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Authors: Hussain, Sabir
Keywords: Natural Sciences
Issue Date: 2007
Abstract: In 1997, S. Kasahara [51] defined an operation α on Topological Spaces and introduced the concept of an α-closed graph of a function. In 1983, D. S. Jankovic [48] defined α-closed set and further investigated functions with α-closed graphs. In 1992, F. U. Rehman and B.Ahmad [91] introduced the notions of γ-interior, γ- boundary and γ-exterior points in the Topological Spaces and studied their properties. They studied properties and characterizations of (γ,β)-continuous mappings introduced by H. Ogata [88]. They also studied some interesting characterizations of (γ,β)-closed (open) mappings in Topological Spaces. In [41], S. Hussain investigated the basic properties of γ-operations in Topological Spaces by introducing γ-limit point, γ-derived set, γ-dense in itself, γ-nbd. and γ-nbd. system. H. Ogata [88], introduced the notions of γ-Ti spaces, for i=0, 1/2, 1, 2 and studied their properties. The properties of (γ,β)-continuous functions have also been studied in General Topological Spaces as well as in γ-T2 spaces. The concept of γ-convergence of a sequence, and its properties have been defined and investigated in γ-T2 spaces. Concepts like γ*-regular space in Topological Spaces have also been defined and their properties in γ-T2 space have been explored in [8]. The study of semi-open sets and their properties was initiated by N. Levine [63] in 1963. The introduction of semi-open 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 defined and examined. Many new properties and characteristics of classical notions have been studied. The purpose of this thesis is to study these notions in terms of γ-operations in Topological Spaces We divide the work into seven chapters. In 1975, Maheshwari and Prasad [67], have defined new axiom called s- regularity, which is strictly weaker than regularity (without T2). In 1982, C. Dorsett [30], defined and investigated a separation axiom called semi-regular space. It is shown [30] that s-regularity is weaker than semi-regularity. A new class of regularity called s*-regular spaces, PΣ and weakly PΣ spaces, locally s-regular space, P-regular space and γ*-regular space have been defined and studied in [19], [52], [59] and [8]. In chapter 1, we discuss the characterizations and properties of γ- convergence, γ*-regular, γ0-compact, γ-locally compact and γ-normal spaces. In section 2, we investigate the characterizations of γ-convergence, γ*-regular spaces defined in [8]. In section 3, we define and discuss the γ0-compact space, which is the generalization of compact space, and study the properties of γ0-compact space in (γ,β)-continuity defined and investigated by H. Ogata [88] and further studied by F. U. Rehman and B. Ahmad [91]. Several properties and characterizations of γ0- compact space have been explored in this section. In section 4, we define and investigate γ-locally compact space in General Topological Space as well as in γ-T2 space [88]. It is interesting to mention that every γ0-compact space is a γ-locally compact space. In section 5, we define γ-normal space which is independent of normal space. We study its properties and characterizations in γ-T1 spaces under (γ,β)- continuous functions defined in [88]. In chapter 2, we define a new space called γ-connected space. It is remarkable that the class of connected spaces is the subclass of class of γ-connected spaces. In section 2, we study the characterizations and properties of γ-connected spaces and then properties under (γ,β)-continuous functions [88]. In section 3, we define and explore the characterizations of γ-components in a space X. In section 4, we define and discuss a new notion called γ-locally connected space which generalizes locally connected space. In section 5 and 6, we define and investigate γs-regular and γs- normal spaces. Here we also study the relation of γ0-compact, γ-T2 spaces and γs- normal spaces. In chapter 3, we define γs-connected space and γs-locally connected space and analyze their many interesting properties and characterizations. We also define and explore the properties of γs-components in a space X. In 1992 (respt. 