Journal of Advances in Mathematics and Computer Science

On Alexandroff Shadow Spaces

Hisham Mahdi *

Depertment of Mathematics, Islamic University of Gaza, P.O. Box 108, Gaza, Palestine.

S. Nada

Department of Mathematics, Menoufia University, Shebeen El-Khom, Egypt.

Riyad Muamar

Department of Mathematics, Al Aqsa University, P.O. Box 4051, Gaza, Palestine.

*Author to whom correspondence should be addressed.

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Abstract

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.

Keywords: Alexandroff spaces, generalized open sets, α-open sets, preopen sets, shadow spaces

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