The de Groot dual of time scale

This thesis consists of five chapters. The first chapter is devoted to the introduction. In the second chapter, the basic definitions of time scale, the forward jump operator, the backward jump operator and the concept of continuity on time scale are given. In the third chapter, topological structure of time scales is investigated. The definition of neighborhood in terms of a time scale, the notions of T-openness and the closedness are given. In the fourth chapter de Groot dual of topology is studied. The theorems and proofs related to the closure, interior, and boundary of a set with respect to the de Groot dual topology are given. The fifth chapter is the original part of this study. In this chapter, the topological properties of a time scale with respect to the de Groot dual of usual topology of set of the real numbers R are studied and the theorems related to the closure, interior, boundary and connectedness of a time scale in dual space are stated and proved. Also, the de Groot dual of a time scale and a time scale with subspace topology induced from the de Groot dual of usual topological space of the real numbers are compared. All obtained results are exemplified by well-known examples of discrete and continuous time scales. In the sixth chapter of this thesis, a brief summary of this study is given and some suggestions are proposed for new investigations.