Countable Coverage Property And Choice Theory

Since Michael set up the pioneering work on continuous selection on the basis of extension problem in continuous functions, selection theory has become one of most interesting subject in general topology. And now selection theory is widely used in many different branches in math. Studying selection problems from the angle of covering properties has become an important part in selection theory. In this paper, based on the former results, the sieves-method will be used to describe the relationship between countable covering properties and selection, in spaces which satisfy appropriate separation properties. This paper will be divided into three chapters.In Chapter one, the backgrounds on selection theory and some preliminaries will be stated.In Chapter two, we give two selection theorems respectively for zero-dimensional completely normal spaces and countably paracompact normal spaces.Finally, in the last chapter, we will include that several countable covering properties are equivalent to each other in topological spaces which are normal; Furthermore, we will characterize this equivalence by selection; Also, we will give up the separate properties to found two selection theorems about countably hereditary mesocompactness and countably hereditary metacompactness.