The Hyperspace Of The Regions Below Of Continuous Maps From The Converging Sequence

This paper consists of two chapters.Chapter 1 consists of introductory materials: the developing history of Infinite-Dimensional Topology; some symbols, conceptions and theorems related to this paper. In section 3, we introduce the study background of this paper, listing several important results the formers obtained and basing on them wehave our own results.In Chapter 2, considering the hyperspace of the regions below of continuous maps from the converging sequence to unit interval I, we get the following results when we regard the region below of a continuous map as a closed subset of its corresponding product space endowed with the Vietoris Topology:Let S = {1,1/2,1/(2²),...,1/∞ = 0} and I = [0,1] be the unit interval. We use ↓USC(S) and ↓C(S) to denote the families of the regions below of all upper semi-continuous maps and of the regions below of all continuous maps from S to I and↓C₀(S) = {↓f ∈↓C(S) : f(0) = 0}. ↓USC(S) endowed with the Vietoris Topology is a topological space. It will be proved that, (↓USC(S),↓C₀(S)) ≈ (Q,s) and (↓USC(S),↓C(S)\ ↓C₀(S)) ≈ (Q,c₀), where Q = [-1,1]^ω is the Hilbert cube and s = (-1,1)^ω, c₀ = {(x_n) ∈ Q : lim x_n = 0}. But we do notknow what ↓ C(S) and the pair (↓USC(S),↓C(S)) are, and whether or not (↓ USC(X)) ≈ Q for any infinite compactum X, which are just problems we put forward but not resolved in 2.4.