Volume 25 , Issue 3 , PP: 296-306, 2025 | Cite this article as | XML | Html | PDF | Full Length Article
Jamal Oudetallah 1 *
Doi: https://doi.org/10.54216/IJNS.250326
This paper is dedicated to studying for the first time the building of a topological space based on the intervals defined over 4-refined neutrosophic real numbers and 5-refined neutrosophic real numbers, where we define a special partial order relation on these rings, and we use it to study the structure of the corresponding intervals generated from this relation. Also, we characterize the formula of open sets through these two topological spaces with some illustrated examples.
4-refined neutrosophic numbers , 5-refined neutrosophic numbers , Partial order relation , Neutrosophic interval topology , Open set
[1] I. M. Hanafy, A. A. Salama and K.M. Mahfouz (2013).Neutrosophic crisp events and its probability. International Journal of Mathematics and Computer Applications Research,3(1),171-178.
[2] Kim, J., Lee, J. G., & Hur, K. (2021). Intuitionistic neutrosophic crisp sets and their application to topology. Infinite Study.
[3] A. M. AL-Odhari. (2015).On infra topological space. International Journal of Mathematical Archive, 6(11), 179-184.
[4] Abobala, M., Bal, M., & Hatip, A. (2021). A review on recent advantages in algebraic theory of neutrosophic matrices. International Journal of Neutrosophic Science, 68-86.
[5] Abobala, M., Ziena, B. M., Doewes, R., & Hussein, Z. (2022). The Representation of Refined Neutrosophic Matrices by Refined Neutrosophic Linear Functions. Int. J. Neutrosophic Sci, 19, 342-349. [6] Ahmad, K. D., Thjeel, N. N., Zahra, M. A., & Jaleel, R. A. (2022). On the Classification of n-Refined Neutrosophic Rings and Its Applications in Matrix Computing Algorithms and Linear Systems. International Journal Of Neutrosophic Science, 18(4).
[7] Abobala, M. (2020). A study of AN H-substructures in n-refined neutrosophic vector spaces. International Journal of Neutrosophic Science, 9, 74-85.
[8] Abobala, M., On Refined Neutrosophic Matrices and Their Applications in Refined Neutrosophic Algebraic Equations, Journal of Mathematics, Hindawi, 2021.
[9] Adeleke, E.O., Agboola, A.A.A., and Smarandache, F., Refined Neutrosophic Rings II", International Journal of Neutrosophic Science, Vol. 2(2), pp. 89-94, 2020.
[10] Hatip, A., and Olgun, N., “On Refined Neutrosophic R-Module", International Journal of Neutrosophic Science, Vol. 7, pp.87-96, 2020.
[11] Smarandache, F., and Abobala, M., n-Refined Neutrosophic Rings, International Journal Of Neutrosophic Science, 2020.
[12] Abobala, M., "On the Characterization of Maximal and Minimal Ideals In Several Neutrosophic Rings", Neutrosophic Sets and Systems, Vol. 45, 2021.
[13] AL-Nafee, A. B., Al-Hamido, R. K., & Smarandache, F. (2019). Separation axioms in neutrosophic crisp topological spaces. Infinite Study.
[14] Saber, W. A., N. M., A. (2023). N-refined Neutrosophic Fuzzy of Some Topological Concepts. International Journal of Neutrosophic Science, (), 80-87.
[15] Riad K. Al-Hamido. (2018). Neutrosophic crisp bi-topological spaces. Neutrosophic Sets and Systems 21, 66-73.
[16] Ali, R., "A Short Note on the Solution of n-Refined Neutrosophic Linear Diophantine Equations", International Journal of Neutrosophic Science, Vol. 15, 2021.
[17] Ibrahim, M., and Abobala, M., "An Introduction to Refined Neutrosophic Number Theory", Neutrosophic Sets and Systems, Vol. 45, 2021.
[18] Abobala, M, "n-Cyclic Refined Neutrosophic Algebraic Systems of Sub-Indeterminacies, an Application to Rings and Modules", International Journal of Neutrosophic Science, Vol. 12, pp. 81-95. 2020.
[19] Mohammed, H. Y., Mohammed, F. M., & Al-mahbashi, G. (2024). On Q^*-closed Sets in Fuzzy Neutrosophic Τopology: Principles, Proofs, and Examples. Neutrosophic Systems with Applications, 20, 67-74. https://doi.org/10.61356/j.nswa.2024.20352
[20] Vetrivel, G., Mullai, M., & Buvaneshwari, R. (2024). Forgotten Topological Index and its Properties on Neutrosophic Graphs. Neutrosophic Systems with Applications, 21, 36-45. https://doi.org/10.61356/j.nswa.2024.21356
[21] Raed Hatamleh , Ayman Hazaymeh.(2025),On Some Topological Spaces Based On Symbolic n-Plithogenic Intervals, International Journal of Neutrosophic Science, Vol. 25 Issue.1 PP. 23-37. Doi: https://doi.org/10.54216/IJNS.250102.
[22] Raed Hatamleh , Ayman Hazaymeh,(2025),On The Topological Spaces of Neutrosophic Real Intervals, International Journal of Neutrosophic Science, Vol. 25 Issue. 1 PP. 130-136, (Doi: https://doi.org/10.54216/IJNS.250111.
