GENERALIZED CLOSED SETS AND THEIR APPLICATIONS IN FUZZY BITOPOLOGICAL SPACES

Abstract

Abstract of thesis presented to the Senate of Universiti Putra Malaysia in fulfilment of the requirement for the degree of Doctor of Philosophy GENERALIZED CLOSED SETS AND THEIR APPLICATIONS IN FUZZY BITOPOLOGICAL SPACES By ALHARBI, AHLAM AHMED M December 2024 Chairman : Adem Kilicman, PhD Faculty : Science The present thesis focuses on studying generalized closed subsets of fuzzy bitopology spaces, which is an emerging field of mathematics, especially in fuzzy bitopology. This field of study combines fuzzy sets, fuzzy bitopology, and generalizations of fuzzy closed sets to provide a thorough structure for describing and analyzing complicated systems. We introduce new results on generalized closed sets and their applicationsin fuzzy bitopological spaces. Thus, more profound insights into continuity, connectedness, and compactness are made possible, and a greater knowledge of functions, relationships, separation axioms, normal space, and regular space within fuzzy bitopological contexts is reached. This, in turn, provides a comprehensive framework or studying and evaluating complex systems. We began the thesis by recalling the fundamental concepts of fuzzy structures and basic theorems for the remainder of the thesis, providing a quick reference for the reader’s convenience. After that, we defined and introduced some kinds of generalized closed subsets of fuzzy bitopology spaces. The definition of a generalized closed setrelies on two factors: openness and closure. We established the open operator in the first fuzzy topology (𝜏𝑖), while the closure operator in the second topology (𝜏𝑗) varies across different types, such as fuzzy closure, fuzzy 𝛼-closure, fuzzy semi-closure, fuzzy pre-closure, and fuzzy 𝛽-closure. This creates different types of generalized closed subsets in fuzzy bitopological spaces, which we call (𝑖, 𝑗) − 𝑓𝑔𝜑−closed to make studying them easier, where 𝑖, 𝑗 ∈ {1, 2} and 𝑖 ≠ 𝑗. Then, we explored various theorems, properties, and their relationships with important counterexamples. On the other hand, delving into each theory or topic with the fuzzy sample (𝑖, 𝑗) − 𝑓𝑔𝜑−closed group is complex, comprehensive, and correct for all the types of generalized closed sets involved.Furthermore, this thesis introduces their application to mathematical structures. We started by introducing generalized neighborhood concepts using the (∈) connection and the quasi-coincident idea (𝑞) in fuzzy bitopological spaces. We also analyze them according to their properties, which are supported by counterexamples. In addition, we presented types of generalized functions, which consist of five sections: generalized continuous, generalized strong continuous, generalized irresolute, and generalized open and closed functions. The last section is the generalized homeomorphism, which includes some theorems, compositions, corollaries with important examples, and diagrams to show relationships between concepts. Moreover, this thesis presents definitions, results, and links related to the concepts of generalized compactness and generalized connectedness. Next, various types of fuzzy generalized separation axioms are studied by explaining several important theorems and counterexamples.In conclusion, new definitions of generalized regular space, generalized normal space, generalized 𝑇3, and generalized 𝑇4 are discussed. In addition, new theorems, basicproperties, and important counterexamples for both are defined. The results of this study are novel and more comprehensive because of the compact structure of generalized closed groups of fuzzy bitopological spaces. Additionally, the application of all previous mathematical concepts offers advantages in certain aspects. Keywords: Fuzzy generalized closed sets, fuzzy generalized neighborhoods, fuzzy generalized functions, fuzzy generalized compactness and connectedness, fuzzy generalized separation axioms.

Keywords: Fuzzy generalized closed sets, fuzzy generalized neighborhoods, fuzzy generalized functions, fuzzy generalized compactness and connectedness, fuzzy generalized separation axioms. SDG: GOAL 4: Quality Education, GOAL 9: Industry, Innovation and Infrastructure.