Math 2.0

General topology Definition

Ultrafilter

Builds on FilterZorn's lemma

Definition

Let XX be a set. An ultrafilter on XX is a filter 𝒰\mathcal{U} on XX such that for every filter β„±\mathcal{F} on XX with π’°βŠ†β„±\mathcal{U} \subseteq \mathcal{F}, one has β„±=𝒰\mathcal{F} = \mathcal{U}.

Remarks

A filter 𝒰\mathcal{U} on XX is an ultrafilter if and only if, for every AβŠ†XA \subseteq X, either Aβˆˆπ’°A \in \mathcal{U} or Xβˆ–Aβˆˆπ’°X \setminus A \in \mathcal{U}.

Example: for a∈Xa \in X, the principal filter 𝒰a={AβŠ†X:a∈A}\mathcal{U}_a = \set{A \subseteq X : a \in A} is an ultrafilter. For each AβŠ†XA \subseteq X, either a∈Aa \in A or a∈Xβˆ–Aa \in X \setminus A.

Non-example: on β„•\N, the FrΓ©chet filter of sets with finite complement is not an ultrafilter. The set EE of even numbers and its complement are both infinite, so neither has finite complement.

By Zorn's lemma, every filter on XX is contained in an ultrafilter. On an infinite set, an ultrafilter that contains the FrΓ©chet filter is not principal, so non-principal ultrafilters exist.

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Element, definition, no evidence levelElementdefinitionFilter, concept, no evidence levelFilterconceptProposition, concept, no evidence levelPropositionconceptSet, definition, no evidence levelSetdefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionFamily of sets, definition, no evidence levelFamily of setsdefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionUnion, definition, no evidence levelUniondefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUnion of a family of sets, definition, no evidence levelUnion of a family of …definitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionAntisymmetric relation, definition, no evidence levelAntisymmetric relationdefinitionReflexive relation, definition, no evidence levelReflexive relationdefinitionTransitive relation, definition, no evidence levelTransitive relationdefinitionFunction, definition, no evidence levelFunctiondefinitionPreorder, definition, no evidence levelPreorderdefinitionChoice function, definition, no evidence levelChoice functiondefinitionPartial order, definition, no evidence levelPartial orderdefinitionAxiom of choice, statement, no evidence levelAxiom of choicestatementMaximal element, definition, no evidence levelMaximal elementdefinitionTotal order, definition, no evidence levelTotal orderdefinitionUpper bound, definition, no evidence levelUpper bounddefinitionChain, definition, no evidence levelChaindefinitionZorn's lemma, statement, no evidence levelZorn's lemmastatementUltrafilter, definition, no evidence levelUltrafilterdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Element definition
  2. Filter concept
  3. Proposition concept
  4. Set definition
  5. Disjunction definition
  6. Implication definition
  7. Predicate definition
  8. Family of sets definition
  9. Ordered pair definition
  10. Union definition
  11. Existential quantifier definition
  12. Universal quantifier definition
  13. Cartesian product definition
  14. Subset definition
  15. Union of a family of sets definition
  16. Uniqueness quantifier definition
  17. Binary relation definition
  18. Antisymmetric relation definition
  19. Reflexive relation definition
  20. Transitive relation definition
  21. Function definition
  22. Preorder definition
  23. Choice function definition
  24. Partial order definition
  25. Axiom of choice statement
  26. Maximal element definition
  27. Total order definition
  28. Upper bound definition
  29. Chain definition
  30. Zorn's lemma statement
  31. Ultrafilter definition

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Ultrafilter
About this page
TagSR56KCFYYS
URIm2://SR56KCFYYS
Kinddefinition
WritingW0 stub
MSC54A20, 03E05
DefinesUltrafilter
UsesFilter, Zorn's lemma
Changed2 November 2026

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