Operation Approach of 𝛿𝑃s -Open Sets in Topological Spaces, Fuzzy Topological Spaces and Nano Topological Spaces

Shanmugapriya H; Guide - Dr. K.Sivakamasundari

Operation Approach of 𝛿𝑃s -Open Sets in Topological Spaces, Fuzzy Topological Spaces and Nano Topological Spaces

dc.contributor.author	Shanmugapriya H
dc.contributor.author	Guide - Dr. K.Sivakamasundari
dc.date.accessioned	2025-08-20T05:13:50Z
dc.date.available	2025-08-20T05:13:50Z
dc.date.issued	2024-11
dc.description.abstract	The scope of this thesis is to introduce a new operation 𝜚 on the collection of 𝛿𝑃𝑆-open sets of 𝑊, 𝛿𝑃𝑆𝑂(𝑊, 𝜏), into the power set 𝒫(𝑊) of 𝑊 in a way that 𝐻 ⊆ 𝐻𝜚 for every 𝐻 ∈𝛿𝑃𝑆𝑂(𝑊), where 𝒫(𝑊) is the power set of 𝑊 and 𝐻𝜚 is the value of 𝐻 under 𝜚 and to study 𝜚-open sets and all its topological properties. The concept of 𝜚-regular spaces and two types of closures which are 𝑐𝑙𝑜𝑠𝑢𝑟𝑒𝜚, 𝜚-closure of sets are analysed. Some of topological properties on 𝜚-open sets such as limit points, derived sets, neighbourhood, interior, kernel,exterior, boundary, frontier and saturated sets are obtained. The idea of grills is extended to describe a new topology connected to grill 𝜚-space. Similar to the operation 𝜚, another operation 𝜄 on the collection of 𝛿-preopen subsets is introduced and 𝜄-open sets for each of 𝛿𝑃-open set are obtained. Yet another concept of open sets called (𝜄)-open sets using operation 𝜄 for the sub collection 𝛿𝑃𝑆𝑂(𝑊) of 𝛿𝑃𝑂(𝑊) is introduced and relations between 𝜚-open sets and (𝜄)-open sets are discussed. 𝜚-𝑇𝑛 and 𝜚-𝑇′𝑛 for 𝑛 = 0,1,2 are studied with their properties and relationships with other spaces. The theory of generalized closed sets is extended to 𝜚-open sets and 𝜚-generalized closed sets and 𝜚-𝑇1/2 space is defined and its properties are studied. Symmetric spaces, 𝜚-𝑅0 and 𝜚-𝑅1 spaces are introduced with their properties. Two types of continuities 𝛿𝑃𝑆-(𝛼𝜚, 𝛽𝜚)-continuity and (𝛼𝜚, 𝛽𝜚)-continuity are introduced and their properties are focused and compared. Bioperation concepts are introduced for 𝜚-operation. In this connection [𝜚, 𝜚′]-open sets and [𝜚, 𝜚′]-regular spaces are defined and studied. Two types of [𝜚, 𝜚′]-closures are obtained and their properties are discussed. Various spaces like submaximal spaces, extremely disconnected spaces are introduced for bioperation and a characterization theorem is proved via convergence. Some interesting concepts like weakly continuity and somewhat continuity are extended to bioperation. In fuzzy topological spaces and in nano topological spaces 𝜚-open sets are established along with their properties. An edge detection was studied using 𝜚-open sets and Fuzzy 𝜚-open sets. An situation was solved using nano 𝜚-open sets.
dc.identifier.uri	https://ir.avinuty.ac.in/handle/123456789/17876
dc.language.iso	en
dc.publisher	Avinashilingam
dc.subject	Mathematics	En
dc.title	Operation Approach of 𝛿𝑃s -Open Sets in Topological Spaces, Fuzzy Topological Spaces and Nano Topological Spaces
dc.type	Learning Object