الفهرس 
Chapter 1 . PreliminariesOnly 14 pages are availabe for public view
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Abstract
The study of mathematics history indicates that all of the mathematics
branches have been established for human needs and applications. Topology is
established as a result of problems in the geometry and mathematical analysis
which led to the discovery of topology and its different branches. For a long
time, the topological structure on a set X is a general tool for constructing
concepts related to the continuous functions, the open and closed functions,
compactness, connectedness and others.
In the last 20th century, new tools for the construction of the topological
concepts have been induced which are more accurate than the members of
topological structure . These tools are called near open sets such as preopen
[37], semi-open [31], semi-preopen [4] -open [41]. Also, there are many
of generalized closed sets in a topological space (X, ) such as g-closed [32],
gs-closed [6], sg-closed [9 -closed [36 -closed [35], gp-closed [42],
gsp-closed [18], spg-closed [53], g*-closed [54] and -closed [55]. These
notions were used for defining what is called near continuous functions by
many authors (cf. [5, 7, 10, 15, 18, 21, 31, 36, 37, 39, 53, 54, 55]).
It should be noted that our study contains three essential concepts. The
first concept is the bitopological space (X, 1, 2) was introduced by Kelly [25]
in 1963, as a generalization of a topological space (X, ). After the publication
of Kelly’s paper many topologists have shown interest in the study of such
spaces. For instance, El-Tantawy and Abu Donia [19] introduced some
concepts in generalized closed sets in bitoplogical spaces such as ij-gs-closed,
ij-sg-closed, ij- -closed, ij- -closed, ij-gp-closed, ij-gsp-closed, ij-spg-closed
and ij- -closed sets.
The second concept is the rough sets theory which was proposed by
Pawlak [45] in the early in 1982. The theory is a mathematical tool to deal with
vagueness and imperfect knowledge by using the concept of the lower and
upper approximations [45], which represent the interior and closure operators
for any clopen topological spaces. If the lower and upper approximations of the
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set are equal to each other, then it is called crisp (exact) set, otherwise, is
known as a rough (inexact) set. Therefore, the boundary region is defined
as the difference between the upper and lower approximations, and then the
accuracy of the set or ambiguity, depending on the boundary region is empty
or not, respectively. A non-empty boundary region of a set means that our
knowledge about the set is not sufficient to define it precisely. The main
aim of the rough set is reducing the boundary region by increasing the
lower approximation and decreasing the upper approximation. Recently, the
near open and near closed sets are considered as tools for measurement,
approximation, and explanation methods in information systems which
presented by Pawlak in the theory of rough sets.
Finally, the third concept is the fuzzy sets which were introduced by
Zadeh [61] in 1965. This concept provides a natural foundation for treating
mathematical the fuzzy phenomena, which exist pervasive in our real world,
and for building new branches of fuzzy mathematics. In 1968, Change [12]
introduced the concept of fuzzy topological spaces as a generalization of
topological spaces. Many topologists have contributed to the theory of fuzzy
topological spaces. Now fuzzy topology has been firmly established as one of
the basic disciplines of fuzzy mathematics. In 1989, Kandil [24] introduced the
concept of fuzzy bitopological spaces as a generalization of fuzzy topological
spaces. The near open and near closed sets are one of the most important
objects in the study of fuzzy topological and fuzzy bitopological spaces.
This thesis is devoted to:
(1) Introduce new types of generalized closed sets in topological spaces.
(2) Create new types of generalized closed sets in bitopological spaces.
(3) Initiate new approximation spaces by using the new generalized closed sets
in topological and bitopological spaces and make comparisons with the
previous approximation spaces.
(4) Introduce new classes of fuzzy sets by using the new generalized closed
sets in fuzzy topological and fuzzy bitopological spaces.
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(5) Give some applications on the new approximation spaces in the rheumatic
fever and the amino acids.
This thesis consists of six chapters:
Chapter one: This chapter presents a brief survey of the main important
definitions that help us to develop this thesis. It contains the basic concepts of
near open and near closed sets, and near continuous functions in each of
topological spaces, bitopological spaces, fuzzy topological spaces and fuzzy
bitopological spaces. Also, it contains the basic concepts and properties of
rough sets theory.
