New Generalizations Of Continuous Functions And Multifunctions Via New Generalizations Of Open Sets

The concept of continuity is a significant and basic topic in the theory of classical point set Topology in addition to several branches of mathematics of course it’s weak and strong forms of continuity are also important. Semi-open sets, a-open sets, preopen sets, semi-preopen sets, b-open sets, b-9-open sets, β-θ-open sets,8-preopen sets, e-open sets, e-open sets, e-θ-open and e-θ-open sets play an important role in generalization of continuity in Topological spaces. Many authors introduced and investigate various types of modifications of generalization continuity by using these sets. A great number of papers dealing with such functions have been appeared, and a good number of them have been extended to the setting of Multifunction and generalized by weaker forms of open sets such as a-open sets, semi-open sets, preopen sets, b-open sets, semi-preopen sets and8-preopen sets. This implies that both functions and Multifunctions are important tools for studying the properties of spaces and for constructing new spaces from previously existing ones. Moreover, generalized open sets play a very important role in general Topology and they are the research topics of many topologists worldwide. Indeed a significant theme in general Topology and real analysis concerns the variously modified forms of continuity.On the other hand, Topology plays a significant role in quantum physics, high energy physics and superstring theory, one can observe the influence of general Topological spaces also in computer science and digital Topology, computational Topology for geometric and molecular design, Thus we studies Some new forms of continuity Which relate to functions and Multifunctions in Topological spaces. This dissertation consists of six chapters as follows:A short historical introduction about several types of generalized strongly continuous functions, some kinds of generalized closed spaces, multiple types of generalized continuous functions in Topological spaces, different forms of generalized upper and lower continuous Multifunctions and the motivation of the dissertation are introduced in chapter One, The general basic definitions and well-known results which are used in the succeeding chapters are given in chapter Two, as well as some of the preliminary definitions and results which are relevant to each chapter are given in the beginning of the corresponding chapter itself.In chapter Three, we introduce and investigate a new strong form of continuity which named strongly θ-e*-continuous functions by using two new strong forms of e*-open sets namely an e*-regular sets and e*-0-open sets. By using these sets we obtain several characterizations and some fundamental properties of strongly0-e-continuous functions. This class is a generalization of both strongly θ-e-continuous functions which are proposed by Ozkoc Murad and Giilhan Aslim, and strongly θ-β-continuous functions which is proposed by Noiri and Popa. Furthermore, the relationships between strongly θ-e*-continuous functions and other well-known types of strong continuity are also discuss, By using the notion of e*-open sets we investigate the relationships between strongly θ-e-continuous functions and separation axioms and introduce some covering properties by using two new forms of e*-open sets are called e*-closed space and countably e*-closed space. Additionally, we investigate the graphs of strongly θ-e*-continuous functions in Topological spaces.In chapter Four, we offer several characterizations of e*-closed space which was introduced in chapter three including characterizations using nets and filter bases, These are parallel to characterizations of other generalizations of compactness such as s-closed spaces, p-closed spaces and f-closed spaces, In addition we study and discuss a new class of continuous functions in Topological spaces is called θ-e*-continuous functions which transforms e*-closed spaces to quasi H-closed spaces and which contains the class of strongly θ-e*-continuous functions. Some characterizations and basic properties regarding θ-e*-continuous functions are obtain, and by means of e*-open sets we introduce and characterize a new regularity axioms and examine the relationships between θ-e*-continuous functions and separation axioms. Moreover, we investigate a comparison among θ-e*-continuous functions and some other well-known forms of continuous functions in Topological spaces, also we discuss the graphs of θ-e*-continuous functions.The notions of e-continuous functions and e-continuous functions are extend to Multifunctions via e-open sets and e*-open sets in chapter Five, as well as, we introduce and study two new classes of continuous Multifunctions in Topological spaces which called upper (lower) e-continuous Multifunctions and upper (lower) e*-continuous Multifunctions by using the concepts of e-open sets and e-open sets which have been suggested by Erdal E. The class of upper (lower) e-continuous Multifunctions is stronger than upper (lower) e-continuous Multifunctions and generalization of upper (lower)8-precontinuous Multifunctions, and the class upper (lower) e-continuous Multifunctions is a generalization of upper (lower) β-continuous Multifunctions and upper (lower) e-continuous Multifunctions. We obtain several characterizations of upper (lower) e-continuous Multifunctions and upper (lower) e*-continuous Multifunction and present some of their fundamental properties. Furthermore, the relationships between upper (lower) e-continuous Multifunctions, upper (lower) e-continuous Multifunctions and other well-known kinds of continuous Multifunctions are also discuss.In chapter Six, we introduce and investigate a new classes of Multifunctions namely upper and lower faintly e-Continuous Multifunctions and upper and lower faintly e*-Continuous Multifunctions, as a generalization of upper and lower e-Continuous Multifunctions and upper (lower) e-Continuous Multifunctions respectively, due to Alaa, M. F. Al-jumaili and Xiao-Song Yang by utilizing the concepts of e-open sets and e-open sets respectively, additionally we obtain some characterizations and several properties concerning of these new generalizations of continuous Multifunctions in Topological spaces.Finally, chapter Seven concludes on the findings in chapters Three, Four, Five and Six, as well reflects the extensions and possible use of these functions and Multifunctions in other areas.