The topological structure of fuzzy sets with sendograph metric

For a non-degenerate convex subset Y of the n-dimensional Euclidean space R n , let F ( 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 F ( Y ) with the topology of sendograph metric is homeomorphic to the separable Hilbert space ℓ 2 if Y is compact; and the space F ( R n ) is also homeomorphic to ℓ 2 .