Full Space And Countable Compact Space, K Semi-continuous Function Inserts

The insertion of function is a classical branch in general topology.Countably mesocom-pactness is an important covering property. In this thesis, we mainly investigate relations be-tween several kinds of topological spaces with countably mesocompaceness and the insertion of semi-continuous functions.In the second chapter of this thesis, we investigate the properties of countably mesocom-pace spaces. And provide several mapping theorems on countably mesocompact spaces. Then discuss the relations between countably mesocompact spaces、monotonically countably meso-compact spaces and the insertion of semi-continuous functions.In chapter three, we introduce the concept of K-perfect.Using the equivalent characteri-zations,we prove some mapping theorems on K-perfect spaces and respectively give the char-acterizations of the version of Urysohn Lemma and the insertion semi-continuous functions of K-perfect spaces.