On R1- Separation Axioms

Asmaa Rheem Kadhim; Rana Bahjat Esmaeel; Abdelaziz E. Radwan

doi:10.30526/38.4.4032

Authors

Asmaa Rheem Kadhim Department of Mathematics, College of Education for Pure Science (Ibn Al-Haitham), University of Baghdad, Baghdad, Iraq. https://orcid.org/0000-0002-4743-6034
Rana Bahjat Esmaeel Department of Mathematics, College of Education for Pure Science (Ibn Al-Haitham), University of Baghdad, Baghdad, Iraq. https://orcid.org/0000-0002-4743-6034
Abdelaziz E. Radwan Department of Mathematics, College Faculty of science, University of Ain shams, Cairo, Egypt https://orcid.org/0000-0002-2473-4872

DOI:

https://doi.org/10.30526/38.4.4032

Keywords:

: Ideal topological space,, R - open set, R_I-T0- space, R_I-T1- space, 〖 R〗_I-T2 – Space.

Abstract

The purpose of this research is to present the new set that is named open set, where the set was defined and the general properties And the basic concepts and properties of the set were explained, such as the open set, the intersection, and the union. also In this paper the separation axioms were defined and studied in ideal topological spaces in a new way. and discuss some properties. Also discuss the relationship of our definition with other definitions and prove some results in the context of separation axioms in ideal topological space on this set, where the axioms of the type - space, -space and – space were defined and the basic concepts and generalizations of these axioms on the set were studied. In addition the relationships between these concepts and their converses on this set under study were discussed, and illustrative examples and proofs of those properties were provided for that. Additionally, a diagram illustrating these concepts is presented

Author Biographies

Asmaa Rheem Kadhim, Department of Mathematics, College of Education for Pure Science (Ibn Al-Haitham), University of Baghdad, Baghdad, Iraq.

math
Rana Bahjat Esmaeel, Department of Mathematics, College of Education for Pure Science (Ibn Al-Haitham), University of Baghdad, Baghdad, Iraq.

math
Abdelaziz E. Radwan, Department of Mathematics, College Faculty of science, University of Ain shams, Cairo, Egypt

math

References

1. Kuratowski K. Topology. New York: Acad. Press. 1933; I.

2. Dontchev, Julian, Maximilian Ganster, and David Rose. Ideal resolvability.Topology and its Applications 1999: 93.1 ;1-16, https://doi.org/10.1016/S0166-8641(97)00257-5 .

3. Mohammed, M. W., A. A. Mohammed. Some Classes in Ideal Topological Spaces.Journal of Education & Science 2023: 32.2. https://doi.org/10.33899/edusj.2023.137766.1318

4. Auda, Ghufran H., Hula M. Salih. New supra open sets with ideal in supra topological space. AIP Conference Proceedings. 2023;Vol. 2834. No. 1. AIP Publishing,. https://doi.org/10.1063/5.0163559

5. Vaidyanathaswamy, R. The localisation theory in set-topology. Proceedings of the Indian Academy of Sciences-Section A. 1944:Vol. 20. Springer India.

6. Al-Saadi, H., A. Al-Omari. Some operators in ideal topological spaces. Missouri Journal of Mathematical Sciences 2018: 30.1; 59-71. https://doi.org/10.35834/mjms/1534384955

7. Mandal, Dhananjoy, M. N. Mukherjee. Certain new classes of generalized closed sets and their applications in ideal topological spaces. Filomat 2015: 29.5; 1113-1120.

8. Atay, A., F. Eren. KURATOWSKI CLOSURE OPERATORS IN SOFT IDEAL TOPOLOGICAL SPACES. Acta Universitatis Apulensis: Mathematics-Informatics 2023: 74. https://doi.org/ I 10.17114/j.aua

9. Tunç, Ayşe Nur, Sena Özen Yıldırım. On a topological operator via local closure function. Turkish Journal of Mathematics and Computer Science 2023: 15.2; 227-https://doi.org/10.47000/tjmcs.1195540

10. Viplavanjali, K. S., et al. Some Kinds of Separation Axioms in Kasaj Topological Spaces. Indian Journal of Science And Technology 2023:16.41;3575-3582.https://doi.org/10.17485/IJST/v16i41.1509

