On Alexandroff Shadow Spaces

Each Alexandroff space X has a corresponding shadow space [X] which is T0 Alexandroff space. In this paper, we study Alexandroff spaces and their properties via their shadow spaces. The definitions and the concepts such as Artinian, Noetherian, maximal points and minimal points, that are defined on T0 Alexandroff space carry over to any Alexandroff space. We prove that an Alexandroff space X is connected (compact) iff its shadow space [X] is connected (compact). Moreover, X need not be scattered or -scattered. We give a study of preopen, semi-open, and α-open sets on X.