الفهرس 
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Abstract
L. Zadeh [35] introduced the theory of fuzzy sets in (1965) in order to be able
to obtain a more distinctive description of some phenomena than the one which
is offered by systems based on classical two-valued logic and classical set theory.
Atanassov [1, 2, 3, 4] introduced the concept of intuitionistic fuzzy sets as
a generalization of fuzzy sets. Coker [5] generalized topological structures in
intuitionistic fuzzy case. The concept of intuitionistic fuzzy topological spaces
was first given by Coker [6, 7].
Flou set stems from linguistic considerations of Yves Gentilhomme [13] about
the vocabulary of a natural language. The mathematical definition of flou sets
and binary operations on its are introduced by E. E. Kerre [21].
In 2005, the suggestion of J. G. Garcia et al. [10] that double set is a more
appropriate name than flou set, and double topology for the flou topology.
In 2007, Kandil et al. [20] proved the 1 − 1 correspondence mapping f between
the set of all flou (double) sets and the set of all intuitionistic sets defined as:
f (A1, A2) = (A1, A2), A2 is the complement of A2. Kandil et al. [19, 20] intro-
duced the concept of double sets, double points, double topological spaces and
continuous functions between these spaces. They also introduced separation ax-
ioms in double topological spaces. In 2009, Kandil et al. [19] introduced the notion
of double compact topological space and studied some fundamental properties of
this notion. In 2014, Kandil et al. [17] introduced some types of compactness in
double topological spaces.
In 1999, Molodtsov [27] introduced the concept of soft set theory as a gen-
eral mathematical tool for dealing with uncertain objects. In [27, 28], Molodtsov
successfully applied the soft set theory in several directions, such as smoothness
of functions, game theory, operations research, Riemann integration, Perron in-
tegration, probability, theory of measurement, and so on. After presentation of
the operations on soft sets [26], the properties and applications of soft set the-
iii
ory have been studied increasingly [15], [23], [28], [32]. In 2011, Shabir and Naz
[33] initiated the study of soft topological spaces. They defined soft topology on
the collection of soft sets over X and defined basic notions of soft topological
spaces such as open soft sets, closed soft sets, soft subspaces, soft closure, soft
nbds of a point, soft separation axioms, soft regular spaces and soft normal spaces
and established their several properties. Hussain and Ahmad [14] investigated
the properties of soft nbds and soft closure operator. They also defined and dis-
cussed the properties of soft interior, soft exterior and soft boundary which are
fundamental for further research on soft topology.
This thesis is devoted to
1. Introducing a note on Hausdorff spaces and a note on soft connected spaces.
2. Investigating some generalized of double separation axioms, introducing the
notion of double connected, and some deviations results in double topological
spaces.
3. Establishing the concept of soft double topology, soft double separation
axioms and generalized soft double sets.
4. Introducing some types of soft double connected spaces with some relations
between them.
5. Introducing the notion of the soft double ideal and soft double compactness.
6. Extending the notion of soft double compactness to soft double spaces via
soft double ideals.
7. Given comparisons between the current results and the previous one by using
counter examples.
This thesis contains 5 chapters:-
Chapter 1 is the introductory chapter. It contains also the basic concepts
and properties of topological spaces such as neighborhoods, closure, interior and
separation axioms. The basic concepts and properties of the double topological
space (DT S, for short) are presented. Further, this chapter contains the basic
notions related to soft sets and soft topological spaces. Also, in this chapter we
illustrate that the sufficiency of the Theorem 9.2.11 [30], is incorrect by giving a
counter Example and we show that, in [25] Remark 4.2, Example 4.3, Theorem
4.6 and Example 4.15 are not true, in general.
Some results of this chapter are published in:
iv
• “O. A. El-Tantawy, S. A. El-Sheikh and S. Hussien, A note on Hausdorff
spaces, South Asian Journal of Mathematics, (2017), 7 (2) 118−129.”
• “O. A. El-Tantawy, S. A. El-Sheikh and S. Hussien, A note on soft connected
spaces and soft paracompact spaces, International Journal of Scientific and
Engineering Research, accepted.”
