{"id":"https://openalex.org/W7213586464","doi":"https://doi.org/10.48550/arxiv.2609.20514","title":"Finite-Point Metrizable Coarsenings: Compatible Gauges, Simplicial Metrics, and Hausdorff Lower Bounds","display_name":"Finite-Point Metrizable Coarsenings: Compatible Gauges, Simplicial Metrics, and Hausdorff Lower Bounds","publication_year":2026,"publication_date":"2026-09-17","ids":{"openalex":"https://openalex.org/W7213586464","doi":"https://doi.org/10.48550/arxiv.2609.20514"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.20514","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.20514","pdf_url":"https://arxiv.org/pdf/2609.20514","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/2609.20514","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5151791198","display_name":"Ahmad ja'afari kalvan","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I1516879","display_name":"Tarbiat Modares University","ror":"https://ror.org/03mwgfy56","country_code":"IR","type":"education","lineage":["https://openalex.org/I1516879"]}],"countries":["IR"],"is_corresponding":false,"raw_author_name":"kalvan, Ahmad ja'afari","raw_affiliation_strings":["Department of Mathematics , Tarbiat Modares University , 14115-134 , Tehran , Iran"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , Tarbiat Modares University , 14115-134 , Tehran , Iran","institution_ids":["https://openalex.org/I1516879"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5073361610","display_name":"Ehsan Shahoseini","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I1516879","display_name":"Tarbiat Modares University","ror":"https://ror.org/03mwgfy56","country_code":"IR","type":"education","lineage":["https://openalex.org/I1516879"]},{"id":"https://openalex.org/I4210146419","display_name":"Institute for Research in Fundamental Sciences","ror":"https://ror.org/04xreqs31","country_code":"IR","type":"facility","lineage":["https://openalex.org/I4210146419"]}],"countries":["IR"],"is_corresponding":false,"raw_author_name":"Shahoseini, Ehsan","raw_affiliation_strings":["Department of Mathematics , Tarbiat Modares University , 14115-134 , Tehran , Iran","School of Mathematics , Institute for Research in Fundamental Sciences (IPM) , P.O. Box: 19395-5746 , Tehran , Iran"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , Tarbiat Modares University , 14115-134 , Tehran , Iran","institution_ids":["https://openalex.org/I1516879"]},{"raw_affiliation_string":"School of Mathematics , Institute for Research in Fundamental Sciences (IPM) , P.O. Box: 19395-5746 , Tehran , Iran","institution_ids":["https://openalex.org/I4210146419"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"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":0.9628999829292297,"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":0.9628999829292297,"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":0.012799999676644802,"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/T10304","display_name":"Geometric and Algebraic Topology","score":0.00559999980032444,"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/metrizable-topologies","display_name":"metrizable topologies","score":0.9210000038146973},{"id":"https://openalex.org/keywords/complete-metrics","display_name":"complete metrics","score":0.5009999871253967},{"id":"https://openalex.org/keywords/non-hausdorff-topology","display_name":"non-Hausdorff topology","score":0.8130000233650208},{"id":"https://openalex.org/keywords/compact-spaces","display_name":"compact spaces","score":0.4309999942779541}],"concepts":[{"id":"https://openalex.org/C105433845","wikidata":"https://www.wikidata.org/wiki/Q1194053","display_name":"Metrization theorem","level":3,"score":0.8582000136375427},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8503000140190125},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.6103000044822693},{"id":"https://openalex.org/C64543145","wikidata":"https://www.wikidata.org/wiki/Q162942","display_name":"Intersection (aeronautics)","level":2,"score":0.538100004196167},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5307000279426575},{"id":"https://openalex.org/C191399826","wikidata":"https://www.wikidata.org/wiki/Q326908","display_name":"Hausdorff space","level":2,"score":0.49720001220703125},{"id":"https://openalex.org/C198043062","wikidata":"https://www.wikidata.org/wiki/Q180953","display_name":"Metric space","level":2,"score":0.47110000252723694},{"id":"https://openalex.org/C141898687","wikidata":"https://www.wikidata.org/wiki/Q1501997","display_name":"Hausdorff distance","level":2,"score":0.4465999901294708},{"id":"https://openalex.org/C62438384","wikidata":"https://www.wikidata.org/wiki/Q331350","display_name":"Simplex","level":2,"score":0.4357999861240387},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.39800000190734863},{"id":"https://openalex.org/C90673727","wikidata":"https://www.wikidata.org/wiki/Q901718","display_name":"Product (mathematics)","level":2,"score":0.35249999165534973},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.3506999909877777},{"id":"https://openalex.org/C176217482","wikidata":"https://www.wikidata.org/wiki/Q860554","display_name":"Metric (unit)","level":2,"score":0.34369999170303345},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.329800009727478},{"id":"https://openalex.org/C24977911","wikidata":"https://www.wikidata.org/wiki/Q1491995","display_name":"Connected space","level":3,"score":0.3140999972820282},{"id":"https://openalex.org/C73225184","wikidata":"https://www.wikidata.org/wiki/Q1591095","display_name":"Hausdorff measure","level":3,"score":0.3057999908924103},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.30410000681877136},{"id":"https://openalex.org/C162392398","wikidata":"https://www.wikidata.org/wiki/Q272404","display_name":"Finite set","level":2,"score":0.29120001196861267},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.2842000126838684},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.26660001277923584},{"id":"https://openalex.org/C17327563","wikidata":"https://www.wikidata.org/wiki/Q1991405","display_name":"Nowhere dense set","level":3,"score":0.2614000141620636},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.257099986076355}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.20514","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.20514","pdf_url":"https://arxiv.org/pdf/2609.20514","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.2609.20514","doi":"https://doi.org/10.48550/arxiv.2609.20514","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.20514","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.20514","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.20514","pdf_url":"https://arxiv.org/pdf/2609.20514","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/W7213586464.pdf","grobid_xml":"https://content.openalex.org/works/W7213586464.grobid-xml"},"referenced_works_count":9,"referenced_works":["https://openalex.org/W2009712628","https://openalex.org/W2018179008","https://openalex.org/W2026671561","https://openalex.org/W2041762991","https://openalex.org/W2059490528","https://openalex.org/W2154131767","https://openalex.org/W2316681925","https://openalex.org/W7151933015","https://openalex.org/W7213258366"],"related_works":["https://openalex.org/W7213258366","https://openalex.org/W7125940397","https://openalex.org/W2964128908","https://openalex.org/W7166740929","https://openalex.org/W4407794707","https://openalex.org/W2549836596","https://openalex.org/W1524371437","https://openalex.org/W2035264382","https://openalex.org/W4396600892","https://openalex.org/W567449939"],"abstract_inverted_index":{"Let":[0],"$(X,\u03c4)$":[1,167],"be":[2],"metrizable":[3,14,114,137],"and":[4,43,115,139,164],"let":[5],"$F=\\{a_1,\\ldots,a_k\\}\\subseteq":[6],"X$,":[7],"where":[8],"$2\\le":[9],"k<\\infty$.":[10],"We":[11],"represent":[12],"all":[13],"topologies":[15],"$\u03c3\\subseteq\u03c4$":[16],"agreeing":[17],"with":[18,30,144],"$\u03c4$":[19],"on":[20],"$X\\setminus":[21],"F$":[22],"by":[23],"compatible":[24,79,85,96],"systems":[25],"of":[26,45,176],"continuous":[27],"gauges":[28],"$s_i:X\\to[0,1]$":[29],"$s_i^{-1}(0)=\\{a_i\\}$.":[31],"The":[32],"condition":[33],"$\\inf_X\\max\\{s_i,s_j\\}>0$":[34],"for":[35,134,150],"$i\\ne":[36],"j$":[37],"is":[38,73,81,95,113,116,124,156,160,168],"equivalent":[39],"to":[40],"both":[41],"Hausdorffness":[42],"metrizability":[44],"the":[46,55,76,91,99,111,154],"prescribed":[47],"gauge":[48],"topology.":[49,93],"A":[50,126],"normalized":[51],"product":[52],"map":[53],"into":[54],"standard":[56],"simplex":[57,68],"gives":[58],"an":[59,131],"explicit":[60],"metric;":[61],"its":[62],"triangle":[63],"inequality":[64],"follows":[65],"from":[66],"a":[67,103,161],"slack":[69],"inequality.":[70],"This":[71],"metric":[72,80],"complete":[74],"whenever":[75],"auxiliary":[77],"bounded":[78],"complete.":[82],"For":[83],"two":[84,100,147],"systems,":[86],"their":[87,117],"coordinatewise":[88],"minimum":[89],"describes":[90],"intersection":[92,112],"It":[94],"exactly":[97],"when":[98],"coarsenings":[101],"have":[102],"common":[104,121],"Hausdorff":[105],"lower":[106,122],"bound;":[107],"in":[108,170],"that":[109],"case":[110,172],"meet.":[118],"Otherwise":[119],"every":[120,135,140],"topology":[123],"non-Hausdorff.":[125],"closed-discrete":[127],"construction":[128],"produces":[129],"such":[130],"obstructed":[132],"pair":[133],"noncompact":[136],"space":[138],"finite":[141],"exceptional":[142,152],"set":[143],"at":[145],"least":[146],"points.":[148],"Consequently,":[149],"these":[151],"sets,":[153],"family":[155],"downward":[157],"directed,":[158],"or":[159],"lattice,":[162],"if":[163,166],"only":[165,175],"compact,":[169],"which":[171],"it":[173],"consists":[174],"$\u03c4$.":[177]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-19T00:00:00"}
