{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:56:38Z","timestamp":1760144198731,"version":"build-2065373602"},"reference-count":16,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2024,3,15]],"date-time":"2024-03-15T00:00:00Z","timestamp":1710460800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Wonkwang University"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Based on equivalence relation R on X, equivalence class [x] of a point and equivalence class [A] of a subset represent the neighborhoods of x and A, respectively. These neighborhoods play the main role in defining separation axioms, metric spaces, proximity relations and uniformity structures on an approximation space (X,R) depending on the lower approximation and the upper approximation of rough sets. The properties and the possible implications of these definitions are studied. The generated approximation topology \u03c4R on X is equivalent to the generated topologies associated with metric d, proximity \u03b4 and uniformity U on X. Separated metric spaces, separated proximity spaces and separated uniform spaces are defined and it is proven that both are associating exactly discrete topology \u03c4R on X.<\/jats:p>","DOI":"10.3390\/axioms13030199","type":"journal-article","created":{"date-parts":[[2024,3,15]],"date-time":"2024-03-15T12:02:39Z","timestamp":1710504159000},"page":"199","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On Proximity Spaces Constructed on Rough Sets"],"prefix":"10.3390","volume":"13","author":[{"given":"Jong Il","family":"Baek","sequence":"first","affiliation":[{"name":"School of Big Data, Financial Statistics, Wonkwang University, Iksan-Daero, Iksan-si 570749, Republic of Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"S. E.","family":"Abbas","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Sohag University, Sohag 82524, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kul","family":"Hur","sequence":"additional","affiliation":[{"name":"Division of Applied Mathematics, Wonkwang University, Iksan-Daero, Iksan-si 570749, Republic of Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9010-0237","authenticated-orcid":false,"given":"Ismail","family":"Ibedou","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Benha University, Benha 13518, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,3,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1007\/BF01001956","article-title":"Rough Sets","volume":"11","author":"Pawlak","year":"1982","journal-title":"Int. J. Inf. Comput. Sci."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"El-Bably, M.K., and Al-Shami, T.M. (2021). Different kinds of generalized rough sets based on neighborhoods with a medical application. Int. J. Biomath., 14.","DOI":"10.1142\/S1793524521500868"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"3045","DOI":"10.3233\/JIFS-210167","article-title":"A topological reduction for predicting of a lung cancer disease based on generalized rough sets","volume":"41","year":"2021","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1016\/S0888-613X(96)00071-0","article-title":"Two views of the theory of rough sets in finite universes","volume":"15","author":"Yao","year":"1996","journal-title":"Int. J. Approx. 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(1937). Sur les Espaces \u00e0 Structures Uniformes et Sur la Topologie G\u00e9n\u00e9rale, Hermann."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1016\/S0165-0114(97)00347-3","article-title":"The theory of global L-neighborhood structures, (III), Fuzzy uniform structures","volume":"98","author":"Bayoumi","year":"1998","journal-title":"Fuzzy Sets Syst."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/3\/199\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:14:27Z","timestamp":1760105667000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/3\/199"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,3,15]]},"references-count":16,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2024,3]]}},"alternative-id":["axioms13030199"],"URL":"https:\/\/doi.org\/10.3390\/axioms13030199","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2024,3,15]]}}}