{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,2]],"date-time":"2025-11-02T05:42:52Z","timestamp":1762062172404,"version":"build-2065373602"},"reference-count":7,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2022,8,5]],"date-time":"2022-08-05T00:00:00Z","timestamp":1659657600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Thailand Science Research and Innovation","award":["FRB650048\/0164"],"award-info":[{"award-number":["FRB650048\/0164"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>A Kuratowski topology is a topology specified in terms of closed sets rather than open sets. Recently, the binary metric was introduced as a symmetric, distributive-lattice-ordered magma-valued function of two variables satisfying a \u201ctriangle inequality\u201d and subsequently proved that every Kuratowski topology can be induced by such a binary metric. In this paper, we define the strong convergence of a sequence in a binary metric space and prove that strong convergence implies convergence. We state the conditions under which strong convergence is equivalent to convergence. We define a strongly Cauchy sequence and a strong complete binary metric space. Finally, we give the strong completion of all binary metric spaces with a countable indexing set.<\/jats:p>","DOI":"10.3390\/axioms11080383","type":"journal-article","created":{"date-parts":[[2022,8,7]],"date-time":"2022-08-07T21:03:50Z","timestamp":1659906230000},"page":"383","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Towards Strong Convergence and Cauchy Sequences in Binary Metric Spaces"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3906-5739","authenticated-orcid":false,"given":"Shubham","family":"Yadav","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics and Humanities, S. V. National Institute of Technology Surat, Surat 395007, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8217-2778","authenticated-orcid":false,"given":"Dhananjay","family":"Gopal","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Guru Ghasidas Vishwavidyalaya (A Central University), Bilaspur 495009, India"}]},{"given":"Parin","family":"Chaipunya","sequence":"additional","affiliation":[{"name":"Department of Mathematics, King Mongkut\u2019s University of Technology Thonburi, Bangkok 10140, Thailand"}]},{"given":"Juan","family":"Mart\u00ednez-Moreno","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Ja\u00e9n Campus Las Lagunillas s\/n, 23071 Ja\u00e9n, Spain"}]}],"member":"1968","published-online":{"date-parts":[[2022,8,5]]},"reference":[{"key":"ref_1","first-page":"107","article-title":"Binary metrics","volume":"274","author":"Samer","year":"2020","journal-title":"Topol. Appl."},{"key":"ref_2","first-page":"414","article-title":"Every finite topology is generated by a partial pseudometric","volume":"22","author":"Richmond","year":"2005","journal-title":"Order"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1080\/00029890.1988.11971974","article-title":"All topologies come from generalized metrics","volume":"95","author":"Kopperman","year":"1988","journal-title":"Am. Math. Mon."},{"key":"ref_4","first-page":"331","article-title":"Metric representations of categories of closure spaces","volume":"37","author":"Kopperman","year":"2011","journal-title":"Topol. Proc."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1007\/s00012-011-0132-5","article-title":"Topologies arising from metrics valued in abelian \u2113-groups","volume":"65","author":"Kopperman","year":"2011","journal-title":"Algebra Universalis"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1016\/j.topol.2014.12.013","article-title":"Completions of partial metric spaces","volume":"182","author":"Ge","year":"2015","journal-title":"Topol. Appl."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Matthews, S.G. (1994). Partial Metric Topology, The New York Academy of Sciences. Annals of the New York Academy of Sciences-Paper Edition.","DOI":"10.1111\/j.1749-6632.1994.tb44144.x"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/8\/383\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:04:30Z","timestamp":1760141070000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/8\/383"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,8,5]]},"references-count":7,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2022,8]]}},"alternative-id":["axioms11080383"],"URL":"https:\/\/doi.org\/10.3390\/axioms11080383","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2022,8,5]]}}}