Of ¦Ò-tight Limited Network And Weak K Mapping

Since 1960s,the theory of generalized metric space has made great progress. It has been one of the hotest topics to seek characterization of images of metric spaces under some certain maps in general topology.The compact finite collection is one of important collections in generalized metric spaces.In this paper,we establish relations between metric spaces andσ-compact finite CS~*-networks,σ-compact finite CS-networks,σ-compact finite sequence neighborhood networks,σ-compact finite cfp-networks.In chapter one,we briefly introduce some basic concepts of generalized metric spaces,including fundamental operation properties and simple characterizations of these spaces,which provide some necessary preparation for the latter two chapters.In chapter two,we investigate spaces with variousσ-compact finite networks, present some of relations among these spaces and introduce relationship betweenσ-compact finite networks and other networks.In chapter three,we study internal characterizations of some weakly k-mapping images of metric spaces,and prove that a space X is an inductive sequentially quotient,weakly k-image of a metric space if and only if X hasσ-compact finite CS~*-networks;X is an inductive sequence-covering,weakly k-image of a metric space if and only if X hasσ-compact finite CS-networks;X is an inductive 1-sequence-covering,weakly k-image of a metric space if and only if X hasσ-compact finite sequential neighborhood-networks;X is an inductive compact-covering, weakly k-image of a metric space if and only if X hasσ-compact finite compact-finite-partition networks.Conclusions and some problems to be studied further are in the fourth chapter.