Abstract The study of semiopen sets and their properties was initiated by N. Levine [39] in 1963. The introduction of semiopen sets raised many basic general topological questions, which has thus far led to a productive study in which many new mathematical tools have been added to the general topology tool box, many new notions have been studied. The purpose of this thesis is to study these new notions. We divide the work into four chapters. In chapter 1, we define semiboundary of a subset A in a space X and study its properties. Moreover, we characterize semicontinuous and semiopen mappings in terms of semiderived and semiboundary. In Chapter 2, as a generalization of sclosed spaces due to G. Di Maio and T. Noiri [17], the notion of locally sclosed spaces has been introduce and investigated. In 1976 Thompson [67] introduced the notion of Sclosed spaces by using the semiopen sets. In [49], Noiri introduced and investigated Sclosed subspaces and locally Sclosed spaces. The class of Sclosed spaces is contained in the class of Sclosed spaces. Noiri [17] further showed that cdcompactness due to Carnahan, weak RScompactness due to Hong and sclosedness are all equivalent. Sections 1 and 2 contain the brief introduction and definitions which are building blocks of notions given in the squel. In section 3, we define and study sclosed subspaces. We also investigate some properties of sets sclosed relative to a topological space. Noiri [49] generalized Sclosed spaces due to Thompson [67] and introduced and investigated the notion of locally spaces. In section 4, we introduce a new class of spaces called locally sclosed spaces and investigate its fundamental properties. We also find some characterizations of locally sclosed spaces in Hausedorff spaces. In section 5, we obtain some preservation theorvation theorems by using quasiirresolute functions and semi Oclosed functions. Semi Operfect function has also been defined and its characteristics have been found. It is observed that the inverse image of sclosed set relative to a space is also sclosed relative to a space. In section 1 of chapter 3, the brief introduction is given. In section 2, we define a new class of topological spaces called s*regular space. This class is contained in the class of almostregular spaces defined by Singal and Arya [63]. Several characterizations and properties of s*regular spaces have been explored in this section. In section 3, PÎ£ and weakly PÎ£ spaces have been studied. It is also pointed out that PÎ£ spaces are weaker than regular spaces and weaklyPÎ£ spaces are weaker than that of PÎ£spaces. We further investigate the properties of PÎ£ spaces and weaklyPÎ£ spaces. Prealmost open, pre almostclosed and regular open functions are also defined and studied to obtain several properties of PÎ£ and spaces. In section 4, we define the notion of locally sregular spaces. It is pointed out that the class of sregular spaces is proper subclass of locally sregular spaces. Some interesting results have also been proved by using weak and strong forms of continuity. In section 5, we define the strong form of regularity called Pregularity which implies semiregularity, due to C. Dorsett [22] as well as almostregularity due to Singal and Arya [63]. Strong scontinuous functions have also been defined and studied in this section. In Chapter 4, we study several weak forms of continuity. In 1963, Levine [39] defined the notion of a semicontinuous function. Since then, this notion has been extensively investigated. Cameron and Woods [12] and Abd ElMonsef et.al [1] have independently defined scontinuous and strongly semicontinuous functions respectively. We introduce and investigate the notion of almost scontinuous functions. It is shown that almost scontinuity is weaker than that of scontinuity. It is also shown that almost scontinuous functions have certain similar properties to those of strong Ocontinuous functions obtained by Long and Herringron [40]. Although, it is shown in remark 4.4.3 that almost scontinuity and strong Ocontinuity are independent of each other. In section 4, we investigate the relationship among almost scontinuous functions and several modifications of continuous functions. It is shown that a surjective almost scontinuous image of a connected space is semiconnected. In section 5, we define sclosed functions and study the properties and characterizations of sopen and sclosed functions, sopen functions were defined by D. E. Cameron and G. Woods [12].
