{"id":"https://openalex.org/W4290465732","doi":"https://doi.org/10.3390/axioms11080383","title":"Towards Strong Convergence and Cauchy Sequences in Binary Metric Spaces","display_name":"Towards Strong Convergence and Cauchy Sequences in Binary Metric Spaces","publication_year":2022,"publication_date":"2022-08-05","ids":{"openalex":"https://openalex.org/W4290465732","doi":"https://doi.org/10.3390/axioms11080383"},"language":"en","primary_location":{"id":"doi:10.3390/axioms11080383","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms11080383","pdf_url":"https://www.mdpi.com/2075-1680/11/8/383/pdf?version=1659687200","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2075-1680/11/8/383/pdf?version=1659687200","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5023154747","display_name":"Shubham Yadav","orcid":"https://orcid.org/0000-0002-3906-5739"},"institutions":[{"id":"https://openalex.org/I42014448","display_name":"Sardar Vallabhbhai National Institute of Technology Surat","ror":"https://ror.org/02y394t43","country_code":"IN","type":"education","lineage":["https://openalex.org/I42014448"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Shubham Yadav","raw_affiliation_strings":["Department of Applied Mathematics and Humanities, S. V. National Institute of Technology Surat, Surat 395007, India"],"raw_orcid":"https://orcid.org/0000-0002-3906-5739","affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics and Humanities, S. V. National Institute of Technology Surat, Surat 395007, India","institution_ids":["https://openalex.org/I42014448"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5045925927","display_name":"Dhananjay Gopal","orcid":"https://orcid.org/0000-0001-8217-2778"},"institutions":[{"id":"https://openalex.org/I26285277","display_name":"Guru Ghasidas Vishwavidyalaya","ror":"https://ror.org/05bvxq496","country_code":"IN","type":"education","lineage":["https://openalex.org/I26285277"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Dhananjay Gopal","raw_affiliation_strings":["Department of Mathematics, Guru Ghasidas Vishwavidyalaya (A Central University), Bilaspur 495009, India"],"raw_orcid":"https://orcid.org/0000-0001-8217-2778","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Guru Ghasidas Vishwavidyalaya (A Central University), Bilaspur 495009, India","institution_ids":["https://openalex.org/I26285277"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5018646265","display_name":"Parin Chaipunya","orcid":"https://orcid.org/0000-0002-0207-6098"},"institutions":[{"id":"https://openalex.org/I60837268","display_name":"King Mongkut's University of Technology Thonburi","ror":"https://ror.org/0057ax056","country_code":"TH","type":"education","lineage":["https://openalex.org/I60837268"]}],"countries":["TH"],"is_corresponding":true,"raw_author_name":"Parin Chaipunya","raw_affiliation_strings":["Department of Mathematics, King Mongkut\u2019s University of Technology Thonburi, Bangkok 10140, Thailand","Department of Mathematics, King Mongkut's University of Technology Thonburi, Bangkok 10140, Thailand"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, King Mongkut\u2019s University of Technology Thonburi, Bangkok 10140, Thailand","institution_ids":["https://openalex.org/I60837268"]},{"raw_affiliation_string":"Department of Mathematics, King Mongkut's University of Technology Thonburi, Bangkok 10140, Thailand","institution_ids":["https://openalex.org/I60837268"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5020498529","display_name":"Juan Mart\u00ednez\u2010Moreno","orcid":"https://orcid.org/0000-0002-3340-2781"},"institutions":[{"id":"https://openalex.org/I191420491","display_name":"Universidad de Ja\u00e9n","ror":"https://ror.org/0122p5f64","country_code":"ES","type":"education","lineage":["https://openalex.org/I191420491"]}],"countries":["ES"],"is_corresponding":false,"raw_author_name":"Juan Mart\u00ednez-Moreno","raw_affiliation_strings":["Department of Mathematics, University of Ja\u00e9n Campus Las Lagunillas s/n, 23071 Ja\u00e9n, Spain"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Ja\u00e9n Campus Las Lagunillas s/n, 23071 Ja\u00e9n, Spain","institution_ids":["https://openalex.org/I191420491"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5018646265"],"corresponding_institution_ids":["https://openalex.org/I60837268"],"apc_list":{"value":1400,"currency":"CHF","value_usd":1515},"apc_paid":{"value":1400,"currency":"CHF","value_usd":1515},"fwci":0.8025,"has_fulltext":true,"cited_by_count":2,"citation_normalized_percentile":{"value":0.69389619,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"11","issue":"8","first_page":"383","last_page":"383"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11777","display_name":"Fixed