{"id":"https://openalex.org/W2589032788","doi":"https://doi.org/10.1515/forma-2016-0013","title":"Compactness in Metric Spaces","display_name":"Compactness in Metric Spaces","publication_year":2016,"publication_date":"2016-09-01","ids":{"openalex":"https://openalex.org/W2589032788","doi":"https://doi.org/10.1515/forma-2016-0013","mag":"2589032788"},"language":"en","primary_location":{"id":"doi:10.1515/forma-2016-0013","is_oa":true,"landing_page_url":"https://doi.org/10.1515/forma-2016-0013","pdf_url":"https://content.sciendo.com/downloadpdf/journals/forma/24/3/article-p167.pdf","source":{"id":"https://openalex.org/S179241345","display_name":"Formalized Mathematics","issn_l":"1426-2630","issn":["1426-2630","1898-9934"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320322","host_organization_name":"De Gruyter Open","host_organization_lineage":["https://openalex.org/P4310320322","https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter Open","De Gruyter"],"type":"journal"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Formalized Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://content.sciendo.com/downloadpdf/journals/forma/24/3/article-p167.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5005167371","display_name":"Kazuhisa Nakasho","orcid":"https://orcid.org/0000-0003-1110-4342"},"institutions":[{"id":"https://openalex.org/I137975476","display_name":"Shinshu University","ror":"https://ror.org/0244rem06","country_code":"JP","type":"education","lineage":["https://openalex.org/I137975476"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Kazuhisa Nakasho","raw_affiliation_strings":["Shinshu University Nagano, Japan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Shinshu University Nagano, Japan","institution_ids":["https://openalex.org/I137975476"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5057236933","display_name":"Keiko Narita","orcid":null},"institutions":[{"id":"https://openalex.org/I4210149837","display_name":"Hirosaki National Hospital","ror":"https://ror.org/03qrrp624","country_code":"JP","type":"healthcare","lineage":["https://openalex.org/I4210137409","https://openalex.org/I4210149837"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Keiko Narita","raw_affiliation_strings":["Hirosaki-city Aomori, Japan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Hirosaki-city Aomori, Japan","institution_ids":["https://openalex.org/I4210149837"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5055549852","display_name":"Yasunari Shidama","orcid":"https://orcid.org/0000-0001-5634-0838"},"institutions":[{"id":"https://openalex.org/I137975476","display_name":"Shinshu University","ror":"https://ror.org/0244rem06","country_code":"JP","type":"education","lineage":["https://openalex.org/I137975476"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Yasunari Shidama","raw_affiliation_strings":["Shinshu University Nagano, Japan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Shinshu University Nagano, Japan","institution_ids":["https://openalex.org/I137975476"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.17474031,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"24","issue":"3","first_page":"167","last_page":"172"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11151","display_name":"Advanced Topology and Set Theory","score":0.9968000054359436,"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":0.9968000054359436,"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/T11671","display_name":"Fuzzy and Soft Set Theory","score":0.9275000095367432,"subfield":{"id":"https://openalex.org/subfields/1803","display_name":"Management Science and Operations Research"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T10896","display_name":"Homotopy and Cohomology in Algebraic Topology","score":0.9193999767303467,"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/compact-space","display_name":"Compact space","score":0.9429880380630493},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7984471917152405},{"id":"https://openalex.org/keywords/metric-space","display_name":"Metric space","score":0.6390354633331299},{"id":"https://openalex.org/keywords/real-line","display_name":"Real line","score":0.5631409883499146},{"id":"https://openalex.org/keywords/norm","display_name":"Norm (philosophy)","score":0.5579751133918762},{"id":"https://openalex.org/keywords/equivalence","display_name":"Equivalence (formal languages)","score":0.488018661737442},{"id":"https://openalex.org/keywords/topological-space","display_name":"Topological space","score":0.4273505210876465},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4024868309497833},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.37955939769744873}],"concepts":[{"id":"https://openalex.org/C18648836","wikidata":"https://www.wikidata.org/wiki/Q381892","display_name":"Compact space","level":2,"score":0.9429880380630493},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7984471917152405},{"id":"https://openalex.org/C198043062","wikidata":"https://www.wikidata.org/wiki/Q180953","display_name":"Metric space","level":2,"score":0.6390354633331299},{"id":"https://openalex.org/C75432250","wikidata":"https://www.wikidata.org/wiki/Q29654355","display_name":"Real