In Chapter 2, we study some topological properties of double topological spaces
(DT −spaces, for short), also we introduce some generalized of double separation
axioms of DT −spaces based on double separation axioms [20]. Moreover, we in-
troduce some types of double connected spaces (D-connected spaces, for short)
such as q-double connected (qD-connected, for short), double C1−connected (DC1
−connected, for short), strongly (q-)double connected (strongly (q)D-connected,
for short), double hyperconnected (D-hyperconnected, for short), q-double hy-
perconnected (qD-hyperconnected, for short), double component (D-component,
for short) and q-double component (qD-component, for short). Some examples
are given to illustrate this notion. In addition, double T 2 −space presented in
DT −spaces. Furthermore, the deviations between the current work and the pre-
vious one [20] are explained by some counter examples.
Some results of this chapter are published in:
• “A. Kandil, O. A. El-Tantawy, S. A. El-Sheikh and S. Hussien, Some gen-
eralized separation axioms of double topological spaces, Asian Journal of
Mathematics and physics, accepted.”
• “A. Kandil, O. A. El-Tantawy, S. A. El-Sheikh and S. Hussien, Double
connected spaces, New Theory, (17) (2017) 1−17.”
In Chapter 3, the notions of soft double sets (SD-sets, for short), soft double
points (SD-points, for short) and soft double mappings are presented. Then, the
concept of soft double topological space (SDT S, for short) is introduced initially.
In addition, we present the concepts of soft double closure (resp. interior), soft
double neighborhoods, soft double separation axioms and soft double continuous
mappings (SD-continuous mappings, for short). The properties of the present
notions are studied and the relationships between them are given. The importance
of this approach is that, the class of soft double topological spaces (SDT −spaces,
for short) is wider and more general than the class of DT −spaces.
Some results of this chapter are published in:
• “O. A. El-Tantawy, S. A. El-Sheikh and S. Hussien, Topology of soft double
sets, Ann. Fuzzy Math. Inform., 12 (5) (2016) 641−657.”
v
• “O. A. El-Tantawy, S. A. El-Sheikh and S. Hussien, Some topological prop-
erties of soft double topological spaces, New Theory, 16 (2017) 27−48.”
The main purpose of Chapter 4 is to introduce the notion of soft double con-
nected (SD-connected, for short), and some types of this notion such as q-soft
double connected (qSD-connected, for short), soft double C1−connected (SDC1−
connected, for short), strongly (q-)soft double connected (strongly (q)SD-connected,
for short), soft double hyperconnected (SD-hyperconnected, for short), q-soft dou-
ble hyperconnected (qSD-hyperconnected, for short), soft double component (SD-
component, for short) and q-soft double component (qSD-component, for short).
Also, in this chapter, we prove that (X, τ , E) is a SD-connected if and only if it is
qSD-connected. Moreover, some basic properties of these concepts have obtained.
Some results of this chapter are published in:
• “O. A. El-Tantawy, S. A. El-Sheikh and S. Hussien, Soft connected of double
spaces, South Asian Journal of Mathematics, 6 (5) (2016) 249−262.”
• “O. A. El-Tantawy, S. A. El-Sheikh and S. Hussien, Generalized closed soft
double sets, Research and Communications in Mathematics and Mathemat-
ics Science, accepted.”
In Chapter 5, we introduced the concepts of soft double ideal (SD−ideal, for
short), soft double compact space (SD−compact space, for short) and we study
some of its basic properties. Also, we define the concept of ideal soft double
compact space (I − SD−compact space, for short), soft double compactively clo-
sure space (SDC−closure, for short), ideal soft double compactively closure space
(I − SDC−closure, for short), soft double closed compact space (SDC−compact
space, for short) and ideal soft double closed compact space (I − SDC−compact
space, for short). However, we illustrate the relationships between them.
The result of this chapter is published in:
“O. A. El-Tantawy, S. A. El-Sheikh and S. Hussien, Compactness of soft dou-
ble topological spaces, International Fuzzy Mathematics Institute, accepted.”