The topological structure of fuzzy sets with endograph metric

For a non-degenerate convex subset Y of the n-dimensional Euclidean space R n , let K ( Y ) be the family of all fuzzy sets of R n , 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 { ( x n ) ∈ Q : sup | x n | < 1 } iff Y is non-compact and locally compact.