Definitions

1. The ultrafilter is taken to have specified sets as elements.
2. A principal (or fixed, or trivial) ultrafilter is a filter containing a least element. (Web site)
3. An ultrafilter is a maximal filter -- for every set, either that set or its complement is in the filter.

Non-Principal Ultrafilters

1. An ultrafilter on ω is Ramsey if and only if it is minimal in the Rudin–Keisler ordering of non-principal ultrafilters. (Web site)

Point

1. For it’s at this point where we have to use the ultrafilter theorem, in order to produce a witnessing ultrafilter.

Collection

1. This collection generates a filter which is contained in some ultrafilter U, by the ultrafilter theorem.

Countable Set

1. Assuming the continuum hypothesis, an ultrafilter whose cartesian square is dominated by only three ultrafilters is constructed on a countable set. (Web site)

Compact Set

1. Proof: Let K be a compact set, and suppose f[ s] is an ultrafilter on K. Then s is an ultrafilter and so converges to, say, x.

Finite Set

1. An ultrafilter on a finite set is necessarily fixed. (Web site)

Index Set

1. If I is an index set, U is an ultrafilter on I, and F i is a field for every i in I, the ultraproduct of the F i with respect to U is a field. (Web site)

Boolean Algebra

1. The Ultrafilter Theorem: Every Boolean algebra has an ultrafilter on it.
2. This Boolean algebra is the finite-cofinite algebra on X. A Boolean algebra A has a unique non-principal ultrafilter (i.e.

Ultrafilters

1. Construction from ultrafilters -- As in the hyperreal numbers, we construct * Q from the rational numbers using an ultrafilter. (Web site)
2. In classical mathematics, the ultrafilter theorem is a theorem about ultrafilters, proved as a standard application of Zorn's lemma.

Measurable Cardinal

1. Equivalently, κ is a measurable cardinal if and only if it is an uncountable cardinal with a ultrafilter. (Web site)
2. Equivalently, κ is a measurable cardinal if and only if it is an uncountable cardinal with a <κ-additive, non-principal ultrafilter. (Web site)

Compactness

1. So we see that compactness methods and ultrafilter methods are closely intertwined.

Space

1. Exercise 7 Show that a space is compact if and only if every ultrafilter has at least one limit. (Web site)
2. Exercise 6 Show that a space is Hausdorff if and only if every ultrafilter has at most one limit. (Web site)

Set

1. In this case, the non-principal ultrafilter is the set of all cofinite sets. (Web site)

Principal

1. By contrast, every ultrafilter of a finite poset (or on a finite set) is principal, since any finite filter has a least element. (Web site)
2. There are two very different types of ultrafilter: principal and free. (Web site)
3. If S is finite, then every ultrafilter on S is principal.

Topological Space

1. A topological space is compact if and only if every ultrafilter on the space is convergent.

Compact Hausdorff Space

1. Every ultrafilter on a compact Hausdorff space converges to exactly one point. (Web site)

Cardinality

1. If a set of uncountable cardinality supports an ultrafilter whose square is dominated by exactly three ultrafilters, then its cardinality is measurable. (Web site)
2. No non-principal ultrafilter on has a base of cardinality < k. (Web site)

Axiom

1. Gödel's ontological proof of God's existence uses as an axiom that the set of all "positive properties" is an ultrafilter. (Web site)
2. It follows from the ultrafilter lemma, which is weaker than the axiom of choice, that all bases of a given vector space have the same cardinality.
3. Now we will prove that the Tychonoff theorem implies the axiom of choice, while the Tychonoff theorem for Hausdorff spaces implies the ultrafilter theorem.

Subset

1. A subset of a topological space is said to be compact if every ultrafilter in it converges to exactly one point.
2. A subset X of the family of structures is said to be almost all of them if X is an element of the ultrafilter. (Web site)

Poset

1. We have seen that forcing can be done using Boolean-valued models, by constructing a Boolean algebra with ultrafilter from a poset with a generic subset. (Web site)

Filter

1. The ultrafilter lemma states that every filter on a set is contained within some maximal (proper) filter -- an ultrafilter. (Web site)
2. In mathematics, the Ultrafilter Lemma states that every filter is a subset of some ultrafilter, i.e.
3. The Ultrafilter Lemma says that every filter is a subset of some ultrafilter. (Web site)

Ultrafilter

1. In order theory, an ultrafilter is a subset of a partially ordered set (a poset) which is maximal among all proper filters. (Web site)
2. Thus, a principal ultrafilter on a set S consists of all sets containing a particular point of S. An ultrafilter on a finite set is principal. (Web site)
3. In the mathematical field of set theory, an ultrafilter on a set X is a collection of subsets of X that is a filter, that cannot be enlarged (as a filter). (Web site)

Categories

1. Information > Science > Mathematics > Finite Set
2. Filter
3. Poset
4. Mathematics > Algebra > Abstract Algebra > Boolean Algebra
5. Information > Science > Mathematics > Subset
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