  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.
