THE HYPERSPACE OF THE REGIONS BELOW LATTICE-VALUE CONTINUOUS MAPS

摘要

Let L = I be the infinite countable product of unite interval I = [0,1] with point-wise order and the metric p((xi), {yi)) = ∑i~∞12i/1(x1-yi〡. For a compact metric space (X, d), we use 4 USC(X) to denote the family of all lower closed sets including X x {0} in the product space X x I and -----> (X) the family of the regions below all lattice-value continuous maps from X to respectively. In this paper, we consider the two spaces topolo-gized as subspaces of the hyperspace Cld(X I) consisting of all non-empty closed sets in X x I endowed with the Vietoris topology. It is shown that (USC(X) (X)) (Q, co) if X = I = [0,1], where Q = [-1,1] is the Hilbert cube and co = {(xn) E Q : lim xn = 0}.