1994), J. Umehara, H. Maki and T. Noir (respt. J. Umehara) [97] (respt. [98]) defined and discussed the properties of ( γ,γ ′)-open sets, ( γ,γ ′)- closure, and ( γ,γ ′)-generalized closed sets in a space X. In chapter 4, we continue to discuss the properties of (γ,γ ′)-open sets, (γ,γ ′)-closure, (γ,γ ′)-generalized closed sets [97] which generalizes the γ-open sets, γ-closure and γ-generalized closed sets defined by H. Ogata [88] and further investigated in [91] and [7]. It is interesting to Remark 4.2.9 that the class of (γ,γ ′)-open sets contains the class of γ-open and the class of γ ′- open sets. In section 2, we define and discuss the properties of (γ,γ ′)-interior, (γ,γ ′)- closure and (γ,γ ′)-boundary. In section 3, we define and explore many interesting properties of τ(γ, γ ′ ) - cl (A) and (γ,γ ′)-generalized closed sets [97]. It is necessary to mention that τ(γ, γ ′ ) - cl (A) generalizes τγ - cl (A) defined by H. Ogata [88]. We also examine the relation of τ(γ, γ ′ ) - cl (A), cl(A), clγ(A ) and cl(γ, γ ′) (A) in Theorem 4.3.14 (1). In section 4, we define and explore the properties of (γ,γ ′)-nbd and (γ,γ ′)-nbd base at x which generalizes γ-nbd and γ-nbd base at x defined in [7]. In section 5, we define (γ,γ ′)-T1 space and describe many of their characterizations and properties. We also define and explore (γ,γ ′)-derived sets which generalizes γ-derived sets defined in [7]. In chapter 5, we define a new class of continuous functions called Bi (γ,β)- continuous functions and investigate several properties and characterizations of Bi (γ,β)-continuity and Bi (γ,β)-open (closed) functions. In 1963, Levine [63] defined semi-open sets in a space X and discussed many of its properties. In 1997 (2005), A. Csaszar [25-26] defined Generalized Topological Spaces. In 1975, Maheshwari and Prasad [61] introduced concepts of semi-T1 spaces and semi-R0 spaces. In 2005, A. Guldurdek and O.B. Ozbakir [40] defined and discussed γ-semi-open sets using γ-open sets in Topological Spaces which are different from the notions of γ-open sets introduced and studied by H. Ogata [88] in 1991. So far several researchers worked on the findings of H. Ogata and a lot of material is available in the literature. In sections 2 and 3 of chapter 6, we introduce the concept of γ*-semi-open (which generalizes γ-open sets defined in [88]), γ*-semi- closed sets and γ*-semi-closure, γ*-semi-interior sets in a space X in the sense of H. Ogata [88]. It is also shown that the concept of semi open sets and γ*-semi-open sets are independent of each other. In view of the findings of [40], we also introduce γ Λγs − set and Λs − set by using γ*-semi-open sets. Moreover, we show that the concepts of g. Λ s − set , g. V s − set , semi-T1 space and semi-R0 space can be generalized by replacing semi-open sets with γ*-semi-open sets for an arbitrary monotone operator γ∈Γ(X). In section 4, we discuss the several properties of γ*-semi- open sets by defining and studying γ*-semi-interior, γ*-semi-closure, γ*-semi- boundary and their relations between them. In 1963, Levine [63] defined the notion of semi-continuous function. Since then, this notion has been extensively investigated. Cameron and Woods [23] and Abd El-Monsef et-al [1] have independently defined s-continuous and strongly continuous functions respectively. In 1994, M. Khan and B. Ahmad [55] introduced almost S- continuous functions. They showed that almost S-continuous have certain similar properties to those of strongly θ-continuous functions obtained by Long and Herrington [65]. In section 2 of chapter 7, we introduce and investigate the notion of γ-semi-continuous function. It is shown that γ-semi-continuous functions have certain similar properties to that of semi continuous functions [63]. Although γ-semi- continuous functions and semi continuous functions are independent of each other. In section 3, we define and explore many interesting properties and characterizations of γ-semi-open (closed) functions. In section 4, we define γ*-irresolute functions and discuss the properties and characterizations in terms of γ*-semi-derived sets and γ- semi-T2 space In section 5, we define and study the γ-pre-semi-open (closed) functions in space X. We explore the properties and characterizations of them in terms of γ*-semi-interior, γ*-semi-closure, (γ,β)-continuous, and (γ,β)-open (closed) functions [4], [88], [91]. In the end, we study the relationship between γ-pre-semi- open (closed) functions and γ*-irresolute functions.
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