The Study On Cs-regular Covers And The Images Of Metric Spaces

Metric spaces can be characterized by the sequences of open covers.As one of the main methods of classifications of spaces,covering system has been accepted widely.This dissertation mainly focuses on the regular families and the characterizations about a sequence of covers of cs-metrizable spaces,did the following two parts of the work.In Chapter 2,we study the problem of characterizing regular families by the appropriate sequences of covers without adding sequence spaces.We introduce the notion of wsn-networks,cs-star networks and point-2-star networks,obtain some properties of cs-regular families and discuss the relations among cs-regular covers,cs-star networks of cs-finite covers and point-2-star networks of cs-finite covers in topological spaces,and give the characterizations about a sequence of covers of spaces with cs-regular cs-networks.In Chapter 3,we seek the internal characteristics of cs-metrizable spaces,and give the characterizations of spaces which have cs-regular cs*-networks.We introduce the notion of k-cs-mapping and let the cs-metrizable spaces can be described directly as the images of metrizable spaces under the certain mappings.Finally,we We introduce the notion of o-star networks and characterize a space with regular cs-networks by the sequence of covers?Our results extend the research from bases to cs-networks,even cs*-networks.It enriches the properties of regular families and cs-regular families.We further discover the roles of cs-covers or cs-fifinite families in general topology,and further confirmed the importance of the covering system in mutual classififications of spaces and mappings.