A Topological Position Of The Set Of Strongly Discontinuous Maps In The Set Of Upper Semi-continuous Maps

Let X = (X,d) be a metric space. We use USC(X) denote the family ofall the upper semi-continuous maps from X to I = [0, 1], SDC(X) denote allthe strong discontinuous maps in USC(X). For an arbitrary f∈USC(X),let↓f demotes hypographs of f and then↓f is a closed set in X×I if and only if f is upper semi-continuous. For then↓A is a family of closed sets in it can be topologized with Fell topology. We denote denote the family of the regions below of USC(X) andequip with Fell topology. As the subspace of↓USC(X),↓SDC(X)can inherit the Fell topology from↓USC(X), we may denote it by↓SDCF(X).In this paper, we proved mainly the following theorem 1.Theorem 1 If X is locally compact, separable metric space with no isolatedpoint, then(↓USCF(X),↓SDCF(X))≈(Q,s).This paper consists three chapters. The main contents are as follow:In chapter 1, we list the most essential and frequently-used notions inthe first section. We introduced the developing history of In?nite-DimensionalTopology in the second section.In chapter 2, we give the fundamental preliminary knowledge in researchof functional spaces in the ?rst section. We introduced the study backgroundof this paper and partial results which obtained by some scholars and give themain results in this paper—theorem 1.In chapter 3, we ?nish the proof of theorem 1 and give tow questions.