On pre­ Open Regular Spaces





Pre­open set, Pre­ closed set, Pre­ open­ function, Pre­ continuous­ function, Pre-irresolute map, Regular space


In this paper, certain types of regularity of topological spaces have been highlighted, which fall within the study of generalizations of separation axioms. One of the important axioms of separation is what is called regularity, and the spaces that have this property are not few, and the most important of these spaces are Euclidean spaces. Therefore, limiting this important concept to topology is within a narrow framework, which necessitates the use of generalized open sets to obtain more good characteristics and preserve the properties achieved in general topology. Perhaps the reader will realize through the research that our generalization preserved most of the characteristics, the most important of which is the hereditary property. Two types of regular spaces have been presented, namely the topological space Rp and the topological space S-Rp. The properties of these two spaces and their relationship with each other, as well as the effect of functions on them, have been studied. In addition several theorems have been proved regarding the sufficient and necessary conditions to make the topological spaces Rp-regular or S-Rp-regular. The above concepts have been linked with a new type of Hausdorff space and the concepts under study are reinforced with examples.


Levine N. Semi-open sets and semi-continuity in topological spaces. Amer Math Mon. 1963; 70(1): 36-41. https://doi.org/10.1080/00029890.1963.11990039.

Kamel GA, Dib KA. Generalized topology and the family of monotonic maps Γ(X). J Egypt Math Soc. 2023 Dec; 31(1): 1-27. https://doi.org/10.1186/s42787-023-00162-5.

Aziz NI, Jasim TH. On Generalized N*-Closed Set in NANO-N* Topological Spaces With Some Properties. 2nd International Conference on Physics and Applied Sciences (ICPAS 2021), College of Education, Mustansiriyah University, Baghdad, Iraq, 26-27 May. J Phys. Conf Ser. 2021; 1963(1). https://doi.org/10.1088/1742-6596/1963/1/012157.

Al- Shami TM, El-Shafi ME. On soft compact and soft Lindelöf spaces via soft pre-open sets. Ann Fuzzy Math. Inform. 2019 Feb; 17(1): 79-100. https://doi.org/10.30948/afmi.2019.17.1.79.

Mashhour AS, Abd El – Monsef ME, Hasanein IA. On Pretopological Spaces. Bull Math Dela Soc Roum. 1984; 28(76): 39-45.

Sierpinski W. General Topology. USA: Dover Publications; 2020 Apr 15. Chap 2, Topological spaces; p. 40-43. .

Hwaidi AH, Hussein JH. λ –Semi – Open Sets in Generalized Open Spaces. Proceedings of the 1st International Conference on Advance Research in Pure and Applied Science (ICARPAS 2021): Third Annual Conference of Al-Muthanna University/College of Science, 24–25 March 2021, Al-Samawah, Iraq. AIP Conf Proc. 2022; 2398(1): 060078. https://doi.org/10.1063/5.0094717.

Al- Omeri WF, Jafari S. Neutrosophic Pre-Continuous Multifunctions and Almost PreContinuous Multifunctions. Neutrosophic Sets Syst. 2019 Oct; 27(1): 53-69.

Ali HJ, Hassan RF. On Light Mapping and Certain Concepts by Using m_X N-Open Sets. Baghdad Sci J. 2020 Jan 2; 17(1(Suppl.)): 371-377. https://dx.doi.org/10.21123/bsj.2020.17.1(Suppl.).0371.

Al-Omeri W, Jafari S. On Generalized Closed Sets and Generalized Pre-Closed Sets in Neutrosophic Topological Spaces. Math. 2018; 7(1): 1-12. https://doi.org/10.3390/math7010001.

Ali AA, Sadek AR. On regular δ-semi-open space. J Interdiscip Math. 2021; 24(4): 953-960. https://doi.org/10.1080/09720502.2021.1885813.

Jasim AH. Results on a Pre-T2 Space and Pre-Stability. Baghdad Sci J. 2019 Mar; 16(1): 111-115. https://doi.org/10.21123/bsj.2019.16.1.0111.

Kadzam AR, Yousif YY. Fibrewise totally separation axioms. J Interdiscip Math. 2022; 25(2): 511-520. https://doi.org/10.1080/09720502.2021.2003012.

Darwesh HM, Namiq SF. Pre-irresolute Functions in Closure Spaces. Eur J Pure Appl Math. 2019; 12(3): 1082-1095. https://doi.org/10.29020/nybg.ejpam.v12i3.3317.

Kannan K, Nagaveni N, Saranya S. On β ̂g- Continuous and β ̂g- Irresolute maps in topological spaces. Procedia Comput Sci. 2015; 47: 368-373. https://doi.org/10.1016/j.procs.2015.03.218.

Sathiyaraj J, Valivel A, Maheshwari OV. On Double Fuzzy M-open Mappings and Double Fuzzy M-closed Mappings. Appl Appl Math Int J (AAM). 2020 Dec; 15(2): 928-941.

Esmaeel RB. On semi- p- open sets. M.Sc. [thesis]. College of Education/ Ibn Al-Haitham: Baghdad University; 2004.

Rao VV, Rao YS. Neutrosophic Pre-open Sets and Pre-closed Sets in Neutrosophic Topology. Int J Chemtech Res. Infinite study. 2017; 10(10): 449-458.

Morris SA. Topology Without Tears. USA: University of New England; 2023.

Kelley JL. General topology. Graduate Texts in Mathematics, 27. USA: Springer Science & Business Media; 1955. Chap 1, Topological Spaces; p. 50-53.

Sadek AR P. P-L. Compact Topological Ring. Iraqi J Sci. 2016; 57(4B): 2754-2759.





How to Cite

On pre­ Open Regular Spaces. Baghdad Sci.J [Internet]. [cited 2024 Jun. 14];21(12). Available from: https://bsj.uobaghdad.edu.iq/index.php/BSJ/article/view/9237