{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,15]],"date-time":"2025-12-15T06:25:39Z","timestamp":1765779939251,"version":"3.48.0"},"reference-count":34,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2025,12,15]],"date-time":"2025-12-15T00:00:00Z","timestamp":1765756800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,12,15]],"date-time":"2025-12-15T00:00:00Z","timestamp":1765756800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"The National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["12101313"],"award-info":[{"award-number":["12101313"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2026,4]]},"DOI":"10.1007\/s40314-025-03460-z","type":"journal-article","created":{"date-parts":[[2025,12,15]],"date-time":"2025-12-15T06:22:59Z","timestamp":1765779779000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Some topologies on the fuzzy real line and their pointwise metrizations"],"prefix":"10.1007","volume":"45","author":[{"given":"Zhenyu","family":"Jin","sequence":"first","affiliation":[]},{"given":"Chong","family":"Shen","sequence":"additional","affiliation":[]},{"given":"Tahair","family":"Rasham","sequence":"additional","affiliation":[]},{"given":"Fu-Gui","family":"Shi","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,12,15]]},"reference":[{"key":"3460_CR1","doi-asserted-by":"publisher","first-page":"182","DOI":"10.1016\/0022-247X(68)90057-7","volume":"24","author":"CL Chang","year":"1968","unstructured":"Chang CL (1968) Fuzzy topological spaces. J Math Anal Appl 24:182\u2013190","journal-title":"J Math Anal Appl"},{"key":"3460_CR2","first-page":"586","volume":"36","author":"P Chen","year":"2007","unstructured":"Chen P (2007) Fu-gui shi, further simplification of erceg\u2019s metric and its properties. Adv Math 36:586\u2013592 ((in Chinese))","journal-title":"Adv Math"},{"key":"3460_CR3","doi-asserted-by":"publisher","first-page":"403","DOI":"10.1016\/1385-7258(82)90034-8","volume":"85","author":"P Dwinger","year":"1982","unstructured":"Dwinger P (1982) Characterizations of the complete homeomorphic images of a completely distributive complete lattice i. Indagationes Math 85:403\u2013414","journal-title":"Indagationes Math"},{"key":"3460_CR4","series-title":"Fuzzy Number","volume-title":"First course on fuzzy theory and applications. Advances in Soft Computing","author":"H Lee Kwang","year":"2005","unstructured":"Lee Kwang H (2005) First course on fuzzy theory and applications. Advances in Soft Computing, vol 27. Fuzzy Number. Springer, Berlin, Heidelberg"},{"key":"3460_CR5","volume-title":"General topology","author":"R Engelking","year":"1977","unstructured":"Engelking R (1977) General topology. Polish Scientific Publisher, Warsaw"},{"key":"3460_CR6","doi-asserted-by":"publisher","first-page":"205","DOI":"10.1016\/0022-247X(79)90189-6","volume":"69","author":"MA Erceg","year":"1979","unstructured":"Erceg MA (1979) Metric in fuzzy set theory. J Math Anal Appl 69:205\u2013230","journal-title":"J Math Anal Appl"},{"key":"3460_CR7","doi-asserted-by":"publisher","first-page":"547","DOI":"10.1016\/0022-247X(78)90148-8","volume":"62","author":"TE Gantner","year":"1978","unstructured":"Gantner TE, Steinlage RC, Warren RH (1978) Compactness in fuzzy topological spaces. J Math Anal Appl 62:547\u2013562","journal-title":"J Math Anal Appl"},{"key":"3460_CR8","doi-asserted-by":"crossref","unstructured":"Gierz G, Hofmann KH, Keimel K, Lawson JD, Mislove MW, Scott DS (2003) Continuous lattices and Domains, vol 93. Cambridge University Press, Encyclopedia of Mathematics and its Applications","DOI":"10.1017\/CBO9780511542725"},{"key":"3460_CR9","doi-asserted-by":"publisher","first-page":"395","DOI":"10.1016\/0165-0114(94)90162-7","volume":"64","author":"A George","year":"1994","unstructured":"George A, Veeramani P (1994) On some results in fuzzy metric spaces. Fuzzy Sets Syst 64:395\u2013399","journal-title":"Fuzzy Sets Syst"},{"key":"3460_CR10","first-page":"933","volume":"3","author":"A George","year":"1995","unstructured":"George A, Veeramani P (1995) Some