{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:13:46Z","timestamp":1760058826677,"version":"build-2065373602"},"reference-count":34,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2025,4,29]],"date-time":"2025-04-29T00:00:00Z","timestamp":1745884800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Graduate Studies and Scientific Research at Jouf University"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In the present paper, we first introduced the notion of k-fuzzy \u03b3-open (k-F-\u03b3-open) sets as a generalized novel class of fuzzy open (F-open) sets on fuzzy topological spaces (FTSs) in the sense of \u0160ostak. The class of k-F-\u03b3-open sets is contained in the class of k-F-\u03b2-open sets and contains all k-F-semi-open and k-F-pre-open sets. Also, we introduced the closure and interior operators with respect to the classes of k-F-\u03b3-closed and k-F-\u03b3-open sets and discussed some of their properties. After that, we defined and studied the notions of F-\u03b3-continuous (resp. F-\u03b3-irresolute) functions between FTSs(M,\u2111) and (N,\u03dc). However, we displayed and investigated the notions of F-almost (resp. F-weakly) \u03b3-continuous functions, which are weaker forms of F-\u03b3-continuous functions. Next, we presented and characterized some new F-functions via k-F-\u03b3-open and k-F-\u03b3-closed sets, called F-\u03b3-open (resp. F-\u03b3-irresolute open, F-\u03b3-closed, F-\u03b3-irresolute closed, and F-\u03b3-irresolute homeomorphism) functions. The relationships between these classes of functions were investigated with the help of some examples. We also introduced some new types of F-separation axioms called k-F-\u03b3-regular (resp. k-F-\u03b3-normal) spaces via k-F-\u03b3-closed sets and discussed some properties of them. Lastly, we explored and studied some new types of F-compactness called k-F-almost (resp. k-F-nearly) \u03b3-compact sets.<\/jats:p>","DOI":"10.3390\/sym17050678","type":"journal-article","created":{"date-parts":[[2025,5,2]],"date-time":"2025-05-02T11:35:13Z","timestamp":1746185713000},"page":"678","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Some Characterizations of k-Fuzzy \u03b3-Open Sets and Fuzzy \u03b3-Continuity with Further Selected Topics"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3818-3597","authenticated-orcid":false,"given":"Fahad","family":"Alsharari","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, Jouf University, Sakaka 72311, Saudi Arabia"}],"role":[{"role":"author","vocab":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1055-9399","authenticated-orcid":false,"given":"Hind Y.","family":"Saleh","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Basic Education, University of Duhok, Duhok 42001, Iraq"}],"role":[{"role":"author","vocab":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2921-3224","authenticated-orcid":false,"given":"Islam M.","family":"Taha","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Sohag University, Sohag 82534, Egypt"}],"role":[{"role":"author","vocab":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,4,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1016\/S0019-9958(65)90241-X","article-title":"Fuzzy Sets","volume":"8","author":"Zadeh","year":"1965","journal-title":"Inf. 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