[23] Abdallah Shihadeh, Khaled Ahmad Mohammad Matarneh, Raed Hatamleh, Mowafaq Omar Al-Qadri, Abdallah Al-Husban,(2024), On The Two-Fold Fuzzy n-Refined Neutrosophic Rings For 2≤n≤3, Neutrosophic Sets and Systems, Vol. 68,pp.8 25.(Doi: 10.5281/zenodo.11406449).
[24] Abdallah Shihadeh, Khaled Ahmad Mohammad Matarneh, Raed Hatamleh, Randa Bashir Yousef Hijazeen, Mowafaq Omar Al-Qadri, Abdallah Al-Husban.(2024). An Example of Two-Fold Fuzzy Algebras Based On Neutrosophic Real Numbers, Neutrosophic Sets and Systems,vol.67, pp. 169-178.(DOI:10.5281/zenodo.11151930).
[25] A. Rajalakshmi, Raed Hatamleh, Abdallah Al-Husban, K. Lenin Muthu Kumaran, M. S. Malchijah raj. (2025). various (ζ1, ζ2) neutrosophic ideals of an ordered ternary semigroups. 32 (3), PP: 400-417.
[26] Raed Hatamleh, Abdallah Al-Husban, N. Sundarakannan, M. S. Malchijah Raj. (2025). Complex cubic intuitionistic fuzzy set applied to subbisemirings of bisemirings using homomorphism. 32 (3), PP: 418-435.
[27] Abu, Ibraheem. , Al-Husban, Abdallah. , J., Lejo. , J., Jamil. , Palanikumar, M., Balaji, G..(2024). Selection process real-life application for new type complex neutrosophic sets using various aggregation operators. International Journal of Neutrosophic Science, 23(4) 136-153.
[28] Selvaraj, S.. , Gharib, Gharib. , Al-Husban, Abdallah. , Al, Maha. , Lenin, K.. , Palanikumar, Murugan. , Sundareswari, K.. (2024) new algebraic approach towards interval-valued neutrosophic cubic vague set based on subbisemiring over bisemiring. International Journal of Neutrosophic Science, 23(4), pp. 272-292.
[29] C. Sivakumar, Mowafaq Omar Al-Qadri, Abdallah shihadeh, Ahmed Atallah Alsaraireh, Abdallah Al-Husban, P. Maragatha Meenakshi, N. Rajesh, M. Palanikumar. (2024). q-rung square root interval-valued sets with respect to aggregated operators using multiple attribute decision making. Journal of, 23(3), pp. 154-174.
[30] Abu, Ibraheem.,T., T.,Al-Husban, Abdallah.,Alahmade, Ayman.,Azhaguvelavan, S.. , Palanikumar, Murugan. (2024) Computer purchasing using new type neutrosophic sets and its extension based on aggregation operators. International Journal of Neutrosophic Science, 24(1), pp. 171-185.
[31] Raja, K., Meenakshi, P. M., Al-Husban, A., Al-Qadri, M. O., Rajesh, N., & Palanikumar, M. (2024). Multi-criteria group decision making approach based on a new type of neutrosophic vague approach is used to select the shares of the companies for purchase. Full Length Article, 23(3), 296-96.
[32] J., Lejo. , Damrah, Sadeq. , M., Mutaz. , Al-Husban, Abdallah. , Palanikumar, M.. (2024). Type-I extension Diophantine neutrosophic interval valued soft set in real life applications for a decision making. International Journal of Neutrosophic Science, 24(4), pp. 151-164 .
[33] Abu, Ibraheem. , Al-Husban, Abdallah. , J., Lejo. , J., Jamil. , Palanikumar, M.. , Balaji, G.. (2024). Selection process real-life application for new type complex neutrosophic sets using various aggregation operators. International Journal of Neutrosophic Science, 23(4), pp. 136-153.
[34] Shihadeh, Abdallah. , Mahmoud, Wael. , Bataineh, Malik. , Al-Tarawneh, Hassan. , Alahmade, Ayman. , Al-Husban, Abdallah. (2024) On the Geometry of Weak Fuzzy Complex Numbers and Applications to the Classification of Some A-Curves. International Journal of Neutrosophic Science, 23(4), pp. 369-375.
[35] Abualhomos, Mayada. , Mahmoud, Wael. , Bataineh, Malik. , Omar, Mowafaq. , Alahmade, Ayman. , Al-Husban, Abdallah. (2024). An Effective Algorithm for Solving Weak Fuzzy Complex Diophantine Equations in Two Variables. International Journal of Neutrosophic Science, 23(4), pp. 386-394.
[36] Abualhomos, Mayada. , Atallah, Ahmed. , Bataineh, Malik. , Abu-Alkishik, Nabeela. , Al-Shbeil, Isra. , Al-Husban, Abdallah.(2024(. On The Real Inner Products and Orthogonality for Plithogenic Vector Spaces of Orders 4 and 5. International Journal of Neutrosophic Science, 24 (1), pp. 94-103.
[37] Abu, Ibraheem. , Al-Husban, Abdallah. , M., Mutaz. , A., Ahmed. , Shatarah, Amani. , Mousa, Norah.(2024) On the Characterization of Some m-Plithogenic Vector Spaces and Their AH-Substructures Under the Condition 6 ≤ dim SPV ≤10. International Journal of Neutrosophic Science, 24(3), pp. 258-267.
[38] Shihadeh, A., Matarneh, K. A. M., Hatamleh, R., Al-Qadri, M. O., & Al-Husban, A. (2024). On The Two-Fold Fuzzy n-Refined Neutrosophic Rings For 2≤ 3. Neutrosophic Sets and Systems, 68, 8-25.
