Metric Space Can Count To 1 And ¦Ò-compact Mappings

The theory of mappings is very important in general topology, it's closely related with the nature of covering and the theory generalized matric space. It has been one of the hotest topics to research the characterization of images of metric spaces, under some certain maps in general topology. In this paper, We chiefly study countable-to-one maps andσ-compact maps, respectively the characterizations of sequence quotient countable-to-one images of metric spaces; sequence-covering countable-to-one images of metric spaces; a sequence quotientσ-compact image of a metric space; a sequence-coveringσ-compact image of a metric space are given.In chapter one, we introduce the background of this thesis and some basic concepts of generalized metric spaces.In chapter two, we investigate spaces with N0-type networks, present the concepts of quasi-cs*-first-countable spaces and quasi-cs-first-countable spaces. Also, give their characterizations by maps. At the some time, we introduce characterizations of sequence quotient countable-to-one and sequence-covering countable-to-one images of metric spaces.In chapter three, we lay out the characterizations of a sequence quotientσ-compact image of a metric space and a sequence-coveringσ-compact image of a metric space. We also discuss relationship between countable-to-one maps andσ-compact maps in separable metric spaces.In the forth chapter, conclusions and some problems to be studied further.