theorems in fuzzy metric spaces. J Fuzzy Math 3:933\u2013940","journal-title":"J Fuzzy Math"},{"key":"3460_CR11","doi-asserted-by":"publisher","first-page":"145","DOI":"10.1016\/0022-247X(67)90189-8","volume":"18","author":"JA Goguen","year":"1967","unstructured":"Goguen JA (1967) $$l$$-fuzzy sets. J Math Anal Appl 18:145\u2013174","journal-title":"J Math Anal Appl"},{"key":"3460_CR12","doi-asserted-by":"publisher","first-page":"737","DOI":"10.1016\/0022-247X(73)90288-6","volume":"43","author":"JA Goguen","year":"1973","unstructured":"Goguen JA (1973) The fuzzy tychonoff theorem. J Math Anal Appl 43:737\u2013742","journal-title":"J Math Anal Appl"},{"key":"3460_CR13","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781139524438","volume-title":"Non-Hausdorff topology and domain theory","author":"J Goubault-Larrecq","year":"2013","unstructured":"Goubault-Larrecq J (2013) Non-Hausdorff topology and domain theory. Cambridge University Press, Cambridge"},{"key":"3460_CR14","doi-asserted-by":"publisher","first-page":"1141","DOI":"10.1016\/j.ins.2007.10.001","volume":"178","author":"H-L Huang","year":"2008","unstructured":"Huang H-L (2008) Fu-gui shi, $$l$$-fuzzy numbers and their properties. Inf Sci 178:1141\u20131151","journal-title":"Inf Sci"},{"key":"3460_CR15","doi-asserted-by":"publisher","first-page":"74","DOI":"10.1016\/0022-247X(75)90039-6","volume":"50","author":"B Hutton","year":"1975","unstructured":"Hutton B (1975) Normality in fuzzy topological spaces. J Math Anal Appl 50:74\u201379","journal-title":"J Math Anal Appl"},{"key":"3460_CR16","doi-asserted-by":"publisher","first-page":"93","DOI":"10.1016\/0165-0114(80)90007-X","volume":"3","author":"B Hutton","year":"1980","unstructured":"Hutton B, Reilly I (1980) Separation axioms in fuzzy topological spaces. Fuzzy Sets Syst 3:93\u2013104","journal-title":"Fuzzy Sets Syst"},{"key":"3460_CR17","first-page":"336","volume":"11","author":"I Kramoosil","year":"1975","unstructured":"Kramoosil I, Michalek J (1975) Fuzzy metric and statistical metric spaces. Kybernetika 11:336\u2013344","journal-title":"Kybernetika"},{"key":"3460_CR18","first-page":"733","volume":"30","author":"J-H Liang","year":"1987","unstructured":"Liang J-H (1987) Pointwise characterizations of fuzzy metrics and its applications. Acta Math Sinica 30:733\u2013741 ((in Chinese))","journal-title":"Acta Math Sinica"},{"key":"3460_CR19","doi-asserted-by":"crossref","unstructured":"Liang J (2001) A note on pointed metric on fuzzy lattices. Sichuan University (Natural Science Edition) 38:483\u2013486 ((in Chinese))","DOI":"10.1163\/156853388X00300"},{"key":"3460_CR20","volume-title":"Fuzzy Topology","author":"Y-M Liu","year":"1997","unstructured":"Liu Y-M, Luo M-K (1997) Fuzzy Topology. World Scientific, Singapore"},{"key":"3460_CR21","doi-asserted-by":"publisher","first-page":"351","DOI":"10.1016\/0165-0114(88)90061-9","volume":"27","author":"NN Morsi","year":"1988","unstructured":"Morsi NN (1988) On fuzzt pseudo-normed vector spaces. Fuzzy Sets Syst 27:351\u2013372","journal-title":"Fuzzy Sets Syst"},{"key":"3460_CR22","first-page":"353","volume":"13","author":"Y-W Peng","year":"1992","unstructured":"Peng Y-W (1992) Pointwise p.q.-metric and the family of induced maps on completely distributive latices. Ann Math Ser 13:353\u2013359 ((in Chinese))","journal-title":"Ann Math Ser"},{"key":"3460_CR23","doi-asserted-by":"publisher","first-page":"518","DOI":"10.1090\/S0002-9939-1953-0058568-4","volume":"4","author":"GN Raney","year":"1953","unstructured":"Raney GN (1953) A subdirect-union representation for completely distributive complete lattices. Proc Am Math Soc 4:518\u2013522","journal-title":"Proc Am Math Soc"},{"key":"3460_CR24","doi-asserted-by":"publisher","DOI":"10.1016\/j.topol.2022.108133","volume":"314","author":"C Shen","year":"2022","unstructured":"Shen C (2022) Fu-gui shi, on quasi-metrizable $$d$$-spaces. Topol Appl 314:108133","journal-title":"Topol Appl"},{"key":"3460_CR25","doi-asserted-by":"publisher","first-page":"1212","DOI":"10.1109\/TFUZZ.2021.3054466","volume":"30","author":"C