Each of the following versions of the Banach-Alaoglu theorem is equivalent to the ultrafilter lemma:
- Any equicontinuous set of scalar-valued maps on a topological vector space (TVS) is relatively compact in the weak-* topology (that is, it is contained in some weak-* compact set).
- The polar of any neighborhood of the origin in a TVS is a weak-* compact subset of its continuous dual space.
- The closed unit ball in the continuous dual space of any normed space is weak-* compact.
- If the normed space is separable then the ultrafilter lemma is sufficient but not necessary to prove this statement.
- A topological space is compact if every ultrafilter on converges to some limit.
- A topological space is compact if and only if every ultrafilter on converges to some limit.
- The addition of the words "and only if" is the only difference between this statement and the one immediately above it.
- The Ultranet lemma: Every net has a universal subnet.
- A topological space is compact if and only if every ultranet on converges to some limit.
- If the words "and only if" are removed then the resulting statement remains equivalent to the ultrafilter lemma.
- A convergence space is compact if every ultrafilter on converges.
- A uniform space is compact if it is complete and totally bounded.
- The Stone–Čech compactification Theorem.
- Each of the following versions of the compactness theorem is equivalent to the ultrafilter lemma:
- The completeness theorem: If is a set of zero-order sentences that is syntactically consistent, then it has a model (that is, it is semantically consistent).
Any statement that can be deduced from the ultrafilter lemma (together with ZF) is said to be weaker than the ultrafilter lemma. A weaker statement is said to be strictly weaker if under ZF, it is not equivalent to the ultrafilter lemma. Under ZF, the ultrafilter lemma implies each of the following statements:
- The Axiom of Choice for Finite sets (ACF): Given and a family of non-empty finite sets, their product is not empty.
- A countable union of finite sets is a countable set.
- However, ZF with the ultrafilter lemma is too weak to prove that a countable union of countable sets is a countable set.
- The Hahn–Banach theorem.
- In ZF, the Hahn–Banach theorem is strictly weaker than the ultrafilter lemma.
- The Banach–Tarski paradox.
- Every set can be linearly ordered.
- Every field has a unique algebraic closure.
- The Alexander subbase theorem.
- Non-trivial ultraproducts exist.
- The weak ultrafilter theorem: A free ultrafilter exists on
- Under ZF, the weak ultrafilter theorem does not imply the ultrafilter lemma; that is, it is strictly weaker than the ultrafilter lemma.
- There exists a free ultrafilter on every infinite set;
- This statement is actually strictly weaker than the ultrafilter lemma.
- ZF alone does not even imply that there exists a non-principal ultrafilter on some set.
The completeness of an ultrafilter on a powerset is the smallest cardinal κ such that there are κ elements of whose intersection is not in The definition of an ultrafilter implies that the completeness of any powerset ultrafilter is at least . An ultrafilter whose completeness is greater than —that is, the intersection of any countable collection of elements of is still in —is called countably complete or σ-complete.
Ordering on ultrafiltersEdit
The Rudin–Keisler ordering (named after Mary Ellen Rudin and Howard Jerome Keisler) is a preorder on the class of powerset ultrafilters defined as follows: if is an ultrafilter on and an ultrafilter on then if there exists a function such that
- if and only if
for every subset
Ultrafilters and are called Rudin–Keisler equivalent, denoted U ≡RK V, if there exist sets and and a bijection that satisfies the condition above. (If and have the same cardinality, the definition can be simplified by fixing )
Ultrafilters on Edit
There are several special properties that an ultrafilter on where extends the natural numbers, may possess, which prove useful in various areas of set theory and topology.
- A non-principal ultrafilter is called a P-point (or weakly selective) if for every partition of such that for all there exists some such that is a finite set for each
- A non-principal ultrafilter is called Ramsey (or selective) if for every partition of such that for all there exists some such that is a singleton set for each
It is a trivial observation that all Ramsey ultrafilters are P-points. Walter Rudin proved that the continuum hypothesis implies the existence of Ramsey ultrafilters. In fact, many hypotheses imply the existence of Ramsey ultrafilters, including Martin's axiom. Saharon Shelah later showed that it is consistent that there are no P-point ultrafilters. Therefore, the existence of these types of ultrafilters is independent of ZFC.
P-points are called as such because they are topological P-points in the usual topology of the space βω \ ω of non-principal ultrafilters. The name Ramsey comes from Ramsey's theorem. To see why, one can prove that an ultrafilter is Ramsey if and only if for every 2-coloring of there exists an element of the ultrafilter that has a homogeneous color.
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- Let be a filter on that is not an ultrafilter. If is such that then has the finite intersection property (because if then if and only if ) so that by the ultrafilter lemma, there exists some ultrafilter on such that (so in particular ). It follows that ∎
- Suppose is filter subbase that is ultra. Let and define Because is ultra, there exists some such that equals or The finite intersection property implies that so necessarily which is equivalent to ∎
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