Quotient S Image Results

Tracing to the first Brague meeting of 1961, Alexandroff imagined that spaces could be studied by mappings . Then in 1966,the imagination was herited and developed by a comprehensive paper named " Mappings and Spaces " and written by Arhangel'skii.According to the conception of Alexandroff-Arhangel'skii,the main content of using mappings to study spaces is to found a comprehensive connection between metric spaces and spaces with specific topology property by means of mappings and studing the inner property of images which are under all kinds of mappings of spaces .Which mapping can preserve the specific spaces is studied also.In this paper,we mainly discuss some propositions about the quotient s-images of metric spaces and locally separable metric spaces.This paper is constituted by five parts.In chapter 1, it shows that the material of related background which is concerned with the development and some gained propositions.In chapter 2, some definitions and theorems are included which are used in this paper.In chapter 3, after the definition of weak Lindelof is introduced , its basic properties are studied . Then we show the inner character description about the quotient s-images of metric spaces .In chapter 4, after introducing the definition of relative cs-networks , we give a judgement theorem and a property theorem on quotient s-image of locally separable metric spaces.In chapter 5,we raise some questions.