The Countable Products Of Some Generalized Metric Spaces,and Point-star Networks And Images Of Metric Spaces

In recent years,the theory of generalized metric spaces has been producing a lot of active spaces in the study.In order to understand properties of these spaces more comprehensively,it is necessary to find out the existing problems.This dissertation first introduces the research background and related concepts,and then mainly does the following three parts of the work:In Chapter 2,we mainly prove that some spaces in three kinds of topological spaces have the property of countable products,which contain point-countable property,like G?-property or like metric space propertiesIn Chapter 3,we introduce certain property(?)with some kinds of finiteness,the notion of mssnc-mappings and ?-cs-mappings.We also discusses the equivalent characterizations of the spaces which has a point-star network consisting of cs-coverings with property(?)and the spaces which has a point-star network consisting of cs*-coverings with property(?).They expand the research from bases to cs*-networks and unify some results of An and Tuyen.So we enrich the idea of mutual classification on mappings and spacesIn Chapter 4,we characterize the spaces which has a point-star network consisting of cs-finite cs-coverings and the spaces which has a point-star network consisting of cs-finite(or sn-finite)cs*-coverings,and also express their as certain images of metric spaces,respectively What's more,we give a partial answer to a question of Tanaka and Ge,and give a counterexample of the relevant results.