Shen","year":"2022","unstructured":"Shen C, Shi Y, Shi FG, Andradi H (2022) Characterizations of pointwise pseudo-metrics via Pointwise Closed-Ball Systems. IEEE Trans Fuzzy Syst 30:1212\u20131223","journal-title":"IEEE Trans Fuzzy Syst"},{"key":"3460_CR26","doi-asserted-by":"publisher","first-page":"718","DOI":"10.1007\/BF03186961","volume":"42","author":"F-G Shi","year":"1997","unstructured":"Shi F-G (1997) Pointwise uniformities and pointwise metrics on fuzzy lattices. Chin Sci Bull 42:718\u2013720","journal-title":"Chin Sci Bull"},{"key":"3460_CR27","doi-asserted-by":"publisher","first-page":"209","DOI":"10.1016\/S0165-0114(00)00013-0","volume":"121","author":"F-G Shi","year":"2001","unstructured":"Shi F-G (2001) Pointwise pseudo-metrics in $$l$$-fuzzy set theory. Fuzzy Sets Syst 121:209\u2013216","journal-title":"Fuzzy Sets Syst"},{"key":"3460_CR28","first-page":"217","volume":"9","author":"F-G Shi","year":"2001","unstructured":"Shi F-G (2001) Product operations of fuzzy pointwise metric spaces. J Fuzzy Math 9:217\u2013222","journal-title":"J Fuzzy Math"},{"key":"3460_CR29","doi-asserted-by":"publisher","first-page":"1218","DOI":"10.1016\/j.fss.2008.09.002","volume":"160","author":"F-G Shi","year":"2009","unstructured":"Shi F-G (2009) $$l$$-fuzzy interiors and $$l$$-fuzzy closures. Fuzzy Sets Syst 160:1218\u20131232","journal-title":"Fuzzy Sets Syst"},{"key":"3460_CR30","doi-asserted-by":"publisher","first-page":"95","DOI":"10.1016\/j.fss.2020.03.015","volume":"399","author":"F-G Shi","year":"2020","unstructured":"Shi F-G (2020) $$l$$-metric on the space of $$l$$-fuzzy numbers. Fuzzy Sets Syst 399:95\u2013109","journal-title":"Fuzzy Sets Syst"},{"key":"3460_CR31","first-page":"15","volume":"2","author":"F-G Shi","year":"2005","unstructured":"Shi F-G (2005) Pointwise pseudo-metric on the $$l$$-real line. Iran J Fuzzy Syst 2:15\u201320","journal-title":"Iran J Fuzzy Syst"},{"key":"3460_CR32","unstructured":"Wang G-J (1988) Theory of $$L$$-fuzzy topological spaces. Shanxi Normal Univ. Publ, House ((In Chinese))"},{"key":"3460_CR33","doi-asserted-by":"publisher","DOI":"10.1016\/j.fmre.2025.02.003","author":"H Wang","year":"2025","unstructured":"Wang H, Shen C, Jin Z-Y (2025) The separation and compactness on fuzzy partial metric spaces. Fund Res Pre-publ. https:\/\/doi.org\/10.1016\/j.fmre.2025.02.003","journal-title":"Fund Res Pre-publ"},{"key":"3460_CR34","doi-asserted-by":"publisher","first-page":"1267","DOI":"10.1016\/j.fss.2008.06.005","volume":"160","author":"J Xiao","year":"2009","unstructured":"Xiao J, Zhu X (2009) Metric topology on the $$l$$-fuzzy real line. Fuzzy Sets Syst 160:1267\u20131276","journal-title":"Fuzzy Sets Syst"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-025-03460-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-025-03460-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-025-03460-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,12,15]],"date-time":"2025-12-15T06:23:02Z","timestamp":1765779782000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-025-03460-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,12,15]]},"references-count":34,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2026,4]]}},"alternative-id":["3460"],"URL":"https:\/\/doi.org\/10.1007\/s40314-025-03460-z","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"value":"2238-3603","type":"print"},{"value":"1807-0302","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,12,15]]},"assertion":[{"value":"14 July 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 September 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 September 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 December 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no Conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"Not applicable.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Data and code availability"}},{"value":"Not applicable.","order":4,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical approval"}}],"article-number":"104"}}