In this paper a survey on tri-quotient maps is given, some relations among tri-quitient maps, open maps, compact-covering maps and inductively perfect maps are discussed, and several interesting questions are posed.

By proving the rationality of quotient mapping s definition,and proving and discussing the continuity and homeomorphism of quotient space mapping,gets the quality of quotient space mapping,provides the precise theoretical foundation for popularization and application of quotient space mapping.

Some discussions on images of locally compact Lindelof spaces are presented by means of compact covering mappings,closed mappings,and quotient mappings,and some characterizations are given to the spaces with a kind of spaces with k system.

In this paper,the relationships between metric spaces and spaces with a σ-compact-finite weak base are established by certain quotient mappings,which partially answers a problem posed by C.

In this paper, we establish the relationships between locally compact metric spaces and all kinds of spaces with mk system by means of quotient mappings, pseudo open mappings and closed mappings.

In this paper by discussion,we obtain that the topology on Y is the largest topology for f being continuous if and only if f is a quotient map.

Used the theorem of exponential correspondence,a theorem on quotient map was generalized to the cases of coinduced topology, and some applications of it in algebraic topology were also discussed.