{"id":"https://openalex.org/W2793829847","doi":"https://doi.org/10.1080/00029890.2018.1420334","title":"On Certain Dense, Uncountable Subsets of the Real Line","display_name":"On Certain Dense, Uncountable Subsets of the Real Line","publication_year":2018,"publication_date":"2018-03-22","ids":{"openalex":"https://openalex.org/W2793829847","doi":"https://doi.org/10.1080/00029890.2018.1420334","mag":"2793829847"},"language":"en","primary_location":{"id":"doi:10.1080/00029890.2018.1420334","is_oa":false,"landing_page_url":"https://doi.org/10.1080/00029890.2018.1420334","pdf_url":null,"source":{"id":"https://openalex.org/S156280010","display_name":"American Mathematical Monthly","issn_l":"0002-9890","issn":["0002-9890","1930-0972"],"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547"],"host_organization_lineage_names":["Taylor & Francis"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The American Mathematical Monthly","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/A5019343744","display_name":"Chung-Wu Ho","orcid":"https://orcid.org/0000-0002-6889-259X"},"institutions":[{"id":"https://openalex.org/I2800396611","display_name":"Evergreen Valley College","ror":"https://ror.org/05jsgsm79","country_code":"US","type":"education","lineage":["https://openalex.org/I2800396611"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Chungwu Ho","raw_affiliation_strings":["Department of Mathematics, Evergreen Valley College, San Jose, CA 95135 Current address: 1545 Laurelwood Crossing Terrace, San Jose, CA 95138"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Evergreen Valley College, San Jose, CA 95135 Current address: 1545 Laurelwood Crossing Terrace, San Jose, CA 95138","institution_ids":["https://openalex.org/I2800396611"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5113586570","display_name":"Seth Zimmerman","orcid":"https://orcid.org/0000-0001-9636-2265"},"institutions":[{"id":"https://openalex.org/I2800396611","display_name":"Evergreen Valley College","ror":"https://ror.org/05jsgsm79","country_code":"US","type":"education","lineage":["https://openalex.org/I2800396611"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Seth Zimmerman","raw_affiliation_strings":["Department of Mathematics, Evergreen Valley College, San Jose, CA 95135 Current address: 525 Whitecap Road, Bellingham, WA 98229"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Evergreen Valley College, San Jose, CA 95135 Current address: 525 Whitecap Road, Bellingham, WA 98229","institution_ids":["https://openalex.org/I2800396611"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5019343744"],"corresponding_institution_ids":["https://openalex.org/I2800396611"],"apc_list":null,"apc_paid":null,"fwci":1.8465,"has_fulltext":false,"cited_by_count":6,"citation_normalized_percentile":{"value":0.83353097,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":96},"biblio":{"volume":"125","issue":"4","first_page":"339","last_page":"346"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T13400","display_name":"Mathematical and Theoretical Analysis","score":0.9585999846458435,"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"}},"topics":[{"id":"https://openalex.org/T13400","display_name":"Mathematical and Theoretical Analysis","score":0.9585999846458435,"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":0.9438999891281128,"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"}},{"id":"https://openalex.org/T11151","display_name":"Advanced Topology and Set Theory","score":0.9391999840736389,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/uncountable-set","display_name":"Uncountable set","score":0.9947382211685181},{"id":"https://openalex.org/keywords/real-line","display_name":"Real line","score":0.7156288027763367},{"id":"https://openalex.org/keywords/measure","display_name":"Measure (data warehouse)","score":0.5858878493309021},{"id":"https://openalex.org/keywords/unit-interval","display_name":"Unit interval","score":0.5804421305656433},{"id":"https://openalex.org/keywords/complement","display_name":"Complement (music)","score":0.5715270042419434},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5431278347969055},{"id":"https://openalex.org/keywords/partition","display_name":"Partition (number theory)","score":0.5122391581535339},{"id":"https://openalex.org/keywords/real-number","display_name":"Real number","score":0.478950560092926},{"id":"https://openalex.org/keywords/integer","display_name":"Integer (computer science)","score":0.47859928011894226},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.4639306664466858},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4601078927516937},{"id":"https://openalex.org/keywords/null-set","display_name":"Null set","score":0.44922202825546265},{"id":"https://openalex.org/keywords/line","display_name":"Line (geometry)","score":0.44553306698799133},{"id":"https://openalex.org/keywords/interval","display_name":"Interval (graph theory)","score":0.4131150245666504},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.3605937361717224},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.33694377541542053},{"id":"https://openalex.org/keywords/countable-set","display_name":"Countable set","score":0.07832375168800354},{"id":"https://openalex.org/keywords/data-mining","display_name":"Data