{"id":"https://openalex.org/W3099155029","doi":"https://doi.org/10.1002/malq.200910014","title":"The Ultrafilter Closure in ZF","display_name":"The Ultrafilter Closure in ZF","publication_year":2010,"publication_date":"2010-05-19","ids":{"openalex":"https://openalex.org/W3099155029","doi":"https://doi.org/10.1002/malq.200910014","mag":"3099155029"},"language":"en","primary_location":{"id":"doi:10.1002/malq.200910014","is_oa":true,"landing_page_url":"https://doi.org/10.1002/malq.200910014","pdf_url":"https://onlinelibrary.wiley.com/doi/pdfdirect/10.1002/malq.200910014","source":{"id":"https://openalex.org/S172351290","display_name":"Mathematical logic quarterly","issn_l":"0942-5616","issn":["0942-5616","1521-3870"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematical Logic Quarterly","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"bronze","oa_url":"https://onlinelibrary.wiley.com/doi/pdfdirect/10.1002/malq.200910014","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5017189560","display_name":"Gon\u00e7alo Gutierres","orcid":"https://orcid.org/0000-0001-9480-498X"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Gon\u00e7alo Gutierres","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5017189560"],"corresponding_institution_ids":[],"apc_list":{"value":2630,"currency":"USD","value_usd":2630},"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.32384372,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"56","issue":"3","first_page":"331","last_page":"336"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11151","display_name":"Advanced Topology and Set Theory","score":0.995199978351593,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11151","display_name":"Advanced Topology and Set Theory","score":0.995199978351593,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11545","display_name":"Advanced Banach Space Theory","score":0.928600013256073,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/ultrafilter","display_name":"Ultrafilter","score":0.9387015700340271},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8641290664672852},{"id":"https://openalex.org/keywords/countable-set","display_name":"Countable set","score":0.7165504693984985},{"id":"https://openalex.org/keywords/axiom-of-choice","display_name":"Axiom of choice","score":0.7058618664741516},{"id":"https://openalex.org/keywords/topological-space","display_name":"Topological space","score":0.7014838457107544},{"id":"https://openalex.org/keywords/closure","display_name":"Closure (psychology)","score":0.686640739440918},{"id":"https://openalex.org/keywords/separation-axiom","display_name":"Separation axiom","score":0.5848950743675232},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.5220941305160522},{"id":"https://openalex.org/keywords/space","display_name":"Space (punctuation)","score":0.5197760462760925},{"id":"https://openalex.org/keywords/t1-space","display_name":"T1 space","score":0.5189558863639832},{"id":"https://openalex.org/keywords/metric-space","display_name":"Metric space","score":0.5140057802200317},{"id":"https://openalex.org/keywords/open-set","display_name":"Open set","score":0.5101454854011536},{"id":"https://openalex.org/keywords/complete-metric-space","display_name":"Complete metric space","score":0.4994010925292969},{"id":"https://openalex.org/keywords/closed-set","display_name":"Closed set","score":0.48868969082832336},{"id":"https://openalex.org/keywords/closure-operator","display_name":"Closure operator","score":0.4688454866409302},{"id":"https://openalex.org/keywords/axiom","display_name":"Axiom","score":0.4423699378967285},{"id":"https://openalex.org/keywords/set-theory","display_name":"Set theory","score":0.4303404688835144},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.42934736609458923},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.37711426615715027},{"id":"https://openalex.org/keywords/isolated-point","display_name":"Isolated point","score":0.34583890438079834},{"id":"https://openalex.org/keywords/topological-vector-space","display_name":"Topological vector space","score":0.2840317189693451}],"concepts":[{"id":"https://openalex.org/C87598191","wikidata":"https://www.wikidata.org/wiki/Q1571831","display_name":"Ultrafilter","level":2,"score":0.9387015700340271},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8641290664672852},{"id":"https://openalex.org/C110729354","wikidata":"https://www.wikidata.org/wiki/Q185478","display_name":"Countable set","level":2,"score":0.7165504693984985},{"id":"https://openalex.org/C151797676","wikidata":"https://www.wikidata.org/wiki/Q179692","display_name":"Axiom of choice","level":4,"score":0.7058618664741516},{"id":"https://openalex.org/C81332173","wikidata":"https://www.wikidata.org/wiki/Q179899","display_name":"Topological space","level":2,"score":0.7014838457107544},{"id":"https://openalex.org/C146834321","wikidata":"https://www.wikidata.org/wiki/Q2979672","display_name":"Closure (psychology)","level":2,"score":0.686640739440918},{"id":"https://openalex.org/C40409056","wikidata":"https://www.wikidata.org/wiki/Q1140985","display_name":"Separation axiom","level":3,"score":0.5848950743675232},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5220941305160522},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.5197760462760925},{"id":"https://openalex.org/C106558176","wikidata":"https://www.wikidata.org/wiki/Q284347","display_name":"T1 space","level":5,"score":0.5189558863639832},{"id":"https://openalex.org/C198043062","wikidata":"https://www.wikidata.org/wiki/Q180953","display_name":"Metric space","level":2,"score":0.5140057802200317},{"id":"https://openalex.org/C42357961","wikidata":"https://www.wikidata.org/wiki/Q213363","display_name":"Open