{"id":"https://openalex.org/W7213872016","doi":"https://doi.org/10.20944/preprints202608.2228.v3","title":"\u03c3-Hereditarily Closure-Preserving I-sn-Networks, I-sn-Metrizability, and Local HCP-Finiteness","display_name":"\u03c3-Hereditarily Closure-Preserving I-sn-Networks, I-sn-Metrizability, and Local HCP-Finiteness","publication_year":2026,"publication_date":"2026-09-21","ids":{"openalex":"https://openalex.org/W7213872016","doi":"https://doi.org/10.20944/preprints202608.2228.v3"},"language":"en","primary_location":{"id":"doi:10.20944/preprints202608.2228.v3","doi":"https://doi.org/10.20944/preprints202608.2228.v3","is_oa":true,"landing_page_url":"https://doi.org/10.20944/preprints202608.2228.v3","pdf_url":null,"source":{"id":"https://openalex.org/S6309402219","display_name":"Preprints.org","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":null,"host_organization_name":null,"host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"posted-content"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.20944/preprints202608.2228.v3","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5047506639","display_name":"Xing-Yu Hu","orcid":"https://orcid.org/0009-0007-9953-9362","observed_orcids":["https://orcid.org/0009-0007-9953-9362","https://orcid.org/0009-0004-4351-413X"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Xing-Yu Hu","raw_affiliation_strings":[],"raw_orcid":"https://orcid.org/0009-0007-9953-9362","affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5047506639"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"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":0.9911999702453613,"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":0.9911999702453613,"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/T12062","display_name":"Approximation Theory and Sequence Spaces","score":0.00139999995008111,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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":0.0010000000474974513,"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/admissible-ideals","display_name":"admissible ideals","score":0.656000018119812},{"id":"https://openalex.org/keywords/free-ultrafilters","display_name":"free ultrafilters","score":0.9660000205039978},{"id":"https://openalex.org/keywords/regular-spaces","display_name":"regular spaces","score":0.45899999141693115}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8744000196456909},{"id":"https://openalex.org/C2776639384","wikidata":"https://www.wikidata.org/wiki/Q840396","display_name":"Ideal (ethics)","level":2,"score":0.7576000094413757},{"id":"https://openalex.org/C110729354","wikidata":"https://www.wikidata.org/wiki/Q185478","display_name":"Countable set","level":2,"score":0.6805999875068665},{"id":"https://openalex.org/C2780841128","wikidata":"https://www.wikidata.org/wiki/Q5073781","display_name":"Characterization (materials science)","level":2,"score":0.5388000011444092},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.492000013589859},{"id":"https://openalex.org/C55526981","wikidata":"https://www.wikidata.org/wiki/Q1149022","display_name":"Fubini's theorem","level":2,"score":0.4325000047683716},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.43220001459121704},{"id":"https://openalex.org/C136134351","wikidata":"https://www.wikidata.org/wiki/Q431937","display_name":"Antichain","level":3,"score":0.42809998989105225},{"id":"https://openalex.org/C31316033","wikidata":"https://www.wikidata.org/wiki/Q1207972","display_name":"Polish space","level":3,"score":0.42750000953674316},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.3953999876976013},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.36910000443458557},{"id":"https://openalex.org/C117354338","wikidata":"https://www.wikidata.org/wiki/Q1165112","display_name":"Singleton","level":3,"score":0.3465999960899353},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.30730000138282776},{"id":"https://openalex.org/C118211362","wikidata":"https://www.wikidata.org/wiki/Q1203540","display_name":"Maximal ideal","level":3,"score":0.29260000586509705},{"id":"https://openalex.org/C162392398","wikidata":"https://www.wikidata.org/wiki/Q272404","display_name":"Finite set","level":2,"score":0.2897999882698059},{"id":"https://openalex.org/C150591576","wikidata":"https://www.wikidata.org/wiki/Q739925","display_name":"Family of sets","level":3,"score":0.2892000079154968},{"id":"https://openalex.org/C2779467367","wikidata":"https://www.wikidata.org/wiki/Q863912","display_name":"Prime ideal","level":3,"score":0.2863999903202057},{"id":"https://openalex.org/C136979486","wikidata":"https://www.wikidata.org/wiki/Q773483","display_name":"Existential