The hyperspace of the regions below of all lattice-value continuous maps and its Hilbert cube compactification

Abstract

Let L be a continuous semilattice. We use USC(X, L) to denote the family of all lower closed sets including X x 0 in the product space X x ΛL and ↓ C(X, L) the one of the regions below of all continuous maps from X to ΛL. USC(X, L) with the Vietoris topology is a topological space and ↓ C(X, L) is its subspace. It will be proved that, if X is an infinite locally connected compactum and ΛL is an AR, then USC(X, L) is homeomorphic to [-1,1]ω. Furthermore, if L is the product of countably many intervals, then ↓ C(X, L) is homotopy dense in USC(X, L), that is, there exists a homotopy h : USC(X, L) × [0,1] → USC(X, L) such that h0 = idUSC(X, L) and ht(USC(X, L)) ⊂↓C(X, L) for any t > 0. But ↓C(X, L) is not completely metrizable.