Several Topological Spaces Based On Convergent Sequences And Semi - Continuous Function Insertion

The insertion of functions is one of classical branches in general topology. It provides important methods to structure continuous and semi-continuous functions.In this thesis, we introduce several new kinds of topological spaces with convergent sequences, which are closed relate to sequentially mesocompactness properties. Then we give relations between these spaces and the insertion of semi-continuous functions.Also, we give some mapping theorems on these spaces.In the first chapter, we introduce the research background of the insertion of functions and list some basic conceptions used through this thesis.In the second chapter, we discuss the properties of countably sequentially mesocompact spaces, and give some characterizations of countably sequentially mesocompact spaces by the insertion of semi-continuous functions and set-valued mappings. Also, we give some mapping theorems on these spaces.In the third chapter, we introduce the conception of monotonically countably sequentially mesocompact spaces, and give characterizations of monotonically countably sequentially mesocompact spaces by the monotonic insertion of semi-continuous functions and set-valued mappings.In the fourth chapter, we introduce the conception of cs-perfect spaces, and give characterizations of cs-perfect spaces by the insertion of semi-continuous functions and set-valued mappings. Also, some mapping theorems are given.