الفهرس | Only 14 pages are availabe for public view |
Abstract In this thesis, a new structure in tritopological spaces which called tri 𝛿𝛽 open set is presented. These sets were generalized the usual notions of near open sets in tritopological spaces. Several topological characterizations and properties of the current new sort of sets were studied. We also characterized this new structure in nano tritopological space by introducing a new type of sets called nano tri 𝛿𝛽 open set. These sets are stronger than any type of the other nano near open sets (𝛽, 𝑏, 𝑆,𝑃, 𝛼). Tritopological approximation space as a generalization of classical approximation space is presented. A new type of tritopological approximation spaces using the near open sets(𝛿𝛽, 𝛽, 𝑏, 𝑆, 𝑃, 𝛼) were introduced by a family of binary relations defined on the universe set and their properties were studied. All these approximations were compared together, and we deduced that our model is the best of proposed models in this study. The thesis has been covered in four Chapters described as follows: Chapter 1 provides some basic definitions and some important theorems that are used throughout the thesis. Besides that, some illustrative examples are given. Chapter 2 is devoted to study tri 𝛿𝛽 open sets in tritopological spaces along with their several properties and characterizations. tri 𝛿𝛽 continuous and tri 𝛿𝛽 irresolute functions and some of their basic properties are discussed. Some new spaces in tritopological spaces, called tri 𝛿𝛽 − 𝑇𝑘 spaces, k = 0, 1, 2 are introduced and their properties and characterizations are analyzed. Chapter 3 mainly concerns with generalizing nano near open sets in nano tritopological spaces by introducing new type of sets called nano tri 𝛿𝛽 open sets. These sets are stronger than any type of the other nano tri near open sets (𝛽, 𝑏, 𝑆, 𝑃, 𝛼). The main properties and the relationships among these sets are discussed. In addition, the various forms of nano tri 𝛿𝛽 open sets corresponding to different cases of approximations are investigated. Moreover, the notion of nano tri 𝛿𝛽 continuous function is presented and compared to the other types of nano tri continuous functions. Chapter 4 presented tritopological approximation space as a generalization of Pawlak classical approximation space. We generalized Pawlak approximation space by family of binary relations to tritopological approximation space. Using the right neighborhoods and the left neighborhood of these relations we generated three topologies and using them we defined the tri-lower and tri-upper approximations of any subset in the universe. A new type of tritopological approximation spaces using the near open sets(𝛿𝛽, 𝛽, 𝑏, 𝑆, 𝑃, 𝛼) were introduced by a family of binary relations defined on the universe set and their properties were studied. We introduced tri 𝛿𝛽 model in tritopological approximation space and deduced that our model is the best of proposed models in this study as tri 𝛿𝛽 boundary region of any vague concept is decreased in a great degree. Some properties of rough sets on tritopological approximation spaces are studied. Finally, a real-life application example for Rheumatic Fever is given to reduce data to determine the least number of tests, but this does not reduce the efficiency of diagnosis |