الفهرس 
Some Basic Concepts in Topological, Bitopological and Tritopological SpacesOnly 14 pages are availabe for public view
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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