Pre-λ~*-Closed Sets And Weak Separability In Alexandroff Spaces

In 1940,Alexandroff obtained Alexandroff space by weakening the union requirements to countable union of open sets were taken to be open.Since then,many scholars have made extensive research on Alexandroff space and obtained many important conclusions.In this paper,on the basis of previous studies,we further study the preopen sets and their properties in Alexandroff space and its open subspace,and use the preopen sets to define some mappings of Alexandroff space,and then study the properties and relations among these mappings.We also define the pre-λ*-closed set and weak separability in Alexandroff space,and investigate their properties and the relationship between these separability.This paper is parted into three chapters.In Chapter 1,we introduce the background and research development of the preopen set in Alexandroff space,and give the main definitions,theorems and related symbols used in the paper.In Chapter 2,we study some properties of preopen sets in Alexandroff space and its open subspace.By using preopen sets,we define precontinuous mapping,weak precontinuous mapping,preopen mapping and weak preopen mapping,and discuss the relations and properties among these mappings.In Chapter 3,we give the concept of the pre-g*-closed set and the pre-λ*-closed set,and then define the pre-Tω space,the pre-Tω/4 space,the pre-T3ω/8 space and the pre-T5ω/8 space.And we give their properties and the relationship between these weak separations by using the pre-λ*-closed set.