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

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, c ₀ ∪ (Q Σ)), i.e., there is a homeomorphism h:↓USCC(X) → Q such that h(↓CC(X)) = c ₀ ∪ (Q Σ), where Q = −1,1]^ω, Σ = {(x _n ) _n∈ℕ ∈ Q: sup|x _n | < 1} and c ₀ = {(x _n ) _n∈ℕ ∈ Σ: lim _n→+∞ x _n = 0}. Combining this statement with a result in our previous paper, we have if X is an infinite compact metric space. We also prove that, for a metric space X, (↓USCC(X), ↓CC(X)) ≈ (Σ, c ₀) if and only if X is non-compact, locally compact, non-discrete and separable. This work was supported by National Natural Science Foundation of China (Grant No. 10471084)