Point Theorems Analysis","score":0.9998000264167786,"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/T11777","display_name":"Fixed Point Theorems Analysis","score":0.9998000264167786,"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":0.9926999807357788,"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/T12062","display_name":"Approximation Theory and Sequence Spaces","score":0.989300012588501,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"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/mathematics","display_name":"Mathematics","score":0.7749940752983093},{"id":"https://openalex.org/keywords/cauchy-sequence","display_name":"Cauchy sequence","score":0.6177761554718018},{"id":"https://openalex.org/keywords/metric-space","display_name":"Metric space","score":0.6167148947715759},{"id":"https://openalex.org/keywords/closed-set","display_name":"Closed set","score":0.4943099021911621},{"id":"https://openalex.org/keywords/topology","display_name":"Topology (electrical circuits)","score":0.48073267936706543},{"id":"https://openalex.org/keywords/binary-number","display_name":"Binary number","score":0.47555893659591675},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.47461429238319397},{"id":"https://openalex.org/keywords/complete-metric-space","display_name":"Complete metric space","score":0.45452845096588135},{"id":"https://openalex.org/keywords/compact-convergence","display_name":"Compact convergence","score":0.43600189685821533},{"id":"https://openalex.org/keywords/metric","display_name":"Metric (unit)","score":0.4340941607952118},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.412702351808548},{"id":"https://openalex.org/keywords/distributive-property","display_name":"Distributive property","score":0.4102635681629181},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.3110431432723999},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.28135085105895996},{"id":"https://openalex.org/keywords/rate-of-convergence","display_name":"Rate of convergence","score":0.14225968718528748},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.11733239889144897}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7749940752983093},{"id":"https://openalex.org/C203192437","wikidata":"https://www.wikidata.org/wiki/Q217847","display_name":"Cauchy sequence","level":2,"score":0.6177761554718018},{"id":"https://openalex.org/C198043062","wikidata":"https://www.wikidata.org/wiki/Q180953","display_name":"Metric space","level":2,"score":0.6167148947715759},{"id":"https://openalex.org/C164953516","wikidata":"https://www.wikidata.org/wiki/Q320357","display_name":"Closed set","level":2,"score":0.4943099021911621},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.48073267936706543},{"id":"https://openalex.org/C48372109","wikidata":"https://www.wikidata.org/wiki/Q3913","display_name":"Binary number","level":2,"score":0.47555893659591675},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.47461429238319397},{"id":"https://openalex.org/C93361540","wikidata":"https://www.wikidata.org/wiki/Q848569","display_name":"Complete metric space","level":3,"score":0.45452845096588135},{"id":"https://openalex.org/C5871375","wikidata":"https://www.wikidata.org/wiki/Q1780715","display_name":"Compact convergence","level":4,"score":0.43600189685821533},{"id":"https://openalex.org/C176217482","wikidata":"https://www.wikidata.org/wiki/Q860554","display_name":"Metric (unit)","level":2,"score":0.4340941607952118},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.412702351808548},{"id":"https://openalex.org/C11821877","wikidata":"https://www.wikidata.org/wiki/Q187959","display_name":"Distributive property","level":2,"score":0.4102635681629181},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3110431432723999},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.28135085105895996},{"id":"https://openalex.org/C57869625","wikidata":"https://www.wikidata.org/wiki/Q1783502","display_name":"Rate of convergence","level":3,"score":0.14225968718528748},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.11733239889144897},{"id":"https://openalex.org/C127162648","wikidata":"https://www.wikidata.org/wiki/Q16858953","display_name":"Channel (broadcasting)","level":2,"score":0.0},{"id":"https://openalex.org/C21547014","wikidata":"https://www.wikidata.org/wiki/Q1423657","display_name":"Operations