Volume 26 • Issue 1 • PP: 322-334 • 2025
The Topology T (PR⋆) ^⊛ in the Frame of Primal Topological Spaces
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In this paper, we will use the family of regular⁺-open subsets to present and examine two new operators (.){PR⁺}⊛ and Cl{PR⁺}⊛. We demonstrate that, in contrast to the operator (.){PR⁺}⊛, the operator Cl{PR⁺}⊛ is a Kuratowski closure operator. We show that each of these operators lies between two previously defined operators where for each subset H⊆S, H_Pᶲ⊆H_{PR⁺}⊛⊆H_PRᶲ and H⊆Cl_Pᶲ(H)⊆Cl_{PR⁺}⊛(H)⊆Cl_{PR}ᶲ(H). Furthermore, we show that the topology, denoted by T_{PR⁺}⊛, which is obtained by Cl_{PR⁺}⊛ is independent from T and it is finer than T_η⁺, where T_η⁺ is the family of all unions of regular⁺-open subsets of (S, T). Then we demonstrate several fundamental results concerning this new structure and present many illustrative examples that relate to our conclusions. Finally, by using the operator Cl_{PR⁺}⊛ we introduce a new notion namely, P-generalized closed sets, and study some of their basic properties.
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References
[1] S. Acharjee, M. Ozkoc¸, F. Y. Issaka, Primal topological spaces, Bol. Soc. Paran. Mat. (3s.)v. (43), 2025, 1-9. DOI: https://doi.org/10.5269/bspm.66792
[2] A. Al-Omari, M. H. Alqahtani, Primal structure with closure operators and their applications, Mathematics, 11 (2023), 4946. https://doi.org/10.3390/math11244946.
[3] O. Alghamdi, On the compactness via primal topological spaces, AIMS Math., 9(11), (2024), 32124-32137. https://doi.org/10.3934/math.20241542.
[4] 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.
[5] 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/sym16040402
[6] Z. A. Ameen, R. A. Mohammed, T. M. Al-shami, B. A. Asaad, Novel fuzzy topologies from old through fuzzy primals, 2023, arXiv: 2308.06637. https://doi.org/10.48550/arXiv.2308.0663.
[7] K. C. Chattopadhyay, O. Nj˚astad, W. J. Thron, Merotopic spaces and extensions of closure spaces, Can. J. Math., 35 (1983), 613–629. https://doi.org/10.4153/CJM-1983-035-6.
[8] K. Chattopadhyay, W. J. Thron, Extensions of closure spaces, Can. J. Math., 29 (1977), 1277–1286. https://doi.org/10.4153/CJM-1977-127.
[9] G. Choquet, Sur les notions de filtre et de grille, Comptes Rendus Acad. Sci. Paris, 224 (1947), 171–173.
[10] J. Dontchev, “Survey on preopen sets,” in Proceedings of the Yatsushiro Topological Conference, Yat- sushiro, Japan, August, (1998), 1–18.
[11] W. Dunham, A new closure operator for non-T1 topologies, Kyungpook Math. J., 22(1), (1982), 55-60.
[12] D. Jankovic, T. R. Hamlett, New topologies from old via ideals, ´ Am. Math. Mon., 97(1990), 295–310. https://doi.org/10.1080/00029890.1990.1199559
[13] K. Kuratowski, Topology, American: Academic Press, 2014.
[14] N. Laxmi, B. Mathad, R. S. Wali, Generalized regular open sets, Malaya j. mat., 8(1), (2020), 563-569.
[15] N. Levine, Generalized closed sets in topology. Rend. Circ. Mat. Palermo 19(1970), 89–96.
[16] A.S. Mashhour, M.E. Abd El-Monsef, El-Deeb S.N. On pre-continuous and weak pre-continuous mappings, Proc. Math. Phys. Soc. Egypt, 53, (1982), 47–53.
[17] PL. Meenakshi, K. Sivakamasundari, Unification of regular star open sets, IJRAR, Special issue, (2019), 20-23.
[18] 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.
[19] S. Modak, Grill-filter space, J. Indian Math. Soc., 80 (2013), 313–320.
[20] L. Nachbin, Topology and order, D. Van Nostrand Inc., 1965.
[21] M. Ozkoc¸, B. K Ostel, On the topology τ ♢R of primal topological spaces, AIMS Math., 9(7), (2024), 17171-17183. doi: 10.3934/math.2024834
[22] S. Pious Missier and M. Annalakshmi, Between regular open sets and open sets, IJMA., 7(5),128-133.
[23] S. Pious Missier, M.Annalakshmi, S.Jackson, Separation axioms via regular *– open set, International Journal of Recent Research Aspects, Special Issue: Conscientious Computing Technologies, April 2018, 766-769.
[24] B. Roy, M. N. Mukherjee, On a typical topology induced by a grill, Soochow J. Math., 33 (2007), 771–786.
[25] B. Roy, M. N. Mukherjee, S. K. Ghosh, On a new operator based on grill and its associated topology, Arab J. Math. Sci., 14 (2008), 21–32.
[26] M. H. Stone, Applications of the theory of Boolean rings to general topology, T. Am. Math. Soc., 41 (1937), 375-381.
[27] N. V. Velicko, H-Closed topological spaces, Amer. Math. Soc. Transl., 78 (1968), 103–118.
[28] S. Willard, General Topology, Addition Wesley (1970).
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