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 T0 spaces. We give a link between finite T0 spaces and the related partial order. Finally, we study some properties of multifunctions and their relationships with connected ordered topological spaces.
Mathematics Subject Classification. 54B10 54D30 (54A05 54G99).
Keywords. $T_0$ spaces, finite $T_0$ spaces, minimal neighborhoods, partially ordered sets, multifunctions, topological dimension, weak continuity.
Comments. This article is in final form.