Chapter two: This chapter consists of seven sections. In section one, we
introduce a new class of sets called *- -closed sets
[41] -closed sets [36]. In addition, some of its properties are
studied. In section two, we use *-open and *-closed sets to define new types
of notions namely, *-closure, *-interior, *-boundary, *-exterior, *-dense,
*-nowhere dense and *-residual. Many characterizations and properties of
the previous notions are studied. In section three, we initiate new separation
axioms namely, 1 5 T , *
1 5 T , 1 5
*T , Te, Te, Tk, Tk, Tl and Tl spaces. Also, we
study the relationships between these separation axioms and give counter
examples. In section four, we construct new types of functions namely,
*-continuous, *-irresolute and pre- *-closed functions. Many of their
properties are studied. In section five, we investigate more properties of
-topological operators depend on -open and -closed sets which introduced
by Kumar [55] under the name of -open and -closed sets. In section six, we
offer new separation axioms namely, 1 6 T , 1 6 T , 1 6 T , Tq, Tq, Tr, Tr, Ts and
Ts spaces. The relationships between these separation axioms are studied and
some counter examples are introduced. In section seven, we investigate some
properties of -continuous, -irresolute and pre- -closed functions.
Some results of this chapter are:
M. A. Abd Allah and A. S. Nawar ” *-closed sets in topological spaces”
Wulfenia Journal, 21(9) (2014), 391-401.
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Chapter three: This chapter consists of eight sections. In section one, we
define a new class of sets called ij- *-closed sets in bitoplogical spaces
which is weaker than ji- -closed sets [23] and stronger than ij- -closed [19].
In addition, some of its properties are investigated. In section two, we use
ij- *-open and ij- *-closed sets to define new types of notions namely,
ij- *-closure, ij- *-interior, ij- *-boundary, ij- *-exterior, ij- *-dense,
ij- *-nowhere dense and ij- *-residual. The characterizations and properties of
these notions are studied. In section three, we give new separation axioms
namely, 1 5 ij -T , *
1 5 ij -T , 1 5
ij- *T , ij-Te, ij- Te, ij-Tk, ij- Tk, ij-Tl and ij- Tl
spaces. The relationships between these separation axioms are studied and
counter examples are presented. In section four, we initiate new types of
functions namely, ij- *-continuous, ij- *-irresolute and ij-pre- *-closed
functions. The main properties of these functions are studied. In section five,
we define a new class of sets called ij- -closed sets in bitopological spaces
which contains ji-semi-closed sets [34] and contained in ij-sg-closed [19]
and study some of its properties. In section six, we use ij- -open and ij- -closed
sets to define new types of notions namely, ij- -closure, ij- -interior,
ij- -boundary, ij- -exterior, ij- -dense, ij- -nowhere dense and ij- -residual.
The characterizations and properties of these notions are investigated. In
section seven, new separation axioms namely, 1 6 ij -T , 1 6 ij -T , 1 6 ij- T , ij-Tq,
ij- Tq, ij-Tr, ij- Tr, ij-Ts and ij- Ts spaces are introduced. The relationships
between these separation axioms are introduced and some of counter examples
are given. Finally, in section eight, we initiate new types of functions namely,
ij- -continuous, ij- -irresolute and ij-pre- -closed functions and investigate
their properties.
Some results of this chapter are:
H. M. Abu Donia, M. A. Abd Allah and A. S. Nawar ”Generalized
*-closed sets in bitopological spaces” Journal of the Egyptian
Mathematical Society, 23 (2015), 527-534.
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H. M. Abu Donia, M. A. Abd Allah and A. S. Nawar ”Generalized
-closed sets in bitopological spaces” submitted.
Chapter four: This chapter consists of two sections. In section one,
we create a new approximation space by using the concept of *-open
sets, which we call *-approximation space. Moreover, we introduce new
concepts based on *-approximation space namely, *-lower approximation,
*-upper approximation, *-boundary, *-rough set, *-dense, *-residual
and *-nowhere dense. The main properties of these concepts are presented.
