This HTML5 document contains 165 embedded RDF statements represented using HTML+Microdata notation.

The embedded RDF content will be recognized by any processor of HTML5 Microdata.

Namespace Prefixes

PrefixIRI
dctermshttp://purl.org/dc/terms/
dbohttp://dbpedia.org/ontology/
n16http://dbpedia.org/resource/File:
foafhttp://xmlns.com/foaf/0.1/
n12https://global.dbpedia.org/id/
dbthttp://dbpedia.org/resource/Template:
rdfshttp://www.w3.org/2000/01/rdf-schema#
n8http://commons.wikimedia.org/wiki/Special:FilePath/
rdfhttp://www.w3.org/1999/02/22-rdf-syntax-ns#
owlhttp://www.w3.org/2002/07/owl#
wikipedia-enhttp://en.wikipedia.org/wiki/
dbphttp://dbpedia.org/property/
dbchttp://dbpedia.org/resource/Category:
provhttp://www.w3.org/ns/prov#
xsdhhttp://www.w3.org/2001/XMLSchema#
wikidatahttp://www.wikidata.org/entity/
dbrhttp://dbpedia.org/resource/

Statements

Subject Item
dbr:Ultrafilter_(set_theory)
rdf:type
owl:Thing
rdfs:label
Ultrafilter (set theory)
rdfs:comment
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.
rdfs:seeAlso
dbr:Boolean_prime_ideal_theorem dbr:Filter_(mathematics)
dbp:name
Proposition The /principle/theorem
foaf:depiction
n8:Filter_vs_ultrafilter.svg
dcterms:subject
dbc:Order_theory dbc:Nonstandard_analysis dbc:Families_of_sets
dbo:wikiPageID
67416519
dbo:wikiPageRevisionID
1108342317
dbo:wikiPageWikiLink
dbr:Cardinal_number dbr:Singleton_set dbr:Family_of_sets dbr:Filter_(mathematics) dbr:Kernel_(set_theory) dbr:Product_topology dbr:Field_(mathematics) dbr:Boolean_algebra dbr:Bijection dbr:Zermelo–Fraenkel_set_theory dbr:Walter_Rudin dbc:Order_theory dbr:Partially_ordered_set dbr:Alfred_Tarski dbr:Maximal_element dbr:Stone–Čech_compactification dbr:Totally_bounded dbr:Uniform_space dbr:Category_of_sets dbr:Equicontinuous dbr:Ultrafilter dbr:Stone–Čech_compactification_Theorem dbr:Finitely_additive dbr:Set_inclusion dbr:Equivalence_relation dbr:Axiom_of_countable_choice dbr:Axiom_of_dependent_choice dbr:Martin's_axiom dbr:Partition_of_a_set dbr:Preorder dbr:Axiom_of_choice dbr:Model_theory dbr:Monad_(category_theory) dbr:FinSet dbr:Ultrafilter_monad dbr:Hamel_basis dbr:Propositional_calculus dbr:Functor dbr:Intersection_(set_theory) dbr:Howard_Jerome_Keisler dbr:Cartesian_product dbr:Discrete_topology dbr:P-point dbr:Saharon_Shelah dbr:Proper_subset dbr:Boolean_space dbr:Hahn–Banach_theorem dbr:Sentence_(mathematical_logic) dbr:Codensity_monad dbc:Nonstandard_analysis dbr:Mary_Ellen_Rudin dbr:Complete_space dbr:ZFC dbr:Linear_order dbr:Net_(mathematics) dbr:Content_(measure_theory) dbr:Independence_(mathematical_logic) dbr:Ultraproducts dbr:Polar_set dbr:Convergence_space dbr:Stone's_representation_theorem_for_Boolean_algebras dbr:Topology n16:Filter_vs_ultrafilter.svg dbr:Grigorii_Fichtenholz dbr:Springer-Verlag dbr:Subset_inclusion dbr:Tychonoff's_theorem dbr:First-order_predicate_calculus dbr:Leonid_Kantorovich dbr:Topological_vector_space dbc:Families_of_sets dbr:Compactness_theorem dbr:Power_set dbr:Compact_space dbr:Krein–Milman_theorem dbr:Mathematics dbr:Ultranet_(math) dbr:Proper_filter dbr:Pi-system dbr:Completeness_theorem dbr:2-valued_morphism dbr:Empty_set dbr:Banach-Alaoglu_theorem dbr:Normed_space dbr:Algebraic_closure dbr:Product_space dbr:Countable_set dbr:Zorn's_lemma dbr:Weak-*_topology dbr:Almost_everywhere dbr:Ramsey's_theorem dbr:Boolean_prime_ideal_theorem dbr:Finite_intersection_property dbr:Measure_(mathematics) dbr:Continuous_dual_space dbr:Set_theory dbr:Alexander_subbase_theorem dbr:Continuum_hypothesis dbr:Measurable_cardinal dbr:Aleph-naught dbr:Hausdorff_space dbr:Ordinal_number dbr:Set_(mathematics) dbr:Bulletin_of_the_American_Mathematical_Society dbr:Banach–Tarski_paradox dbr:Minimal_element dbr:Felix_Hausdorff
owl:sameAs
wikidata:Q106671513 n12:Fn1t2
dbp:wikiPageUsesTemplate
dbt:Cite_book dbt:Short_description dbt:Pi dbt:Dolecki_Mynard_Convergence_Foundations_Of_Topology dbt:Math_theorem dbt:Mathematical_logic dbt:Schubert_Topology dbt:Schechter_Handbook_of_Analysis_and_Its_Foundations dbt:Citation_needed dbt:Arkhangel'skii_Ponomarev_Fundamentals_of_General_Topology_Problems_and_Exercises dbt:Citation dbt:Császár_General_Topology dbt:Bourbaki_General_Topology_Part_I_Chapters_1-4 dbt:Reflist dbt:About dbt:Sfn dbt:Rp dbt:Annotated_link dbt:Narici_Beckenstein_Topological_Vector_Spaces dbt:Vanchor dbt:Anchor dbt:Set_theory dbt:Em dbt:Nlab dbt:Clarify dbt:See_also dbt:Visible_anchor dbt:Cite_journal dbt:Math dbt:Dugundji_Topology dbt:Dixmier_General_Topology dbt:Joshi_Introduction_to_General_Topology
dbo:thumbnail
n8:Filter_vs_ultrafilter.svg?width=300
dbp:date
July 2016
dbp:id
ultrafilter
dbp:reason
A function m can certainly be defined in that way. However, this is pointless unless such an m can be shown to have some useful properties . They should be stated here.
dbp:title
Ultrafilter
dbo:abstract
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 Ultrafilters have many applications in set theory, model theory, and topology.
dbp:mathStatement
Every proper filter on a set is contained in some ultrafilter on 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 contain the Fréchet filter on as a subset. is sequential.
prov:wasDerivedFrom
wikipedia-en:Ultrafilter_(set_theory)?oldid=1108342317&ns=0
dbo:wikiPageLength
47379
foaf:isPrimaryTopicOf
wikipedia-en:Ultrafilter_(set_theory)