On bi-Alexandroff spaces

This thesis consists of five chapters. The first chapter is devoted to the general introduction to the thesis. In the chapter two, we recall the basic concepts of the partially ordered sets and its relation with Alexandroff spaces and investigate the fact concerning Alexandroff space and their relation to other topologies. In the chapter three, we give some definitions and theorems related to Alexandroff space. In the chapter four, the definitions of δ(1,2) -open set and δ(a,b) -open set in Bi-Alexandroff space are introduced. Then, some of the main properties of this set, such as δ(1,2)-int (A), δ(1,2)-cl (A), δ(1,2) -lim(A), δ(1,2) -nbh(A) and δ(1,2) – T1–space, are investigated. Also, a new definition of δ(1,2)-continuous for Bi− Alexandroff space is introduced. Furthermore, some results for continuous, open, and closed functions in Bi-Alexandroff spaces is given. Finally, chapter five contains some conclusions and future work.