mining","score":0.05505695939064026}],"concepts":[{"id":"https://openalex.org/C142399903","wikidata":"https://www.wikidata.org/wiki/Q1128796","display_name":"Uncountable set","level":3,"score":0.9947382211685181},{"id":"https://openalex.org/C75432250","wikidata":"https://www.wikidata.org/wiki/Q29654355","display_name":"Real line","level":2,"score":0.7156288027763367},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.5858878493309021},{"id":"https://openalex.org/C128394911","wikidata":"https://www.wikidata.org/wiki/Q1987578","display_name":"Unit interval","level":2,"score":0.5804421305656433},{"id":"https://openalex.org/C112313634","wikidata":"https://www.wikidata.org/wiki/Q7886648","display_name":"Complement (music)","level":5,"score":0.5715270042419434},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5431278347969055},{"id":"https://openalex.org/C42812","wikidata":"https://www.wikidata.org/wiki/Q1082910","display_name":"Partition (number theory)","level":2,"score":0.5122391581535339},{"id":"https://openalex.org/C94020503","wikidata":"https://www.wikidata.org/wiki/Q12916","display_name":"Real number","level":2,"score":0.478950560092926},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.47859928011894226},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4639306664466858},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4601078927516937},{"id":"https://openalex.org/C55375708","wikidata":"https://www.wikidata.org/wiki/Q1201815","display_name":"Null set","level":3,"score":0.44922202825546265},{"id":"https://openalex.org/C198352243","wikidata":"https://www.wikidata.org/wiki/Q37105","display_name":"Line (geometry)","level":2,"score":0.44553306698799133},{"id":"https://openalex.org/C2778067643","wikidata":"https://www.wikidata.org/wiki/Q166507","display_name":"Interval (graph theory)","level":2,"score":0.4131150245666504},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.3605937361717224},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.33694377541542053},{"id":"https://openalex.org/C110729354","wikidata":"https://www.wikidata.org/wiki/Q185478","display_name":"Countable set","level":2,"score":0.07832375168800354},{"id":"https://openalex.org/C124101348","wikidata":"https://www.wikidata.org/wiki/Q172491","display_name":"Data mining","level":1,"score":0.05505695939064026},{"id":"https://openalex.org/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","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},{"id":"https://openalex.org/C188082640","wikidata":"https://www.wikidata.org/wiki/Q1780899","display_name":"Complementation","level":4,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C127716648","wikidata":"https://www.wikidata.org/wiki/Q104053","display_name":"Phenotype","level":3,"score":0.0},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1080/00029890.2018.1420334","is_oa":false,"landing_page_url":"https://doi.org/10.1080/00029890.2018.1420334","pdf_url":null,"source":{"id":"https://openalex.org/S156280010","display_name":"American Mathematical Monthly","issn_l":"0002-9890","issn":["0002-9890","1930-0972"],"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547"],"host_organization_lineage_names":["Taylor & Francis"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The American Mathematical Monthly","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[{"display_name":"Reduced inequalities","score":0.5,"id":"https://metadata.un.org/sdg/10"}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":6,"referenced_works":["https://openalex.org/W1593345651","https://openalex.org/W2010646630","https://openalex.org/W2600879927","https://openalex.org/W3202069268","https://openalex.org/W4234707536","https://openalex.org/W4299390900"],"related_works":["https://openalex.org/W2949853646","https://openalex.org/W2908537796","https://openalex.org/W2914045375","https://openalex.org/W2384513610","https://openalex.org/W2963855116","https://openalex.org/W4251301317","https://openalex.org/W2074456561","https://openalex.org/W2981961718","https://openalex.org/W2793829847","https://openalex.org/W2157014216"],"abstract_inverted_index":{"It":[0],"is":[1,13,22,57,96],"a":[2,37,58,93],"challenge":[3],"to":[4,78],"define":[5,92],"an":[6],"uncountable":[7,24,50],"set":[8,100],"of":[9,39,63,70,84,101],"real":[10,17,41,55],"numbers":[11],"that":[12,48,68,95],"dense":[14],"in":[15],"the":[16,40,79,85,99],"line":[18,42],"and":[19,25,51],"whose":[20],"complement":[21],"also":[23],"dense.":[26],"In":[27],"this":[28],"article":[29],"we":[30],"specify,":[31],"for":[32,61],"any":[33],"positive":[34],"integer":[35],"k,":[36],"partition":[38],"into":[43],"k":[44],"+":[45],"1":[46,103],"sets":[47,72],"are":[49],"dense;":[52],"moreover,":[53],"every":[54],"number":[56],"condensation":[59],"point":[60],"each":[62,83],"these":[64,71],"sets.":[65],"We":[66,90],"show":[67],"one":[69,75],"has":[73,87],"measure":[74,88,102],"when":[76],"restricted":[77],"unit":[80],"interval,":[81],"while":[82],"others":[86],"zero.":[89],"then":[91],"function":[94],"continuous":[97],"on":[98],"but":[104],"discontinuous":[105],"elsewhere.":[106]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2020,"cited_by_count":2},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":1}],"updated_date":"2026-03-27T05:58:40.876381","created_date":"2025-10-10T00:00:00"}
