Exploring $ \delta\beta $-Connectedness within Pythagorean Fuzzy Nano Topologies

Abstract

The concept of fuzzy numbers, which represents non-probabilistic uncertainty, enables fuzzy set theory to effectively handle uncertain circumstances in a set format, a feat that classical set theory struggles with. Topologists have adopted fuzzy sets and integrated them with topological concepts to expand and apply these innovations to meet human needs and drive development. The intuitionistic fuzzy set concept highlights the importance of non-membership in scenarios, complementing the fuzzy concept. Furthermore, Pythagorean fuzzy sets, which consider both membership and non-membership, are particularly effective in refusing indeterminacy and addressing uncertain scenarios that intuitionistic fuzzy sets may not fully encompass. Nano topology can be combined with Pythagorean fuzzy sets to create Pythagorean fuzzy nano topology, which allows for the study of uncertain spatial relationships. This study introduces the concept of Pythagorean fuzzy nano (resp. $ \delta $, $ \delta \mathcal{P} $, $ \delta \mathcal{S} $, $ \delta \alpha $ and $ \delta \beta $) connected and respective disconnectedness in Pythagorean fuzzy nano topological spaces. We also present several properties and theorems related to these concepts, specifically in the context of Pythagorean fuzzy nano connected spaces. Artificial Intelligence can use Pythagorean fuzzy sets to analyze and process, enabling more accurate decision-making in uncertain environments. We have provided an illustrative example of decision-making in a Hybrid model biometric authentication security system.