On gamma-Ps-operations in topological spaces

• Main Author: Asaad, Baravan Abdulmuhsen
• Format: Thesis
• Language: English; English
• Published: 2015
• Subjects: QA Mathematics
• Online Access: http://etd.uum.edu.my/5800/

Topology is one of the focus areas in mathematics. Recently, topology has become an important component in applied mathematics due to its vast applications in understanding real life problems. The basic concept of topological space (X, Ƭ) deals with open sets. Operations on Ƭ have been investigated by numerous researchers. Among these operations are γ-open, γ-preopen, γ-semiopen, γ-b-open, γ-β-open and α-γ-open which involve Ƭγ-Ps-interior and Ƭγ-Ps-closure. However, no one has attempted to define new class of set using operation γ on the topology Ƭ by combining the existing operations. This study, therefore, aims to define new classes of sets, construct new classes of functions, and introduce new types of separation axioms and spaces using the concept of γ-open sets. The new classes developed are γ-regular-open and γ-Ps-open sets. By applying γ-Ps-open sets and their complements, the notions of Ƭγ-Ps-closure, Ƭγ-Ps-interior, Ƭγ-Ps-derived set and Ƭγ-Ps-boundary of a set are established. The notions of γ-Ps-open and Ƭγ-Ps-closure sets are then used to define a new class of γ-Ps-open sets called γ-Ps-generalised closed sets. Moreover, several new classes of functions called γ-Ps-continuous, (γ,β)-Ps-continuous and (γ,β)-Ps-irresolute functions in term of γ-Ps-open sets are introduced. Furthermore, other types of γ-Ps-functions such as β-Ps-open and (γ,β)-Ps-open are constructed. In addition, some new classes of γ-Ps-separation axioms are established by using γ-Ps-open and its complement as well as γ-Ps-generalised closed sets. The relationships and properties of each class of sets, γ-Ps-functions and γ-Ps-separation axioms are also established. In conclusion, this study has succeeded in defining new classes of sets using operation γ on the topology Ƭ .