δβ-open Sets in Pythagorean Neutrosophic Nano Topological Spaces and Their Application via Machine Learning

Mohanarao Navuluri; Shantha lakshmi K

Authors

Mohanarao Navuluri Department of Mathematics, Government College of Engineering, Bodinayakkanur- 625582, Theni, India; (Mohanarao Navuluri)
Shantha lakshmi K Department of Mathematics, M.Kumarasamy College of Engineering, Karur- 639 113, India; (Shantha lakshmi K)

Keywords:

Pythagorean neutrosophic nano open set, Pythagorean neutrosophic nano δ open set, Pythagorean neutrosophic nano δ interior, Pythagorean neutrosophic nano δ closure.

Abstract

This paper introduces a novel class of generalized open sets, namely Pythagorean neutrosophic
nano δ-open sets and their variants (δ-pre, δ-semi, δα, and δβ), within Pythagorean neutrosophic nano topo
logical spaces, extending classical topological concepts to handle uncertainty, indeterminacy, and vagueness
simultaneously. The fundamental properties of these generalized open and closed sets are rigorously analyzed,
and corresponding closure and interior operators are defined to provide essential tools for topological analysis
under uncertainty. Furthermore, the hierarchical interrelations among these sets are established, demonstrating
that δ-open ⇒ δα-open ⇒ δS-open / δP-open ⇒ δβ-open, with counterexamples confirming that converses
do not hold. Finally, a machine learning application is presented to demonstrate the practical utility of con
structing Pythagorean neutrosophic nano topological spaces from real-world datasets for classification and
decision-making under incomplete information.

DOI 10.5281/zenodo.19701310