line","level":2,"score":0.5631409883499146},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.5579751133918762},{"id":"https://openalex.org/C2780069185","wikidata":"https://www.wikidata.org/wiki/Q7977945","display_name":"Equivalence (formal languages)","level":2,"score":0.488018661737442},{"id":"https://openalex.org/C81332173","wikidata":"https://www.wikidata.org/wiki/Q179899","display_name":"Topological space","level":2,"score":0.4273505210876465},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4024868309497833},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.37955939769744873},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1515/forma-2016-0013","is_oa":true,"landing_page_url":"https://doi.org/10.1515/forma-2016-0013","pdf_url":"https://content.sciendo.com/downloadpdf/journals/forma/24/3/article-p167.pdf","source":{"id":"https://openalex.org/S179241345","display_name":"Formalized Mathematics","issn_l":"1426-2630","issn":["1426-2630","1898-9934"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320322","host_organization_name":"De Gruyter Open","host_organization_lineage":["https://openalex.org/P4310320322","https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter Open","De Gruyter"],"type":"journal"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Formalized Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:92ea8ddd8b044e2f87524d3080d46e98","is_oa":false,"landing_page_url":"https://doaj.org/article/92ea8ddd8b044e2f87524d3080d46e98","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Formalized Mathematics, Vol 24, Iss 3, Pp 167-172 (2016)","raw_type":"article"},{"id":"pmh:oai:repozytorium.uwb.edu.pl:11320/5551","is_oa":true,"landing_page_url":"http://hdl.handle.net/11320/5551","pdf_url":null,"source":{"id":"https://openalex.org/S4306400933","display_name":"The Repository of University in Bialystok (University in Bialystok)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210094198","host_organization_name":"University Clinical Hospital In Bialystok","host_organization_lineage":["https://openalex.org/I4210094198"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Article"}],"best_oa_location":{"id":"doi:10.1515/forma-2016-0013","is_oa":true,"landing_page_url":"https://doi.org/10.1515/forma-2016-0013","pdf_url":"https://content.sciendo.com/downloadpdf/journals/forma/24/3/article-p167.pdf","source":{"id":"https://openalex.org/S179241345","display_name":"Formalized Mathematics","issn_l":"1426-2630","issn":["1426-2630","1898-9934"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320322","host_organization_name":"De Gruyter Open","host_organization_lineage":["https://openalex.org/P4310320322","https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter Open","De Gruyter"],"type":"journal"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Formalized Mathematics","raw_type":"journal-article"},"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/10","score":0.4099999964237213,"display_name":"Reduced inequalities"}],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2589032788.pdf","grobid_xml":"https://content.openalex.org/works/W2589032788.grobid-xml"},"referenced_works_count":10,"referenced_works":["https://openalex.org/W118985555","https://openalex.org/W192809544","https://openalex.org/W2105559527","https://openalex.org/W2107221565","https://openalex.org/W2114890787","https://openalex.org/W2126555322","https://openalex.org/W2295728669","https://openalex.org/W2740943156","https://openalex.org/W4236726598","https://openalex.org/W4285719527"],"related_works":["https://openalex.org/W2121543990","https://openalex.org/W1981078319","https://openalex.org/W2402800040","https://openalex.org/W2000020320","https://openalex.org/W4252969132","https://openalex.org/W2785558381","https://openalex.org/W4282929111","https://openalex.org/W1996361334","https://openalex.org/W2494355497","https://openalex.org/W4206811082"],"abstract_inverted_index":{"Summary":[0],"In":[1,26,55,86,108,128],"this":[2],"article,":[3],"we":[4,30,59,90,112,132],"mainly":[5],"formalize":[6,61,97,113,133],"in":[7,45,48,83,93,105,120,141],"Mizar":[8],"[2]":[9],"the":[10,23,27,56,62,70,87,98,109,117,129,134,142],"equivalence":[11,71,99,135],"among":[12],"a":[13],"few":[14],"compactness":[15,92,101,104,137,140],"definitions":[16],"of":[17,35,43,50,52,64,72,100,116,122,124,136],"metric":[18,36,46,84],"spaces,":[19,21],"norm":[20,94,106],"and":[22,41,67,78,102,138,154],"real":[24,118,125,143],"line.":[25,144],"first":[28],"section,":[29,58,89,111,131],"formalized":[31],"general":[32],"topological":[33,114],"properties":[34,115],"spaces.":[37,85,95],"We":[38,96],"discussed":[39],"openness":[40],"closedness":[42],"subsets":[44],"spaces":[47],"terms":[49,121],"convergence":[51,123],"element":[53],"sequences.":[54,127],"second":[57],"firstly":[60],"definition":[63],"sequentially":[65],"compact,":[66],"then":[68],"discuss":[69,91],"compactness,":[73,75,77],"countable":[74],"sequential":[76,103,139],"totally":[79],"boundedness":[80],"with":[81],"completeness":[82],"third":[88],"space.":[107],"fourth":[110],"line":[119],"number":[126],"last":[130],"These":[145],"formalizations":[146],"are":[147],"based":[148],"on":[149],"[20],":[150],"[5],":[151],"[17],":[152],"[14],":[153],"[4].":[155]},"counts_by_year":[{"year":2022,"cited_by_count":1}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}