收敛序列上的连续函数下方图形超空间

Let S={1,1/2,1/2 2,…,1/∞=0} and I = 0,1] be the unit interval. We use ↓USC(S) and ↓C(5) 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. A pair of topological spaces (X, Y) means that X is a topological space and Y is its subspace. Two pairs of topological spaces (X, Y) and (A, B) are called pair-homeomorphic (≈) if there exists a homeomorphism h:X→A from X onto A such that h{Y) = B. It is proved that, (↓USC(S), ↓C0(S))≈(Q,s) and (↓USC(5), ↓C(5)\↓C0(S))≈(Q,c0), where Q =-1, l]ω is the Hilbert cube and s=(-1,1)ω, c0 = {(xn)∈Q:lim n→∞ xn=0}. But we do not know what (↓USC(S),↓C(5)) is.

The Hyperspace of the Regions Below of Continuous Maps from the Converging Sequence