{"id":"https://openalex.org/W2398189839","doi":"https://doi.org/10.2307/2273130","title":"\u03a3<sub>1</sub>-separation","display_name":"\u03a3<sub>1</sub>-separation","publication_year":1979,"publication_date":"1979-09-01","ids":{"openalex":"https://openalex.org/W2398189839","doi":"https://doi.org/10.2307/2273130","mag":"2398189839"},"language":"en","primary_location":{"id":"doi:10.2307/2273130","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2273130","pdf_url":null,"source":{"id":"https://openalex.org/S177675236","display_name":"Journal of Symbolic Logic","issn_l":"0022-4812","issn":["0022-4812","1943-5886"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Symbolic Logic","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5005199708","display_name":"Fred G. Abramson","orcid":null},"institutions":[{"id":"https://openalex.org/I43579087","display_name":"University of Wisconsin\u2013Milwaukee","ror":"https://ror.org/031q21x57","country_code":"US","type":"education","lineage":["https://openalex.org/I43579087"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Fred G. Abramson","raw_affiliation_strings":["University of Wisconsin-Milwaukee, Milwaukee, Wisconsin 53201"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Wisconsin-Milwaukee, Milwaukee, Wisconsin 53201","institution_ids":["https://openalex.org/I43579087"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5005199708"],"corresponding_institution_ids":["https://openalex.org/I43579087"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.32613552,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"44","issue":"3","first_page":"374","last_page":"382"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11151","display_name":"Advanced Topology and Set Theory","score":0.9975000023841858,"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":0.9975000023841858,"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/T13400","display_name":"Mathematical and Theoretical Analysis","score":0.9879999756813049,"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"}},{"id":"https://openalex.org/T12002","display_name":"Computability, Logic, AI Algorithms","score":0.9769999980926514,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8609093427658081},{"id":"https://openalex.org/keywords/countable-set","display_name":"Countable set","score":0.8235777020454407},{"id":"https://openalex.org/keywords/disjoint-sets","display_name":"Disjoint sets","score":0.8076989650726318},{"id":"https://openalex.org/keywords/separation","display_name":"Separation (statistics)","score":0.7252951264381409},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6612077951431274},{"id":"https://openalex.org/keywords/sequence","display_name":"Sequence (biology)","score":0.5986694693565369},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.5376272201538086},{"id":"https://openalex.org/keywords/transitive-relation","display_name":"Transitive relation","score":0.5330549478530884},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.5094221234321594},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.4983024597167969},{"id":"https://openalex.org/keywords/closed-set","display_name":"Closed set","score":0.4177047312259674},{"id":"https://openalex.org/keywords/mutual-fund-separation-theorem","display_name":"Mutual fund separation theorem","score":0.4124707579612732},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.060545384883880615}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8609093427658081},{"id":"https://openalex.org/C110729354","wikidata":"https://www.wikidata.org/wiki/Q185478","display_name":"Countable set","level":2,"score":0.8235777020454407},{"id":"https://openalex.org/C45340560","wikidata":"https://www.wikidata.org/wiki/Q215382","display_name":"Disjoint sets","level":2,"score":0.8076989650726318},{"id":"https://openalex.org/C2776061190","wikidata":"https://www.wikidata.org/wiki/Q7451805","display_name":"Separation (statistics)","level":2,"score":0.7252951264381409},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6612077951431274},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.5986694693565369},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.5376272201538086},{"id":"https://openalex.org/C191399111","wikidata":"https://www.wikidata.org/wiki/Q64861","display_name":"Transitive relation","level":2,"score":0.5330549478530884},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5094221234321594},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.4983024597167969},{"id":"https://openalex.org/C164953516","wikidata":"https://www.wikidata.org/wiki/Q320357","display_name":"Closed set","level":2,"score":0.4177047312259674},{"id":"https://openalex.org/C160471959","wikidata":"https://www.wikidata.org/wiki/Q17082552","display_name":"Mutual fund separation theorem","level":3,"score":0.4124707579612732},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.060545384883880615},{"id":"https://openalex.org/C2780821815","wikidata":"https://www.wikidata.org/wiki/Q5340806","display_name":"Portfolio","level":2,"score":0.0},{"id":"https://openalex.org/C106159729","wikidata":"https://www.wikidata.org/wiki/Q2294553","display_name":"Financial economics","level":1,"score":0.0},{"id":"https://openalex.org/C54355233","wikidata":"https://www.wikidata.org/wiki/Q7162","display_name":"Genetics","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"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/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.2307/2273130","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2273130","pdf_url":null,"source":{"id":"https://openalex.org/S177675236","display_name":"Journal of Symbolic Logic","issn_l":"0022-4812","issn":["0022-4812","1943-5886"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Symbolic Logic","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":8,"referenced_works":["https://openalex.org/W65053296","https://openalex.org/W154666901","https://openalex.org/W195189594","https://openalex.org/W2008744795","https://openalex.org/W2081160992","https://openalex.org/W2131495252","https://openalex.org/W2530006810","https://openalex.org/W2905135108"],"related_works":["https://openalex.org/W2054347355","https://openalex.org/W26693340","https://openalex.org/W2621137221","https://openalex.org/W4295724527","https://openalex.org/W2088461731","https://openalex.org/W2009403891","https://openalex.org/W132109778","https://openalex.org/W1574269463","https://openalex.org/W2088585064","https://openalex.org/W1999685870"],"abstract_inverted_index":{"Abstract":[0],"Let":[1,135],"A":[2,24,27,34,38,63,82,93,100,104,114,116],"be":[3,137],"a":[4,31,65,120,148],"standard":[5,129],"transitive":[6],"admissible":[7,140],"set.":[8],"\u03a3":[9,22,151],"1":[10,23,33,152],"-separation":[11,153],"is":[12,30,64,98,105,131,147,156],"the":[13,84],"principle":[14],"that":[15,40],"whenever":[16],"X":[17,41],"and":[18,44,110],"Y":[19,45],"are":[20],"disjoint":[21],"subsets":[25],"of":[26,37,67,72,89,123,150],"then":[28,54,83,115],"there":[29,146],"\u22bf":[32],"subset":[35],"S":[36,43,47],"such":[39],"\u2286":[42],"\u2229":[46],"=":[48,112,125],"\u2205.":[49],"Theorem.":[50,134],"If":[51,56,103],"satisfies":[52],"\u03a3-separation,":[53],"(1)":[55],"\u3008T":[57],"n":[58,86,96],"\u2223n":[59],"&lt;":[60],"\u03c9)":[61],"\u03f5":[62],"sequence":[66],"trees":[68],"on":[69],"\u03c9":[70],"each":[71],"which":[73],"has":[74,117],"at":[75],"most":[76],"finitely":[77],"many":[78],"infinite":[79,90],"paths":[80,91],"in":[81,92,99,118],"function":[85],"\u21a6":[87],"(set":[88],"through":[94],"T":[95],")":[97],".":[101,133],"(2)":[102],"not":[106],"closed":[107],"under":[108],"hyperjump":[109],"\u03b1":[111,132,136],"On":[113],"it":[119],"nonstandard":[121],"model":[122,149],"V":[124],"L":[126],"whose":[127,154],"ordinal":[128,141],"part":[130],"any":[138],"countable":[139],"greater":[142],"than":[143],"\u03c9.":[144],"Then":[145],"height":[155],"\u03b1.":[157]},"counts_by_year":[],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}