The Topological Structure Of Continuous Function Space Of Non-compact Space With The Fell Topology

This thesis consists of three chapters.In Chapter 1,we introduce the historical development of Infinite-Dimensional Topology first.Then we list some symbols,conceptions and theorems related to this paper.In Chapter 2,we mainly discuss the structure of function spaces.Then we introduce the research background of this paper and list its main results.In Chapter 3,we give the proof of the main result.For a topological space X,let USC(X)and C(X)be the collections of all upper semi-continuous functions and continuous functions from X to I=[0,1], respectively.For a map f:X→I,let↓f={(x,t)∈X×I:t≤f(x)},↓USC(X)={↓f:f∈USC(X)},↓C(X)={↓f:f∈C(X)}.We mainly discuss the topological structure of the pair(↓USCF(X),↓CF(X)),that is (↓USC(X),↓C(X))with Fell topology,when X is a non-compact separable and metrizable space with a dense subset of isolated points.