$ \delta \beta $-OPEN SETS IN PYTHAGOREAN FUZZY NANO TOPOLOGICAL SPACES

Year 2026, Volume: 16 Issue: 4 , 482 - 495 , 07.04.2026

Mohanarao Navuluri K. Shantha Lakshmi Vadivel A , Sivakumar Varudaraj

https://izlik.org/JA45RY84WR

Abstract

The purpose of this paper is to define and study a new class of sets called Pythagorean fuzzy nano $ \delta $ (resp. $ \delta $ pre, $ \delta $ semi, $ \delta \alpha $ and $ \delta \beta$)-open sets in Pythagorean fuzzy nano topological spaces. Analyse the basic properties of Pythagorean fuzzy nano $ \delta $ (resp. $ \delta $ pre, $ \delta $ semi, $ \delta \alpha $ and $ \delta \beta$)-open (resp. closed) sets. We also used them to introduce the new notions like Pythagorean fuzzy nano $ \delta $ (resp. $ \delta $ pre, $ \delta $ semi, $ \delta \alpha $ and $ \delta \beta$)-closure (resp. interior) and investigate their relations with already existing well known sets. We apply entropy measure for decision making problem of selecting the optimum wastewater treatment method for the dying factories based on the required criteria.

Keywords

Pythagorean fuzzy nano open set , Pythagorean fuzzy nano $\delta$ open set , Pythagorean fuzzy nano $ \delta $ interior , Pythagorean fuzzy nano $ \delta $ closure

References

• [1] Ajay, D., Joseline Charisma, J., (2020), Pythagorean nano topological space, International Journal of Recent Technology and Engineering, 8, pp. 3415-3419.
• [2] Ajay, D., Joseline Charisma, J., (2020), On weak forms of Pythagorean nano open sets, Advances in Mathematics: Scientific Journal, 9, pp. 5953-5963.
• [3] Atanassov, K. T., (1983), Intuitionistic fuzzy sets, VII ITKR’s Session, Sofia (deposed in Central Sci.-Technical Library of Bulg. Acad. of Sci., 1697/84) (in Bulgarian).
• [4] Chang, C. L., (1968), Fuzzy topological spaces, J. Math. Anal. Appl., 24 , pp. 182-190.
• [5] Coker, D., (1997), An introduction to intuitionistic fuzzy topological spaces, Fuzzy sets and systems, 88 , pp. 81-89.
• [6] Lellis Thivagar, M., Richard, C., (2013), On nano forms of weekly open sets, International journal of mathematics and statistics invention, 1(1), pp. 31-37.
• [7] Lellis Thivagar, M., Jafari, S., Sutha Devi, V., Antonysamy, V., (2018), A novel approach to nano topology via neutrosophic sets, Neutrosophic Sets and Systems, 20, pp. 86-94.
• [8] Olgun, M., Unver, M., Yardimci, S., (2019), Pythagorean fuzzy topological spaces, Complex & Intelligent Systems, pp. 177-183. https://doi.org/10.1007/s40747-019-0095-2.
• [9] Padma, A., Saraswathi, M., Vadivel, A., Saravanakumar, G., (2019), New Notions of Nano M-open Sets, Malaya Journal of Matematik, S(1), pp. 656-660.
• [10] Pankajam, V., Kavitha, K., (2017), δ-open sets and δ-nano continuity in δ-nano topological spaces, International Journal of Innovative Science and Research Technology, 2(12), pp. 110-118.
• [11] Wu X., Zhu Z., Wang T, Zhang X., (2025), Picture Fuzzy Interactional Aggregation Operators via Strict Triangular Norms and Applications to Multi-Criteria Decision Making, Appl. Comput. Math., Vol.24, No.1, 2025, pp.62-100
• [12] Pankajam, V., Kavitha, K., (2017), δ-open sets and δ-nano continuity in δ-nano topological spaces, International Journal of Innovative Science and Research Technology, 2(12), pp. 110-118.
• [13] Zhou J., Wu X., Chen C., (2025), Investigating Various Intuitionistic Fuzzy Information Measures and Their Relationships, Appl. Comput. Math., Vol.24, No.4, 2025, pp.579-607
• [14] Shiventhiradevi Sathaananthan, Tamilselvan, S., Vadivel, A., Saravanakumar, G., (2020), Fuzzy Z closed sets in double fuzzy topological spaces, AIP Conf Proc., 2277 , pp. 090001.
• [15] Shiventhiradevi Sathaananthan, Vadivel, A., Tamilselvan, S., Saravanakumar, G., (2020), Generalized fuzzy Z closed sets in double fuzzy topological spaces, Adv. Math: Sci. J., 9(4), pp. 2107-2112.
• [16] Vadivel, A., John Sundar, C., Kirubadevi, K., Tamilselvan, S., (2022), More on Neutrosophic Nano Open Sets, International Journal of Neutrosophic Science (IJNS), 18(4), pp. 204-222.
• [17] Yager, R. R., (2013), Pythagorean fuzzy subsets, In: Proceedings of the joint IFSA world congress NAFIPS annual meeting, 57-61.
• [18] Yager, R. R., (2014), Pythagorean membership grades in multicriteria decision making, IEEE Trans Fuzzy Syst. 22 (4), 958-965.
• [19] Zadeh, L. A., (1965), Fuzzy sets, Information and control, 8, pp. 338-353.