The Hyperspace Of The Regions Below Of Upper Semi-continuous Functions

This paper consists of two chapters.Chapter 1 introduce the developing history of Infinite-Dimensional Topology, some symbols, conceptions and theorems related to this paper, list several important results obtained by some famous scholars.The chapter 2 firstly introduce Curtis-Schori-West Hyperspace Theorem. Let X be a compactum and L be a subset of real set R. We use Cld(X) to denote the hyperspace of X, and use ↓ USC(X, L) to denote the hyperspace of the regions below of upper semi-continuous functions from X to L. The content of Curtis-Schori-West Hyperspace Theorem is:Cld(X) ≈ Q if and only if X is a non — degenerate Peano continuum. Curtis-Schori-West Hyperspace Theorem has an equivalent proposition, that is:↓ USC(X, {0,1}) ≈ Q(?){pt} if and only if X is a non — degenerate Peano continuum, here {pt} denote the space consisting of only one point. Considering L = {0, 1/2,1}, we prove that↓ USC(X, L) ≈ Q × {0,1} (?){pt} if and only if X is a non-degenerate Peano continuum.This result is the generalization of the equivalence proposition of the Curtis-Schori-West Hyperspace Theorem. When L = {1,2, ..., n} and n > 3, it can be proved similarly that ↓ USC(X, L) ≈ Q × {0,1,..., n - 2} (?){pt}. But when L is an infinite countable set, we do not know whether ↓ USC(X, L) has the similar structure.