set","level":2,"score":0.5101454854011536},{"id":"https://openalex.org/C93361540","wikidata":"https://www.wikidata.org/wiki/Q848569","display_name":"Complete metric space","level":3,"score":0.4994010925292969},{"id":"https://openalex.org/C164953516","wikidata":"https://www.wikidata.org/wiki/Q320357","display_name":"Closed set","level":2,"score":0.48868969082832336},{"id":"https://openalex.org/C160771489","wikidata":"https://www.wikidata.org/wiki/Q10564851","display_name":"Closure operator","level":3,"score":0.4688454866409302},{"id":"https://openalex.org/C167729594","wikidata":"https://www.wikidata.org/wiki/Q17736","display_name":"Axiom","level":2,"score":0.4423699378967285},{"id":"https://openalex.org/C153046414","wikidata":"https://www.wikidata.org/wiki/Q12482","display_name":"Set theory","level":3,"score":0.4303404688835144},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.42934736609458923},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.37711426615715027},{"id":"https://openalex.org/C174265945","wikidata":"https://www.wikidata.org/wiki/Q1142889","display_name":"Isolated point","level":4,"score":0.34583890438079834},{"id":"https://openalex.org/C47776270","wikidata":"https://www.wikidata.org/wiki/Q1455249","display_name":"Topological vector space","level":3,"score":0.2840317189693451},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C34447519","wikidata":"https://www.wikidata.org/wiki/Q179522","display_name":"Market economy","level":1,"score":0.0},{"id":"https://openalex.org/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1002/malq.200910014","is_oa":true,"landing_page_url":"https://doi.org/10.1002/malq.200910014","pdf_url":"https://onlinelibrary.wiley.com/doi/pdfdirect/10.1002/malq.200910014","source":{"id":"https://openalex.org/S172351290","display_name":"Mathematical logic quarterly","issn_l":"0942-5616","issn":["0942-5616","1521-3870"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematical Logic Quarterly","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1002/malq.200910014","is_oa":true,"landing_page_url":"https://doi.org/10.1002/malq.200910014","pdf_url":"https://onlinelibrary.wiley.com/doi/pdfdirect/10.1002/malq.200910014","source":{"id":"https://openalex.org/S172351290","display_name":"Mathematical logic quarterly","issn_l":"0942-5616","issn":["0942-5616","1521-3870"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematical Logic Quarterly","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320320339","display_name":"University of Birmingham","ror":"https://ror.org/03angcq70"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3099155029.pdf","grobid_xml":"https://content.openalex.org/works/W3099155029.grobid-xml"},"referenced_works_count":6,"referenced_works":["https://openalex.org/W642569514","https://openalex.org/W951868816","https://openalex.org/W1541012876","https://openalex.org/W1608084331","https://openalex.org/W1983991632","https://openalex.org/W2026473093"],"related_works":["https://openalex.org/W3157862209","https://openalex.org/W1436669948","https://openalex.org/W2058564026","https://openalex.org/W4390760006","https://openalex.org/W4388633357","https://openalex.org/W2011502444","https://openalex.org/W4288949362","https://openalex.org/W1551033973","https://openalex.org/W3099155029","https://openalex.org/W2996306627"],"abstract_inverted_index":{"Abstract":[0],"It":[1,37,181],"is":[2,38,69,88,96,128,158,182,195],"well":[3],"known":[4],"that,":[5],"in":[6,114,197],"a":[7,132,165],"topological":[8,50,82,133,201],"space,":[9],"the":[10,26,41,45,56,66,72,89,97,145,151,169,190],"open":[11,46,146],"sets":[12,47,147],"can":[13,62],"be":[14],"characterized":[15,149],"using":[16],"?lter":[17],"convergence.":[18,153],"In":[19],"ZF":[20],"(":[21,103],"Zermelo\u2010Fraenkel":[22],"set":[23,163,187],"theory":[24],"without":[25],"Axiom":[27,170],"of":[28,171,177,200],"Choice":[29,173],"),":[30],"we":[31,61],"cannot":[32],"replace":[33],"filters":[34],"by":[35,150],"ultrafilters.":[36],"proven":[39],"that":[40,65,74,160],"ultra?lter":[42,152],"convergence":[43],"determines":[44],"for":[48,80,136,175],"every":[49,81,162],"space":[51,83,134],"if":[52,55,161],"and":[53,94,121],"only":[54],"Ultrafilter":[57],"Theorem":[58,68],"holds.":[59],"More,":[60],"also":[63,183],"prove":[64],"Ultra?lter":[67,98],"equivalent":[70],"to":[71,119,130],"fact":[73],"u":[75,95,101,138,192],"X":[76,79,84,102,109,115,135,139,142],"=":[77,193],"k":[78,87,141,194],",":[85,143],"where":[86],"usual":[90],"Kuratowski":[91],"closure":[92],"operator":[93],"Closure":[99],"with":[100],"A":[104,122],"):=":[105],"{":[106],"x":[107,120],"\u2208":[108,123],":":[110],"(\u2203":[111],"U":[112,117,124],"ultrafilter":[113],")[":[116],"converges":[118],"]}.":[125],"However,":[126],"it":[127,157],"possible":[129],"built":[131],"which":[137,186],"\u2260":[140],"but":[144],"are":[148],"To":[154],"do":[155],"so,":[156],"proved":[159],"has":[164],"free":[166],"ultra?lter,":[167],"then":[168],"Countable":[172],"holds":[174],"families":[176],"non\u2010empty":[178],"finite":[179],"sets.":[180],"investigated":[184],"under":[185],"theoretic":[188],"conditions":[189],"equality":[191],"true":[196],"some":[198],"subclasses":[199],"spaces,":[202,206],"such":[203],"as":[204],"metric":[205],"second":[207],"countable":[208],"T0\u2010spaces":[209],"or":[210],"{\u211d}":[211],"(\u00a9":[212],"2010":[213],"WILEY\u2010VCH":[214],"Verlag":[215],"GmbH":[216],"&amp;":[217],"Co.":[218],"KGaA,":[219],"Weinheim)":[220]},"counts_by_year":[],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2020-11-23T00:00:00"}
