Generalized \(\mathcal{I}_\mathcal{p}\)-Closed Sets in Ideal Topological Spaces

S. Rakshana *

Department of Mathematics, Faculty of Science, Eastern University, Sri Lanka.

P. Elango

Department of Mathematics, Faculty of Science, Eastern University, Sri Lanka.

*Author to whom correspondence should be addressed.


Abstract

The main focus of this study is to introduce a new category of generalized closed sets, referred to as \(\mathcal{I}_\mathcal{p}\)- closed sets, within the framework of ideal topological spaces. By using a few instances, we demonstrate \(\mathcal{I}_\mathcal{p}\)- closed sets and establish some fundamental properties of \(\mathcal{I}_\mathcal{p}\)-closed sets. We also investigate the relationship between \(\mathcal{I}_\mathcal{p}\)-closed sets and other classes of generalized closed sets in ideal topological spaces, such as \(\mathcal{I}_\mathcal{g}\)- closed sets, \(\alpha\)\(\mathcal{I}_\mathcal{g}\)-closed sets, and  \(\mathcal{I}_\mathcal{r}\)\(\mathcal{g}\)-closed sets. Then, we focus on the topological implications of \(\mathcal{I}_\mathcal{p}\)-closed sets and investigate how they relate to the concepts of \(\mathcal{I}_\mathcal{p}\)-continuous map,  \(\mathcal{I}_\mathcal{p}\)-irresolute map, and a strongly \(\mathcal{I}_\mathcal{p}\)-continuous map. First and foremost, we define the \(\mathcal{I}_\mathcal{p}\)-continuous map, investigate the behavior of \(\mathcal{I}_\mathcal{p}\)- continuous map with respect to \(\mathcal{I}_\mathcal{p}\)-closed sets, and derive several important properties of \(\mathcal{I}_\mathcal{p}\)-continuous map. Further, we studied their relationships with other classes of continuous maps in ideal topological spaces. Nevertheless, we defined the definitions of \(\mathcal{I}_\mathcal{p}\)-irresolute maps and strongly \(\mathcal{I}_\mathcal{p}\)-continuous maps in ideal topological spaces. We explored the connections with the notions of \(\mathcal{I}_\mathcal{p}\)-continuous map,  \(\mathcal{I}_\mathcal{p}\)-irresolute map, and a strongly \(\mathcal{I}_\mathcal{p}\)-continuous map. Our results provide new insights into the study of ideal topological spaces.

Keywords: Preopen set, deals, \(\mathcal{I}_\mathcal{p}\)-closed set, \(\mathcal{I}_\mathcal{p}\)-continuous map, \(\mathcal{I}_\mathcal{p}\)-irresolute map.


How to Cite

Rakshana, S., & Elango, P. (2023). Generalized \(\mathcal{I}_\mathcal{p}\)-Closed Sets in Ideal Topological Spaces. Advances in Research, 24(4), 38–46. https://doi.org/10.9734/air/2023/v24i4946

Downloads

Download data is not yet available.

References

Levine N. Generalized closed set in topology. Rend. Circ.Math. Palermo. 1970;19:89-96.

Rajamani M, Indumathi V, Krishnaprakash S. Iπg-closed sets and Iπg-continuity. Journal of Advanced research in pure math. 2010:2(4):63-72.

Vithyasangaran K, Elango P. Study of a* -Homeomorphisms by a* -Closed Sets. Advances in Research. 2019;19(3):1-6.

Wadei AL-Omeri, Takashi Noiri T. AGI*-sets, BGI*-sets and δβI-open sets in ideal topological spaces. International Journal of Advances in Mathematics. 2018;4:25-33.

Khan M, Hamza M. Isg-closed sets in Ideal Topological Spaces. Global Journal of Pure and Applied Mathematics. 2011;7(1):89-99.

Lellis Thivagar M, Santhini C. New approach of ideal topological generalized closed sets. Bol. Soc.

Paran. Mat. 2013;31(2):191-204.

Kuratowski K. Topology, Vol.I. Academic Press, New York;1966.

Vaidynathaswamy R. The localization theory in set topology. Proc. Indian Acad. Sci. Math. Sci. 1945;20:51-61.

Wadei AL-Omeri, Noiri T. Ig-Closed Sets Via Ideal Topological Spaces. Missouri J. Math. Sci. 2019;31(2):174-191.

Wadei AL-Omeri, Noiri T. On almost e-I- continuous functions. Demonstratio Mathematica. 2021;54(1):168-177.

Wadei AL-Omeri, Noiri T. On semi-I-open sets, pre-I-open sets and e-I-open sets in

ideal topological spaces. Boletim da Sociedade Paranaense de Matem´ atica. 2023;41:1-8.

Dontshev J, Ganster M, Noiri T. Unified approach of generalized closed sets via topological ideals. Math.Japan. 1999;4(9):395-401.

Navaneethakrishnan M, Sivaraj D. Regular generalized closed sets in ideal topological spaces. Journal of Advanced Research in Pure Mathematics. 2010;2(3):24-33.

Maragathavalli D, Vinothini D. -Generalized Closed Sets in ideal topological space. IOSR

Journal of Mathematics. 2014;2(10):33-38.

Renu Thomas, Janaki C. Some new sets in Ideal topological spaces. International Research Journal of Engineering and Technology (IRJET). 2015;2(9):2395-2456.

Jankovic D, Hamlett TR. New topologies from old via ideals. Amer. Math. Monthly. 1990;97(4):295- 310.

Jafari S, Viswanathan K, Jayasudha J. Decomposition of -Continuity in ideal topological spaces. International Journal of Mathematics Archive. 2012;3(6):2340-2345.

Vaidyanathaswamy R. Set Topology. Chelsea Publishing Company;1946.

Mashhour AS, Abd El-Monsef ME, El-Deeb SN. On precontinuous and weak precontinuous

mappings. Proc. Math. Phys. Soc. Egypt. 1982;53:47-53.

Njastad O. On some classes of nearly open sets. Pacific J. Math. 1965;15:961-970.

Stone MH. Applications of the theory of Boolean rings to general topology. Trans. Amer. Math. Soc. 1937;41:375-481.

Rakshana S, Elango P. Ip-Closed Maps in Ideal Topological Spaces. Abstracts of the Proceedings of ARS-FOS-2021, Eastern University, Sri Lanka. ARS-2021-FS-M-04.