The topological structure of fuzzy sets with endograph metric

For a non-degenerate convex subset Y of the n-dimensional Euclidean space Rn, let K(Y) be the family of all fuzzy sets of Rn, which are upper-semicontinuous, fuzzy convex and normal with compact supports contained in Y. We show that the space K(Y) with the topology of endograph metric is homeomorphic to the Hilbert cube Q=[-1,1]ω iff Y is compact; and the space K(Y) is homeomorphic to {(xn)∈Q:sup|xn|<1} iff Y is non-compact and locally compact.