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.

Semi-openness and almost-openness of induced mappings on interval;

In this paper, we point out and correct the errors in the paper by He Ming, entiled “The double induced mapping on the L Fuzzy set” (Kexue Tongbao, 1986, Vol, 31, No.

The author discuss the dynamical connections,such as Property P between the induced map and the original maps on inverse limit space and prove that the induced map has Property P if and only if all original maps have also.

We study the connections on chaotic properties in the sense of Li and Yorke, between the induced map and the original maps.

In this paper, we mainly discuss the dynamical properties of the induced map g∞ on the inverse limit space X∞ including weak specification and uniform positive entropy.

The notion of perfectly base-paracompact spaces is introduced and the following results are proved:(1) Let f:Z→Y be a perfect mapping,if Y is a perfectly base-paracompact space,then X is perfectly base-paracompact;(2)Let X is a perfectly base-paracompact.

The result is that the ppl-and wppl-space are preserved by the inverse image of perfect mapping.

The notion of countably base-mesocompact spaces is introduced and the followingresults are proved:1)Let be a perfect mapping.

Some properties of TL-subgroups and TL-normal subgroups in bi-induced mappings are proved.

In this paper,it proved some properties of anti L-subgroups and anti L-normal subgroups in bi-induced mappings.