Locally Separable Metric Spaces And Weaker Metric Topology Spaces

In the paper [1], Zhou Lizhen gives characterizations of images of locally separable metric spaces under pseudo-sequence-covering s-mappings, but the proof is not perfect. And in the paper [2], J. Chaber gives characterizations of perfect images and open and compact images of spaces that can be mapped onto metrizable spaces by a mapping with fibers having a given property V.In this paper we introduce the concept of cs*-network of a subset to give new characterizations of pseudo-sequence-covering s-images of locally separable metric spaces. And we characterize pseudo-sequence-covering compact-images of locally separable metric spaces. We also consider sequence-covering compact-images and sequentially-quotient compact-images of weaker metric topology spaces.In this dissertation, we discuss some images of locally separable metric spaces and weaker metric topology spaces. It consists of four chapters.In the first chapter, we introduce the background and the objective of this dissertation.In the second chapter, we begin to point out that the proof of Theorem 12 ([1]) is not perfect. Then we characterize pseudo-sequence-covering s-images of locally separable metric spaces. We characterize pseudo-sequence-covering compact-images of locally separable metric spaces with the help of a lemma of [9]. By the characterization it is easy to get the characterization of quotient compact-images of locally separable metric spaces, which is given by Lin Shou et al ([3]).In the third chapter, we investigate spaces with a weaker metric topology that can be mapped onto metrizable spaces by a one-to-one mapping. We give characterizations of sequence-covering compact-images and sequence-quotient compact-images of weaker metric topology spaces.Conclusions and some problems to be studied further are in the fourth chapter.