jaem TWMS Journal of Applied and Engineering Mathematics 2146-1147 2587-1013 Işık University Press Mathematical Logic, Set Theory, Lattices and Universal Algebra Topology Matematiksel Mantık, Kümeler Teorisi, Kafesler ve Evrensel Cebir Topoloji $ \delta \beta $-OPEN SETS IN PYTHAGOREAN FUZZY NANO TOPOLOGICAL SPACES https://orcid.org/0009-0001-5654-5486 Navuluri Mohanarao Government College of Engineering, Bodinayakkanur https://orcid.org/0009-0006-0532-0793 Lakshmi K. Shantha M. Kumarasamy College of Engineering https://orcid.org/0000-0001-5970-035X A Vadivel Arignar Anna Government Arts College https://orcid.org/0009-0003-9713-0388 Varudaraj Sivakumar Arignar Anna Government Arts College 04 07 2026 16 4 482 495 12 29 2024 10 13 2025 Copyright © 2010, TWMS Journal of Applied and Engineering Mathematics 2010 TWMS Journal of Applied and Engineering Mathematics

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.

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