الفهرس 
Basic ConceptsOnly 14 pages are availabe for public view
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Abstract
Molodtsov [1] inaugurated the introductory conception of soft set theory in 1999 and offered the first result of the theory. He has enticed numerous experimenters’ line of work this theory. In [2], he swimmingly utilized the soft theory in various orientations such smoothness of functions, game theory, operation research, Riemann integration, Perron integration, probability, and measure theory.
Over the past few years, P. K. Maji et al. [3] and I. Zorlutuna et al [4] have authored a multitude of papers that delve into the intricate realm of soft set theory, unravelling its potential applications across diverse fields, and further inquiry soft interior point and soft neighborhood and he first examined the compactness of soft topological spaces. Shabir and Naz [5] presented the idea of soft topological spaces derived from a predetermined set of parameters within an initial universe. Furthermore, Maji et al. [6] had proposed many operations for soft sets and so far, had revealed some basic features of these operations.
Aygunoglu and Aygun [7] showed the soft continuity of soft mapping, soft product topology and examined soft compactness and generalized Tychono Theorem to the soft topological space. Some results were given by Min [8] on soft topological spaces. In their study, Maji et al. [9] introduced the idea of the fuzzy soft sets, a unique combination of fuzzy sets and soft sets. Building upon this idea, Tanay and Kandemir explored the topological structure of fuzzy soft set in [10], providing a solid theoretical foundation for further investigation into this concept.
Kannan [11] assigned soft generalized closed sets and soft generalized open in soft topological spaces. Kharal A. and Ahmad [12] showed mappings on soft sets. I. Arockiarani and A. Arokialancy [13]
assigned soft β-open sets and continued to examine weak forms of soft open sets in soft topological spaces. Later, Akdag and Ozkan [14] assigned soft α-open and continuous. As well as examining the idea of a b-open set in soft sets [15]. Connectedness [16] was a powerful aid in topology S. Hussain [17] assigned and explored the features of soft connected space in soft topological spaces. Hameed, S. Z., Hussein, A. K [18] assigned the soft Ƅc-open set. Mahanta and Das [19] showed and characterized various forms of soft functions, such semicontinuous, irresolute and semi-open soft functions. S. Muthuvel, R. Parimelazhagan assigned b^*-continuous functions in topological space [20]. A. Poongothai, R. Parimelazhagan assigned sƄ^*-closed [21], strongly Ƅ^*-continuous [22] and separation axioms [23]. N. Gomathi [24] showed an Sb* Homeomorphism. The theory of selection rules is an area of mathematics with a rich history, dating back to the work of Menger et al [25] and Hurawicz [26, 27]. The systematic examination of selection principles began with the work of Scheepers [28]. This theory has great connections to sundry offshoots of mathematics, such as set theory, general topology, game theory, Ramsey theory, uniform spaces, hyperspaces, topological sets and etc.
For more information on the theory of the selection principle, see Kocinac [29]; LjDR [30], Babinkostova, L., Kocinac, L. D. & Scheepers, M. [31], A. E. Radwan et al. [32], Sakai et [33]. Soft ideals were first devised by Kandil et al. [34]. The demonstrates the notion of a soft local function. These concepts were debated with a view to find new soft topologies from the original one, named soft topological spaces with soft ideal (D,₸,Δ,ᶅ). H. I. Mustafa and Sleim [35] presented the notion of soft ideals in soft topological spaces. They introduced the notion of soft generalized closed sets in connection with soft ideals, thoroughly explored their properties, and extended the notion of soft generalized closed sets.
This thesis consists of six chapters:
In chapter one, we present and detail a series of fundamental concepts and establish results pertaining to both the theory of soft sets as well as soft topological space. These core foundations will be integral for subsequent discussions throughout this thesis.
In chapter two, we identify the soft Ƅ^*-closed and soft Ƅ^*-open sets. As well as we examine some characteristics and characterizations of these sets. We examine some of the new kinds of continuous functions named soft Ƅ^*-continuous functions and present the soft Ƅ^*-irresolute, debated their relations with soft continuous. We give examples and counterexamples with other soft continuous functions. Some regarding characteristics are discussed.
