THE REGIONS BELOW COMPACT-SUPPORTED UPPER SEMICONTINUOUS MAPS

Abstract. Let X be a non-compact locally compact separable metric space. We use ↓USCC(X) to denote the family of the regions below of all compact-supported upper semi-continuous maps from X to I = [0, 1]. We may topol-ogize ↓USCC(X) by the Hausdorff metric. It is proved in this paper that ↓USCC(X) ≈ Σ if X is non-discrete, and ↓USCC(X) ≈ Qf if X is discrete, where Σ = {(xn) ∈ Q: sup |xn | < 1} is the radial interior of the Hilbert cube Q = [−1, 1]ω and Qf = {(xn) ∈ Q: xn = 0 except for finitely many n}. 1. Introduction an