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] 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. © 2009 Elsevier B.V. All rights reserved.