A topological position of the set of continuous maps in the set of upper semicontinuous maps

摘要

Let (X, ρ) be a metric space and ↓USCC(X) and ↓CC(X) be the families of the regions below all upper semi-continuous compact-supported maps and below all continuous compact-supported maps from X to I = [0, 1], respectively. With the Hausdorff-metric, they are topological spaces. In this paper, we prove that, if X is an infinite compact metric space with a dense set of isolated points, then (↓USCC(X), ↓CC(X)) ≈ (Q, c0 ∪ (Q Σ)), i.e., there is a homeomorphism h :↓USCC(X) → Q such that h(↓CC(X)) = c0 ∪ (Q Σ), where Q = [-1, 1]ω, Σ = {(xn)n∈N∈ Q : sup|xn| 1} and c0 = {(xn)n∈N∈Σ : limn→+∞ xn = 0}. Combining this statement with a result in our previous paper, we have (↓USCC(X), ↓CC(X)) ≈■ (Q, c0 ∪ (Q Σ)), if the set of isolanted points is dense in X, (Q, c0), otherwise, if X is an infinite compact metric space. We also prove that, for a metric space X, (↓USCC(X), ↓CC(X)) ≈ (Σ,c0) if and only if X is non-compact, locally compact, non-discrete and separable.