In chapter three, we provide identification of soft Ƅ^*-closed and soft Ƅ^*-open sets. This is paired with an in-depth examination of specified characteristics and characterizations inherent to these particular sets. We also explore certain innovative categories of continuous functions, specifically those referred to as soft Ƅ^*-continuous functions. Additionally, this chapter presents an analysis on relations between the conceptualised soft Ƅ^*-irresolute and conventional aspects of soft continuity. Through the provision of exemplary comparison samples and counterexamples involving other types of soft continuity-related functions, further understanding can be fostered. In conclusion, we engage in a focused discussion about various pertinent characteristics associated with these theories.
In chapter four, we examine some new kinds of soft separation axioms named soft strongly Ƅ^*-separation axioms. We display that, the features of soft strongly Ƅ^*-T_i space (i =0, 1,2) are soft topological features under the bijection, soft irresolute and soft continuous mapping. Furthermore, the feature of being soft strongly Ƅ^*-regular and soft strongly Ƅ^*-normal are soft topological features under bijection, soft continuous functions. As well as a new set of generalized soft open sets in soft generalized topological spaces as a generalization of compact spaces and soft Lindelӧf spaces, we identify the concept of soft strongly Ƅ^*-compact and soft strongly Ƅ^*- Lindelӧf spaces and we supply multiple interesting examples. As well as we indicate that the established sets remain unchanged when subjected to soft strongly Ƅ^*-irresolute mappings, and we investigate specific outcomes related to an expanded soft topology with the showing soft spaces. As well as we inquiry the features and characterizations of soft strongly Ƅ^*-connected spaces in soft topological spaces and discuss and identify their relationship with soft connectedness.
In chapter five, a new set of soft Menger and soft Hurewicz properties are shown, we identify the concept of soft strongly Ƅ^*-Menger property and present soft strongly Ƅ^*- Hurewicz property and examine some of their basic properties. We use this property to identify a soft strongly b^*-Menger game and soft strongly b^*-Hurewicz game and by using covering and soft strongly b^*-compactness to identify a new game referred to as by G_(b^* ) (T Ʌ,J ß). We check the features and characterizations of this game.
In chapter six, we present the soft strongly Ƅ^*-closed and open sets with regard to a soft ideal in soft topological spaces. We check the behavior of soft strongly Ƅ^*-closed sets with regard to a soft ideal relative to the union, intersection, and soft subspaces. We further examine the relationship between soft closed, soft generalized closed and soft strongly Ƅ^*-closed sets with regard to a soft ideal. As well as we identify the soft strongly Ƅ^* ᶅ-continuous functions and soft strongly Ƅ^* ᶅ-irresolute and present the soft strongly Ƅ^* ᶅ-open and soft strongly Ƅ^* ᶅ-closed with some features.
The results of this thesis are published in the journals (or are under publication):
Hameed, Saif Z., Fayza A. Ibrahem, and Essam A. El-Seidy. ”On soft b*-closed sets in soft topological space.” International Journal of Nonlinear Analysis and Applications 12.1 1235-1242., 2021.
Hameed, Saif Z., Abdelaziz E. Radwan, and Essam El-Seidy, On soft b*-continuous functions in soft topological spaces, Measurement: Sensors 27 (2023) 100777.
Hameed, Saif Z., Abdelaziz E. Radwan, and Essam El-Seidy. ” On soft strongly Ƅ^*-separation axioms.” Accepted in Iraqi Journal of Science 2023, to published in June 2024.
Hameed, Saif Z., Abdelaziz E. Radwan, and Essam A. El-Seidy. ” On soft strongly Ƅ^*-closed sets and soft strongly Ƅ^*-continuous functions in soft topological space.” Under the Publication, 2023.
Hameed, Saif Z., Abdelaziz E. Radwan, and Essam El-Seidy. ” On Soft Strongly b^*-Compactness and Soft Strongly b^*-Connectedness in Soft Topological Spaces”. Accepted in Scalable Computing: Practice and Experience, 2024, to published in September 2024.
Essam El-Seidy, Hameed, Saif Z. and Abdelaziz E. Radwan. ” Infinite game via soft strongly Ƅ^*-open cover”. Under the Publication, 2023.
Abdelaziz E. Radwan, Essam El-Seidy and Hameed, Saif Z., ” On soft strongly Ƅ^*-closed sets via soft ideal”. Under the Publication, 2023.