Volume 27 • Issue 1 • PP: 59-72 • 2026
A New Operator via Regular Open Sets in a New Topological Structure
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In this paper, we will use the family of regular open sets in a topological space (Z, τ ) to define an operator ΦR : 2Z → 2Z by ΦR(F) = {s ∈ Z : ∃ D ∈ RO(Z, s) with (D − F )c /∈ P} in frame of primal topological spaces. Then we introduce the notion of topology δ-compatible for a primal in a primal topological space and study some of its properties. Finally, we use the concept of δ-semi-open sets to provide additional properties for the operators (⋄ R) and ΦR(F ), and we add many illustrative examples that help clarify the relationships between the concepts that are presented.
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References
[1] S. Acharjee, M. ¨Ozkoc¸, F. Y. Issaka, Primal topological spaces, Bol. Soc. Paran. Mat. (3s.)v. 2025 (43)1-9, http://dx.doi.org/10.5269/bspm.66792
[2] O. Alghamdi, A. Al-Omari, M. H. Alqahtani, Novel operators in the frame of primal topological spaces, AIMS Mathematics, 9 (2024), 25792–25808.
[3] M. Alotaibi, A. Alzahrani, Primal-A study of soft topological spaces and their applications, Mathematics, 9(12) (2021), 1560.
[4] A. Bhattacharya, S. Choudhury, Neutrosophic soft set and its applications in decision-making, Soft Computing, 25(9) (2021), 7071-7082.
[5] R. A. Mohammed, T. M. Al-Shami, Fuzzy soft sets and their applications in decision-making problems, Information Sciences, 582 (2022), 564-576.
[6] T. M. Al-Shami, Z. A. Ameen, R. Abu-Gdairi, A. Mhemdi, On primal soft topology, Mathematics, 11(2023), 2329 ,https://doi.org/10.3390/math.11102329.
[7] F. Alsharari, H. Alohali, Y. Saber, F. Smarandache, An introduction to single-valued neutrosophic primal theory, Symmetry, 16(4) (2024), 402. https://doi.org/10.3390/sym1604040.
[8] Z. A. Ameen, R. A. Mohammed, T. M. Al-shami, B. A. Asaad, Novel fuzzy topologies formed by fuzzy primal frameworks, J. Intell. Fuzzy Sys., pre-press, 2024, 1–10. https://doi.org/10.3233/JIFS 238408.
[9] G. Choquet, Sur les notions de filtre et de grille, Comptes Rendus Acad. Sci. Paris, 224 (1947), 171–173.
[10] M. Hosny, Rough sets theory via new topological notions based on ideals and applications, AIMS Mathematics, 7 (2021), 869–902. https://doi.org/10.3934/math.2022052.
[11] R. A. Hosny, B. A. Asaad, A. A. Azzam, T. M. Al-shami, Various topologies generated from Ej- neighbourhoods via ideals, Complexity, 2021 (2021), 4149368. http://doi.org/10.1155/2021/4149368.
[12] D. Jankovic, T. R. Hamlett, New topologies from old via ideals, Am. Math. Mon., 97 (1990), 295–310. https://doi.org/10.1080/00029890.1990.11995593.
[13] K. Kuratowski, Topology, American: Academic Press, 2014.
[14] J. A. Zadeh, The Topology Generalized fuzzy sets and their applications, Journal of Fuzzy Logic and Intelligent Systems. 12 (1) (2023), 1-10.
[15] S. Modak, Topology on grill-filter space and continuity, Bol. Soc. Parana. Mat., 31 (2013), 219–230. http://doi.org/10.5269/bspm.v31i2.16603.
[16] S. Modak, Grill-filter space, J. Indian Math. Soc., 80 (2013), 313–320.
[17] M. ¨Ozkoc¸, B. K ¨Ostel. On the topology τ ⋄ R of primal topological spaces, AIMS Mathematics, 9 (2024), 17171-17183. http://doi.org/10.3934/math.2024834.
[18] J.H. Park, B.Y. Lee and M.J. Son, On δ-semi-open sets in topological space, J. Indian Acad. Math., 19(1997), 59 - 67
[19] Z. Pawlak, Rough sets, International Journal of Computer and Information Sciences, 11 (1982), 341– 356. https://doi.org/10.1007/BF01001956.
[20] B. Roy, M. N. Mukherjee, On a typical topology induced by a grill, Soochow J. Math., 33 (2007), 771– 786.
[21] M. H. Stone, Applications of the theory of Boolean rings to general topology, T. Am. Math. Soc., 41.(1937), 375-381https://doi.org/10.2307/1989788.
[22] N. V. Velicko, H-Closed topological spaces, Amer. Math. Soc. Transl., 78 (1968), 103–118.
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