Weak Notions of open and closed Sets in Fuzzy Topological Spaces

Abstract

In the realm of mathematical abstraction and spatial reasoning, the exploration of fuzzy topology has been a captivating and intellectually stimulating journey. This thesis represents the culmination of extensive research, analysis, and contempla tion, providing a comprehensive examination of fuzzy sets and their topological implications. The genesis of this academic endeavor lies in a profound fascination with the nuanced and elusive nature of fuzzy sets. The inherent flexibility of fuzzy topology, offering a powerful framework for modeling uncertainty and imprecision, beckoned as a field ripe for exploration. The quest to understand the intricate interplay between mathematical abstraction and real-world uncertainty has been a driving force throughout this research. This thesis set out with a dual purpose: to contribute to the theoretical foundations of fuzzy topology and to shed light on the practical applications of novel concepts within this domain. The objectives were meticulously crafted to blend a deep dive into existing literature with the formulation of new notions, aiming to enrich our understanding of spatial relationships in the context of fuzzy sets. The structure of this thesis is designed as a guided expedition through the complex landscape of fuzzy topology. Each chapter unfolds progressively, building upon the foundations laid in the preceding ones. From an exploration of fundamental concepts to the definition and analysis of semi alpha regular weakly closed and open sets, the reader is taken on a journey toward a comprehensive understanding of these novel notions. In first chapter background and preliminaries of fuzzy sets and some basic operations on fuzzy sets, intuitionistic fuzzy sets, rough sets, a detailed comparison of fuzzy sets and rough sets and fuzzy topology and its application are discussed. In the second chapter Fuzzy semi alpha regular weakly closed sets in fuzzy set topology with its basic properties are introduced and explained with examples. And in the third chapter Fuzzy semi alpha regular weakly open sets and fuzzy semi alpha regular weakly interior, neighbourhood and limit points in vii fuzzy topological spaces are introduced. As this work is presented to the academic community, it is with the aspiration that it stands not only as a contribution to the theoretical landscape of fuzzy topology but also as an inspiration for future researchers. The dynamic nature of fuzzy mathematics offers a continual avenue for exploration, and this thesis serves as a modest endeavor to propel the field forward. As I reflect on the challenging yet rewarding nature of this journey, I am reminded that this thesis signifies not an end but a new beginning. The pursuit of knowledge in fuzzy mathematics is ongoing, and it is my hope that this work sparks curiosity and further inquiry into the fascinating realm of spatial reasoning in fuzzy sets