𝔑 𝑆 𝔇𝔏𝔑ℓ and 𝔑𝐐𝐒𝔇𝔏𝔑ℓ- Spaces in 𝔇𝔏-Nano Topological Space

Objectives: The purpose of this study is to propose a new kind of Nano Ideal topological space called 𝔇𝔏-Nano topological space, explores several novel forms of open and closed sets in the context of 𝔇𝔏-Nano topological space, and generate 𝔑 𝑆 𝔇𝔏𝔑ℓ and 𝔑𝐐𝐒𝔇𝔏𝔑ℓ-spaces in 𝔇𝔏-Nano topological space. Methods: The 𝔇𝔏-lower, 𝔇𝔏-upper and 𝔇𝔏-boundary regions have been established for any subset of the specified universal set and any binary relation defined on it. The associated 𝔇𝔏-Nano topological space can be determined by these regions in addition with universal set and the empty set. Findings: The relationship between 𝔇𝔏-Nano open sets and other sets in 𝔇𝔏- Nano topological spaces are found. Additionally, we figure out the forms of 𝔑 𝑆 𝔇𝔏𝔑ℓ and 𝔑𝐐𝐒𝔇𝔏𝔑ℓ-spaces. Novelty: Nano-open sets were used in the formation of normal spaces in Nano topological spaces, whereas in the current researchwork𝔑𝔇𝔏 and𝔑𝑆̄𝔇𝔏-open sets have been used to generate 𝔑 𝑆 𝔇𝔏𝔑ℓ and 𝔑𝐐𝐒𝔇𝔏𝔑ℓ -spaces. Keywords: 𝔇𝔏-Nano topological space; 𝔇𝔏-lower, 𝔇𝔏-upper; 𝔇𝔏-boundary region, 𝔑 𝑆 𝔇𝔏𝔑ℓ -space, 𝔑𝐐𝐒𝔇𝔏𝔑ℓ-space