quantification","level":2,"score":0.27799999713897705},{"id":"https://openalex.org/C46333567","wikidata":"https://www.wikidata.org/wiki/Q2138875","display_name":"Regular local ring","level":4,"score":0.2639999985694885}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.20944/preprints202608.2228.v3","doi":"https://doi.org/10.20944/preprints202608.2228.v3","is_oa":true,"landing_page_url":"https://doi.org/10.20944/preprints202608.2228.v3","pdf_url":null,"source":{"id":"https://openalex.org/S6309402219","display_name":"Preprints.org","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":null,"host_organization_name":null,"host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"posted-content"}],"best_oa_location":{"id":"doi:10.20944/preprints202608.2228.v3","doi":"https://doi.org/10.20944/preprints202608.2228.v3","is_oa":true,"landing_page_url":"https://doi.org/10.20944/preprints202608.2228.v3","pdf_url":null,"source":{"id":"https://openalex.org/S6309402219","display_name":"Preprints.org","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":null,"host_organization_name":null,"host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"posted-content"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W7214596439","https://openalex.org/W7213233210","https://openalex.org/W7204838729","https://openalex.org/W2337977798","https://openalex.org/W2355096979","https://openalex.org/W4319993490","https://openalex.org/W2062295640","https://openalex.org/W1975002882","https://openalex.org/W2356224369","https://openalex.org/W2391579865"],"abstract_inverted_index":{"Let":[0],"\\(\\mathcal{I}\\)":[1,143],"be":[2,128],"an":[3,89,140],"admissible":[4,27,141,155],"ideal":[5,142,159,201,210,215],"on":[6,32],"\\(\\mathbb{N}\\).":[7],"Zhou--Liu--Liu--Lin":[8],"asked":[9],"whether":[10],"every":[11,26,49,68,93],"regular":[12,103],"space":[13,43,62,94,104],"with":[14,95,105,178],"a":[15,33,42,60,96,102,106],"\\(\\sigma\\)-hereditarily":[16],"closure-preserving":[17],"\\(\\mathcal{I}\\)-\\(sn\\)-network":[18,98],"is":[19,23,57,87,99,151],"\\(\\mathcal{I}\\)-\\(sn\\)-metrizable.":[20],"The":[21,29,122],"answer":[22],"affirmative":[24],"for":[25,144,154,172,191],"ideal.":[28],"proof":[30],"rests":[31],"stronger":[34],"local":[35,149,161,189],"countability":[36,123],"lemma.":[37],"If":[38,71],"\\(y_n\\to_{\\mathcal{I}}":[39],"x\\)":[40],"in":[41,65,126],"\\(X\\)":[44],"and":[45,113,182,203,212],"\\(\\{n:y_n\\ne":[46],"x\\}\\in\\mathcal{I}^{+}\\),":[47],"then":[48,83,118],"point-discrete":[50],"family":[51],"of":[52,55,111,116,136,196,218],"\\(\\mathcal{I}\\)-sequential":[53,90],"neighborhoods":[54],"\\(x\\)":[56,76],"countable.":[58],"In":[59],"\\(T_1\\)":[61],"this":[63,164],"applies":[64],"particular":[66],"to":[67,75,130],"HCP":[69],"family.":[70],"no":[72],"\\(\\mathcal{I}\\)-convergent":[73],"sequence":[74],"has":[77,160],"\\(\\mathcal{I}\\)-positive":[78],"support":[79],"away":[80],"from":[81],"\\(x\\),":[82],"the":[84,167,192,207,213,219],"singleton":[85],"\\(\\{x\\}\\)":[86],"itself":[88],"neighborhood.":[91],"Consequently":[92],"\\(\\sigma\\)-point-discrete":[97],"\\(\\mathcal{I}\\)-\\(snf\\)-countable.":[100],"For":[101],"\\(\\sigma\\)-HCP":[107],"\\(\\mathcal{I}\\)-\\(sn\\)-network,":[108],"Ge's":[109],"characterization":[110,115],"\\(sn\\)-metrizability":[112],"Zhou--Liu--Liu--Lin's":[114],"\\(\\mathcal{I}\\)-\\(sn\\)-metrizability":[117],"settle":[119],"Problem~5.2":[120],"affirmatively.":[121],"bound":[124],"cannot":[125],"general":[127],"strengthened":[129],"finiteness.":[131],"A":[132],"countable":[133],"Rudin\u2013Keisler":[134],"antichain":[135],"free":[137],"ultrafilters":[138],"gives":[139],"which":[145],"\\(\\mathrm{HCF}(\\mathcal{I})\\)":[146],"fails.":[147],"Thus":[148],"HCP-finiteness":[150,190],"not":[152],"automatic":[153],"ideals.":[156],"Every":[157],"\\(\\mathfrak{p}^{-}\\)":[158],"HCP-finiteness.":[162],"Beyond":[163],"basic":[165],"class,":[166],"paper":[168],"develops":[169],"permanence":[170],"results":[171,187],"Fubini":[173,176,194],"products,":[174],"heterogeneous":[175],"sums":[177],"cross-row":[179],"Kat\u02c7etov":[180],"absorption,":[181],"suitable":[183],"increasing":[184],"unions.":[185],"These":[186],"establish":[188],"finite":[193,205],"powers":[195],"\\(\\mathrm{Fin}\\),":[197],"Kat\u02c7etov's":[198],"first":[199],"limit-stage":[200],"\\(\\mathrm{Fin}^{\\omega}\\)":[202],"its":[204],"successors,":[206],"tree-derived":[208],"rank-\\(\\omega\\)":[209],"\\(\\mathcal{H}_{<\\omega}\\),":[211],"inductive-limit":[214],"\\(\\mathrm{Fin}_{\\omega}\\).":[216],"All":[217],"listed":[220],"examples":[221],"except":[222],"\\(\\mathrm{Fin}\\)":[223],"fail":[224],"\\(\\mathfrak{p}^{-}\\).":[225]},"counts_by_year":[],"updated_date":"2026-11-09T13:22:03.289127","created_date":"2026-09-22T00:00:00"}