management","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"score":0.0},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.0},{"id":"https://openalex.org/C31258907","wikidata":"https://www.wikidata.org/wiki/Q1301371","display_name":"Computer network","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/axioms11080383","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms11080383","pdf_url":"https://www.mdpi.com/2075-1680/11/8/383/pdf?version=1659687200","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:6c77134037be4ec481983771906b0aea","is_oa":true,"landing_page_url":"https://doaj.org/article/6c77134037be4ec481983771906b0aea","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":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Axioms, Vol 11, Iss 8, p 383 (2022)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2075-1680/11/8/383/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/axioms11080383","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Axioms; Volume 11; Issue 8; Pages: 383","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/axioms11080383","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms11080383","pdf_url":"https://www.mdpi.com/2075-1680/11/8/383/pdf?version=1659687200","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1308479839","display_name":null,"funder_award_id":"FRB650048/0164","funder_id":"https://openalex.org/F4320329659","funder_display_name":"Thailand Science Research and Innovation"},{"id":"https://openalex.org/G1669993805","display_name":null,"funder_award_id":"FRB650048/0164","funder_id":"https://openalex.org/F4320322818","funder_display_name":"King Mongkut's University of Technology Thonburi"},{"id":"https://openalex.org/G1767018141","display_name":null,"funder_award_id":"FRB650048","funder_id":"https://openalex.org/F4320329659","funder_display_name":"Thailand Science Research and Innovation"},{"id":"https://openalex.org/G3202123143","display_name":null,"funder_award_id":"FRB650048","funder_id":"https://openalex.org/F4320322818","funder_display_name":"King Mongkut's University of Technology Thonburi"}],"funders":[{"id":"https://openalex.org/F4320322818","display_name":"King Mongkut's University of Technology Thonburi","ror":"https://ror.org/0057ax056"},{"id":"https://openalex.org/F4320329659","display_name":"Thailand Science Research and Innovation","ror":null}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4290465732.pdf","grobid_xml":"https://content.openalex.org/works/W4290465732.grobid-xml"},"referenced_works_count":5,"referenced_works":["https://openalex.org/W1973191593","https://openalex.org/W2046423866","https://openalex.org/W2047277075","https://openalex.org/W2072198207","https://openalex.org/W4235411478"],"related_works":["https://openalex.org/W3021341953","https://openalex.org/W4317914668","https://openalex.org/W1539964918","https://openalex.org/W3103277056","https://openalex.org/W4291017225","https://openalex.org/W3117679268","https://openalex.org/W1988206529","https://openalex.org/W343637954","https://openalex.org/W2771922180","https://openalex.org/W2804034501"],"abstract_inverted_index":{"A":[0],"Kuratowski":[1,40],"topology":[2,5,41],"is":[3,81],"a":[4,23,32,47,59,62,87,92,110],"specified":[6],"in":[7,61],"terms":[8],"of":[9,28,58,104],"closed":[10],"sets":[11],"rather":[12],"than":[13],"open":[14],"sets.":[15],"Recently,":[16],"the":[17,55,75,101],"binary":[18,48,63,95,106],"metric":[19,64,96,107],"was":[20],"introduced":[21],"as":[22],"symmetric,":[24],"distributive-lattice-ordered":[25],"magma-valued":[26],"function":[27],"two":[29],"variables":[30],"satisfying":[31],"\u201ctriangle":[33],"inequality\u201d":[34],"and":[35,66,91],"subsequently":[36],"proved":[37],"that":[38,68],"every":[39],"can":[42],"be":[43],"induced":[44],"by":[45],"such":[46],"metric.":[49],"In":[50],"this":[51],"paper,":[52],"we":[53,99],"define":[54,86],"strong":[56,69,79,93,102],"convergence":[57,70,80],"sequence":[60,90],"space":[65],"prove":[67],"implies":[71],"convergence.":[72,84],"We":[73,85],"state":[74],"conditions":[76],"under":[77],"which":[78],"equivalent":[82],"to":[83],"strongly":[88],"Cauchy":[89],"complete":[94],"space.":[97],"Finally,":[98],"give":[100],"completion":[103],"all":[105],"spaces":[108],"with":[109],"countable":[111],"indexing":[112],"set.":[113]},"counts_by_year":[{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":1}],"updated_date":"2026-05-06T08:25:59.206177","created_date":"2025-10-10T00:00:00"}