Volume 27 • Issue 1 • PP: 59-72 • 2026
A New Operator via Regular Open Sets in a New Topological Structure
Open Access & Copyright
© 2026 The Author(s). Published by ASPG. This article is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
View PDF Abstract
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.
Keywords
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.
Cite This Article
Choose your preferred format
Publisher's Note
The statements, opinions, and data presented in this article are solely those of the author(s) and do not necessarily represent those of ASPG, the journal, or its editors. ASPG and the editors disclaim responsibility for any harm arising from the use of any ideas, methods, instructions, or products described in this article, to the fullest extent permitted by applicable law.