The Mapping Theorems For Some Generalized Metric Properties And Countable Properties

Mappings and spaces are a huge and promising topic.For more than 20 years,the theory of generalized metric spaces has been growing and growing,producing many active spaces,and also obtaining their rich mapping theorems.It is necessary to understand the mapping properties of these spaces more comprehensively and find out the existing problems.The main results in this dissertation are the followings:Firstly,we select 30 generalized metric properties or countable properties,and find out their invariance or inverse invariance on the 8-class mappings in the published literature.Secondly,we mainly prove that point-G? properties,(?)0-snf-countability,weak quasi-first-countability,quasi-first-countability,csf-countability,snf-countability,gf-countability and sof-countability are invariant and inversely invariant under closed finite-to-one mappings.In addition,the closed finite-to-one mappings preserve g-second countable spaces,sn-second countable spaces,spaces with a point-countable wcs*-network,sn-symmetric spaces,etc.Thirdly,we prove that mappings preserve sequential separability,quotient mappings preserve kR-sapces,perfect mappings preserve c-semi-stratifiable spaces,(mod k)-metrizable spaces and quasi-(mod k)-metrizable spaces are inversely invariant under perfect mappings.Fourthly,we discuss the applications of the closed finite-to-one mappings in symmetric products.Finally,we give counterexamples to show that some topological properties are not preserved by determining mappings,which give negative answers to the mapping questions posed by Lin Shou and Shen Rongxin.