{"id":"https://openalex.org/W7202315328","doi":"https://doi.org/10.48550/arxiv.2608.11023","title":"On Ishiki's Conjecture: $\\mathrm{Met}(D)$ Is Not Completely Metrizable for $\\lvert D\\rvert=\\aleph_1$","display_name":"On Ishiki's Conjecture: $\\mathrm{Met}(D)$ Is Not Completely Metrizable for $\\lvert D\\rvert=\\aleph_1$","publication_year":2026,"publication_date":"2026-08-11","ids":{"openalex":"https://openalex.org/W7202315328","doi":"https://doi.org/10.48550/arxiv.2608.11023"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2608.11023","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.11023","pdf_url":"https://arxiv.org/pdf/2608.11023","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2608.11023","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5071370486","display_name":"Tomoki Uda","orcid":"https://orcid.org/0000-0002-6073-1572","observed_orcids":["https://orcid.org/0000-0002-6073-1572"]},"institutions":[{"id":"https://openalex.org/I119031078","display_name":"Nanzan University","ror":"https://ror.org/042b6j696","country_code":"JP","type":"education","lineage":["https://openalex.org/I119031078"]}],"countries":["JP"],"is_corresponding":true,"raw_author_name":"Uda, Tomoki","raw_affiliation_strings":["Nanzan University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Nanzan University","institution_ids":["https://openalex.org/I119031078"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5071370486"],"corresponding_institution_ids":["https://openalex.org/I119031078"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11151","display_name":"Advanced Topology and Set Theory","score":0.9829000234603882,"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":0.9829000234603882,"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":0.008999999612569809,"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/T11777","display_name":"Fixed Point Theorems Analysis","score":0.0012000000569969416,"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/completely-metrizable-spaces","display_name":"completely metrizable spaces","score":0.859000027179718},{"id":"https://openalex.org/keywords/closed-subspaces","display_name":"closed subspaces","score":0.7049999833106995},{"id":"https://openalex.org/keywords/lean-4-formalization","display_name":"Lean 4 formalization","score":0.8050000071525574}],"concepts":[{"id":"https://openalex.org/C105433845","wikidata":"https://www.wikidata.org/wiki/Q1194053","display_name":"Metrization theorem","level":3,"score":0.817300021648407},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8165000081062317},{"id":"https://openalex.org/C41608201","wikidata":"https://www.wikidata.org/wiki/Q980509","display_name":"Embedding","level":2,"score":0.5514000058174133},{"id":"https://openalex.org/C112313634","wikidata":"https://www.wikidata.org/wiki/Q7886648","display_name":"Complement (music)","level":5,"score":0.5187000036239624},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.501800000667572},{"id":"https://openalex.org/C32834561","wikidata":"https://www.wikidata.org/wiki/Q660730","display_name":"Subspace topology","level":2,"score":0.4869999885559082},{"id":"https://openalex.org/C87117476","wikidata":"https://www.wikidata.org/wiki/Q362383","display_name":"Cardinality (data modeling)","level":2,"score":0.4611999988555908},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4544000029563904},{"id":"https://openalex.org/C95611797","wikidata":"https://www.wikidata.org/wiki/Q17502105","display_name":"Infimum and supremum","level":2,"score":0.45100000500679016},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.4336000084877014},{"id":"https://openalex.org/C164953516","wikidata":"https://www.wikidata.org/wiki/Q320357","display_name":"Closed set","level":2,"score":0.4120999872684479},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.37720000743865967},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.36399999260902405},{"id":"https://openalex.org/C17327563","wikidata":"https://www.wikidata.org/wiki/Q1991405","display_name":"Nowhere dense set","level":3,"score":0.33799999952316284},{"id":"https://openalex.org/C42357961","wikidata":"https://www.wikidata.org/wiki/Q213363","display_name":"Open set","level":2,"score":0.337799996137619},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.31869998574256897},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3151000142097473},{"id":"https://openalex.org/C153782549","wikidata":"https://www.wikidata.org/wiki/Q175116","display_name":"Discrete space","level":2,"score":0.3068999946117401},{"id":"https://openalex.org/C81332173","wikidata":"https://www.wikidata.org/wiki/Q179899","display_name":"Topological space","level":2,"score":0.3061999976634979},{"id":"https://openalex.org/C153046414","wikidata":"https://www.wikidata.org/wiki/Q12482","display_name":"Set theory","level":3,"score":0.2994000017642975},{"id":"https://openalex.org/C45340560","wikidata":"https://www.wikidata.org/wiki/Q215382","display_name":"Disjoint sets","level":2,"score":0.2800000011920929},{"id":"https://openalex.org/C13336665","wikidata":"https://www.wikidata.org/wiki/Q125977","display_name":"Vector space","level":2,"score":0.25699999928474426}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2608.11023","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.11023","pdf_url":"https://arxiv.org/pdf/2608.11023","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2608.11023","doi":"https://doi.org/10.48550/arxiv.2608.11023","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2608.11023","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2608.11023","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.11023","pdf_url":"https://arxiv.org/pdf/2608.11023","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7202315328.pdf","grobid_xml":"https://content.openalex.org/works/W7202315328.grobid-xml"},"referenced_works_count":1,"referenced_works":["https://openalex.org/W4404223827"],"related_works":["https://openalex.org/W1980787000","https://openalex.org/W2055018325","https://openalex.org/W835798799","https://openalex.org/W3047568838","https://openalex.org/W4407794707","https://openalex.org/W1529408443","https://openalex.org/W4246847200","https://openalex.org/W3131390145","https://openalex.org/W7098880873","https://openalex.org/W1242601853"],"abstract_inverted_index":{"For":[0],"a":[1,53,70],"discrete":[2,20],"topological":[3],"space":[4],"$D$,":[5],"let":[6],"$\\mathrm{Met}(D)$":[7,36],"denote":[8],"the":[9,19,24,28],"set":[10,54],"of":[11,58,73],"metrics":[12],"on":[13],"$D$":[14],"that":[15,35,61],"are":[16],"compatible":[17],"with":[18,23],"topology,":[21],"equipped":[22],"topology":[25],"induced":[26],"by":[27,51],"supremum":[29],"distance.":[30],"Ishiki's":[31],"Conjecture":[32],"5.1":[33],"asserts":[34],"is":[37,62],"not":[38,63],"completely":[39],"metrizable":[40],"when":[41],"$\\lvert":[42],"D\\rvert":[43],"=":[44],"\\aleph_1$.":[45],"We":[46],"prove":[47],"this":[48],"in":[49,81],"ZFC":[50],"constructing":[52],"$A":[55],"\\subseteq":[56],"[0,1]$":[57],"cardinality":[59],"$\\aleph_1$":[60],"$F_\u03c3$":[64],"and":[65],"embedding":[66],"its":[67],"complement":[68],"as":[69],"closed":[71],"subspace":[72],"$\\mathrm{Met}(D)$.":[74],"The":[75],"proof":[76],"has":[77],"also":[78],"been":[79],"formalised":[80],"Lean":[82],"4.":[83]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-08-13T00:00:00"}