In addition, we compare between our approximation and Kozae et al.’s
approximation [29 -approximation with respect to the accuracy.
We get from this comparison that our approximation is better than Kozae
et al.’s -approximation. Furthermore, an application of
*-approximation space in rheumatic fever is initiated. In section two,
we use 12- *-open sets to introduce a new type of approximations called
12- *-approximation space. Also, we define new concepts based on
12- *-approximation space namely, 12- *-lower approximation, 12- *-upper
approximation, 12- *-boundary, 12- *-rough set, 12- *-dense, 12- *-residual
and 12- *-nowhere dense. The main properties of these concepts are
investigated. Also, we measure the accuracy of our approximation and make
a comparison with the other approximations. from this comparison, we
conclude that our approximation is better than the other approximations.
Also, we apply the rheumatic fever’s example on ij- *-approximation spaces.
The results of this chapter are:
H. M. Abu Donia, M. A. Abd Allah and A. S. Nawar ”New
Generalization of Rough Set Approximations and Its Applications”
European Journal of Scientific Research, 130 (4) (2015), 360-375.
H. M. Abu Donia, M. A. Abd Allah and A. S. Nawar ” *-Rough
Approximation Spaces” Ayer Journal, 2 (2015), 252-272.
Chapter five: This chapter consists of two sections. In section one,
we initiate a new approximation space by using the concept of -open sets
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called -approximation space. Also, we define new concepts based on
-approximation space namely, -lower approximation, -upper approximation,
-boundary, -rough set, -dense, -residual and -nowhere dense. The main
properties of these concepts are constructed. Also, we compare between our
approximation and Kozae et al.’s approximation [29] and semi-approximation
with respect to the accuracy. We get from this comparison that our
approximation is better than Kozae et al.’s approximation and semiapproximation.
Moreover, an application of -approximation space in amino
acids is created. In section two, we use 12- -open sets to introduce a new
type of approximations called 12- -approximation space. Also, we give
new concepts based on 12- -approximation space namely, 12- -lower
approximation, 12- -upper approximation, 12- -boundary, 12- -rough set,
12- -dense, 12- -residual and 12- -nowhere dense. The main properties
of these concepts are offered. Also, we measure the accuracy of our
approximation and make a comparison with the other approximations. From
this comparison, we conclude that our approximation is better than the other
approximations.
The results of this chapter are:
M. A. Abd Allah and A. S. Nawar ” -Rough Approximation Spaces”
Applied Mathematics and Information Sciences, 10 (4) (2016),
1593-1601.
O. Tantawy, M. A. Abd Allah and A. S. Nawar ”Generalization of
ij- -open sets and its
applications” Accepted for Publication in Journal of Intelligent and
Fuzzy Systems.
Chapter six: This chapter consists of six sections. In section one, we
introduce fuzzy *-open and fuzzy *-closed sets in fuzzy topological spaces
and investigate many of their properties. In section two, we construct fuzzy
*-continuous and fuzzy *-irresolute functions and some properties of these
functions are obtained. In section three, we initiate new separation axioms
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namely, fuzzy 1 5 T , fuzzy *
1 5 T and fuzzy 1 5
*T and some of their properties are
studied. In section four, we introduce new classes of fuzzy sets in fuzzy
bitopological spaces called ij-fuzzy *-open and ij-fuzzy *-closed sets. The
main properties of these concepts are investigated. In section five, we define
new types of fuzzy functions namely, ij-fuzzy *-continuous and ij-fuzzy
*-irresolute functions. The main properties of these functions are presented.
In section six, new separation axioms namely, ij-fuzzy 1 5 T , ij-fuzzy *
1 5 T and
ij-fuzzy 1 5
*T are presented and some of their properties are studied.
The results of this chapter are:
O. Tantawy, M. A. Abd Allah and A. S. Nawar ” *-closed sets in fuzzy
topological spaces” submitted.
O. Tantawy, M. A. Abd Allah and A. S. Nawar ” *-closed sets in fuzzy
bitopological spaces” submitted.