The Hyperspace Of The Regions Below Of Continuous Maps From A Compactum To The Closed Unit Interval

This paper consists of three chapters.In Chapter 1, we first introduce the developing history of Infinite-Dimensional Topology. And then we list some symbols, conceptions and theorems related to this paper. Finally, we introduce the study background of this paper and list its main results.In Chapters 2 and 3, we mainly discuss the structure of the hyperspace of the regions below of continuous maps from a compactum (compact metrizable space) to the closed unit interval, here we regard the region below of a continuous map as a closed subset of its corresponding product space endowed with the Vietoris Topology( this toplogy will be introduced in detail in Section 1.3).Let X be a metrical space. We use USC(X) and C(X), respectively, to denote the families of all upper semi-continuous maps and all continuous maps from X to I = [0,1]. We use X' to denote the set consisting of all limit points of X. For any f G USC(X), let ↓f = {(x,λ) (?) X× I : A ≤ f(x)}, ↓USC(X) = {↓f:f(?) USC(X)}, ↓C(X) = {↓f:(?) C(X)} and ↓C₀(X) = {↓f (?)|C(X) : f(a) = 0 for any a (?) X'}. ↓USC(X) endowed with the Vietoris topology is a topological space which contains ↓C(X) , ↓Co(X) and ↓C_>o(X) as its subspaces.In Chapter 2, we mainly prove that if X is a compactum without a dense set of isolated points, then there exists a homeomorphismh :↓USC(X) → Q = [-1, l]^ω such that hIn Chapter 3, we mainly discuss the case of X being compactum with a dense set of isolated points. If X is finite, it is proved that ↓USC(X) =↓C(X) ≈ (is homeomorphic to )I^|X|, where \X\ is the cardinal number of X. If X is infinite , it is proved that there are two homeomorphisms h₁ h₂ :↓USC(X) → Q =...