The Expansions Of Set-valued Mappings On K-semi-stratifiable Spaces And Generalized Semi-stratifiable Spaces

Insertions and expansions of functions are classical problems in general top ology.Many important topological properties and insertions of functions and ex pansion problems of functions are closely related.In recent years,the connecti ons between insertions of functions and spaces with certain stratifiable structure s have attracted attention of topologists.In this paper,by introducing the conce pt of k-upper-semi-continuous set-valued mappings,we will give some characte rizations of K-MCM and K-semistratifiable spaces by the expansions of set-val ued mappings,and answer a question raised by Li-hong Xie and Peng-fei Yan.On the other hand,á.Császár,the hungarian mathematician,proposed the concept of generalized topological spaces in 1997.As an important generalizat ion of the classical general topology,the general topology theory has been rapi dly developed in recent years.Many important concepts of general topology ar e extended to generalized topological theory.We will introduce the concept of generalized semi-stratifiable spaces based on á.Császár et al,and give some c onnections between this type of spaces and insertions of functions.This thesis is organized as follows:In the first chapter,we introduce the background knowledge and related c oncepts.In the second chapter,we introduce the concept of K-upper-semi-continuo us set-valued mappings and establish some relations between K-semistratifiable and K-MCM spaces and the expansions of set valued mappings.In the last chapter,the concept of ?-semistratifiable spaces is introduced.Also,we give some characterizations of ?-semistratifiable spaces,and obtain some connections between ?-semistratifiable spaces and insertions of functions.