11. Karim, MA Abdel, and A. I. Nasir. "Separation Axioms via Ǐ Semi g Open Sets." Ibn AL-Haitham Journal For Pure and Applied Science, 2020: 33.1; 157-161. https://doi.org/10.30526/33.1.2382

12. Mohammad, Muna Jabbar, Faik Jameel Hasan Mayah. On Pre-δ-Separation Axioms in Ideal Topological Spaces.Journal of Wasit for science and medicine 2023:16.1;7-12. https://doi.org/10.31185/jwsm.342

13. Jackson, S., T. G. Denosha, P. Mariappan. Separation Axioms through Nano JD open set. Indian Journal of Science and Technology 2024: 17;9-13. https://doi.org/10.17485/IJST/v17sp1.252

14. Yalaz, Ferit, and Aynur Keskin KaymakcÄs. A New Separation Axiom in Ideal Topological Spaces.Conference Proceeding Science and Technology: International ONLINE Conference on Mathematical Advances and Applications (ICOMAA-2022), Istanbul. Vol. 5. No. 1. 2022.

15. Saeed, Sufyan G., R. B. Esmaeel. Separation axioms via αg Ị-open set. Journal of Physics: Conference Series. 2020: Vol.1591.No.1.IOP Publishing, https://iopscience.iop.org/article/10.1088/1742-6596/1591/1/012099

16. Dontchev,Julian. On Hausdorff Spaces via Topological Ideals and I‐irresolute Functions. Annals of the New York Academy of Sciences 1995: 767.1; 28-38. https://doi.org/10.1111/j.1749-6632.1995.tb55891.x

17. Banerjee, Amar Kumar, Jagannath Pal. New separation axioms in generalized bitopological spaces. Mathematical Sciences, 2020: 14.2 ;185-192. https://doi.org/10.1007/s40096-020-00330-z

18. Mohammad, R. J. Esmaeel. R. B.On Separation Axioms With Soft-J-Semi-g-Open Sets. IOP Conference Series: Materials Science and Engineering 2020:Vol. 871. No. 1. IOP Publishing, https://doi.org/10.1088/1757-899X/871/1/012052

19. Esmaeel, R. B.,R. J. Mohammad.On Nano soft-ℐ-semi-g-closed sets. Journal of Physics: Conference Series. 2020:Vol. 1591. No. 1. IOP Publishing. https://doi.org/ 10.1088/1742-6596/1591/1/012071

20. Renuka, J. Joycy, J. Arul Jesti. Some Separation Axioms In Nano Ideal Topological Spaces. Journal of Namibian Studies: History Politics Culture, 2023: 35;2502-2508. https://doi.org/10.59670/jns.v35i.4029

21. Karim, MA Abdel, A. I. Nasir. Some Game via Ἷ-Semi-g-Separation Axioms. Baghdad Science Journal , 2020: 17.3 https://doi.org/10.21123/bsj.2020.17.3.0861

22. Mahmood, S. I. On generalized regular continuous functions in topological spaces. Ibn Al-Haitham Journal for pure and applied science 2017: 25.3; 377-376.

23. Nasir, NA Jabbar AI, N. A. Jabbar. Some Types of Compactness in Bitopological Spaces. Ibn Al-Haitham Journal For Pure and Applied Sciences 2010:23.1.

24. Mahmoud, Balqees K., Yousif Y. Yousif. CutPoints and Separations in Alpha-Connected Topological Spaces. Iraqi Journal of Science 2021: 24;3091-3096. https://doi.org/10.24996/ijs.2021.62.9.24

25. Al-talkany, Yiezi K. Mahdi, Suadud H. Al-ismaily. On separation axioms and continuity with respect to some types of sets in ideal topological space.International Journal of Pure and Applied Mathematics 2018: 119.10; 409-422.

26. Al-shami, Tareq M. Soft somewhat open sets: Soft separation axioms and medical application to nutrition. Computational and Applied Mathematics 2022: 41.5; 216. https://doi.org/10.1007/s40314-022-01919-x

27. El-Atik, Abd El-Fattah A., Hanan Z. Hassan. Some nano topological structures via ideals and graphs. Journal of the Egyptian Mathematical Society 2020: 28.1; 41. DOI https://doi.org/10.1186/s42787-020-00093-5