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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.” |