Exploring the 𝜹 – Semiclosed Sets in Topological Spaces and Applications in Decision Making Problem
No Thumbnail Available
Date
2025-08
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
Avinashilingam
Abstract
This thesis is dedicated to the novel concept 𝜆𝑔 𝛿𝑠 - closed sets after defining Λ𝛿𝑆 – sets and (Λ,𝛿𝑆) – closed sets. Using Λ𝛿𝑆 – sets and (Λ,𝛿𝑆) – closed sets many characterizations of 𝛿 – semi D, 𝛿 – semi T, 𝛿 – semi R, extremally disconnected, hyperconnected and submaximal spaces are obtained. Some weaker forms of (Λ,𝛿𝑆) – open sets in the name of s(Λ,𝛿𝑆) – open, p(Λ,𝛿𝑆) – open, 𝛼(Λ,𝛿𝑆) – open and 𝛽(Λ,𝛿𝑆) – open sets are introduced and their properties are studied. 𝜆𝑔 𝛿𝑠 – closed sets are defined using the operator Λ and 𝛿 – semi closed sets which is situated precisely between δ - semiclosed sets and δgs - closed sets. The behaviour of 𝜆𝑔 𝛿𝑠 – closed sets in various spaces such as semi 𝑇1, semi-regular space, almost weakly Hausdorff spaces, and 𝑇3 4⁄ – spaces are discussed and interesting characterizations are obtained. The properties of 𝜆𝑔 𝛿𝑠 – closed sets are discussed and the new concept is compared with other closed sets in literature. Properties associated with 𝜆𝑔 𝛿𝑠- closed sets including neighborhoods, limit points, derived sets, frontiers, boundaries, exteriors, and saturated sets are rigorously analyzed. 𝜆𝑔 𝛿𝑠- open sets are investigated through grills in topological spaces. Furthermore, we examine the behaviour of λg δs- continuous functions, showing that while composition is not generally preserved, suitable modifications to the continuity conditions are given to the restore composition. This work also addresses various types of 𝜆𝑔 𝛿𝑠- continuity and irresoluteness. Two types of homeomorphisms namely λg δS – homeomorphisms and λg δS∗ - homeomorphisms, are developed and their properties obtained. Notably, it is observed that the collection of all λg δS∗ - homeomorphisms forms a group when composed together. The study of 𝜆𝑔 𝛿𝑠- closed sets are extended to two new frameworks: Hesitant fuzzy soft topological spaces and Pythagorean nano topological spaces. Lastly, we propose and illustrate an application of these concepts in multi-attribute decision making (MADM) in Pythagorean nano topological spaces, supported by a real-life example.
Description
Keywords
Mathematics