On finite T_0 topological spaces

Finite topological spaces became much more essential in topology, with the development of computer science. The task of this paper is to study and investigate some properties of such spaces with the existence of an ordered relation between their minimal neighborhoods. We introduce notations and elementary facts known as Alexandroff space. The family of minimal neighborhoods forms a unique minimal base. We consider T_0 spaces. We give a link between finite $T_0$ spaces and the related partial order. Finally, we study some properties of multifunctions and their relationships with connected ordered topological spaces.