Minimal Structure and Related Alexandroff Spaces

Abstract

Alexandroff topological space is a kind of topology which satisfies a stronger condition. Namely, arbitrary intersections of open sets is open. The main aim of this thesis is to study the concepts of an Alexandroff topological spaces which related to a family of subsets of a nonempty set called minimal structure. We study some families of a new type of minimal structure and construct a new collection of sets related to this minimal structure such as ∧m and Vm which will be Alexandroff spaces. We study some of topological properties using previous studies of Alexandroff spaces. We give a characterization of the order which generated by these topological spaces. We prove that the topologies generated by this Order are the same topologies Λm on X. Finally we study some mappings related to these new type of topologies related to m-structure.