Semi-continuous Function Is Inserted Into Outer Space

The insertion of function is a classical branch in general topology. In this thesis, using the insertion of semi-continuous function, we give the some characterizations of some classical spaces. Also, we introduce monotonic cb-spaces, which show that they are closely relation to the insertion of continuous function and have many interesting properties.In the first chapter of this thesis, we introduce the background of the insertion of function.In the second chapter of this thesis, we investigate the relations between decreasing se-quences of sets and the insertion of semi-continuous functions, and give some characteriza-tions of countably metacompact spaces, countably paracompact spaces, monotonically count-ably paracompact spaces (MCP), monotonically countably metacompact spaces (MCM), per-fect normal spaces and stratifiable spaces.In chapter three, we introduce a monotone version of cb-spaces, monotone cb-spaces (mcb-spaces). This property is shown to relate closely to insertion of continuous functions. We give some characterizations of mcb-spaces and investigate the relations among mcb-spaces and cb-spaces, countably compact spaces, stratifiable spaces, MCP and MCM.In the fourth chapter, we give some characterizations of stratifiable and semi-stratifiable spaces by the insertion of semi-continuous set-valued mappings. Also, we introduce the K-lower and K-upper set-valued mappings and, using them, give some characterizations of K-semi-stratifiable spaces andκ-MCM (κβ).