The topological structure of fuzzy sets with sendograph metric

For a non-degenerate convex subset Y of the n  -dimensional Euclidean space RnRn, let F(Y)F(Y) be the family of all fuzzy sets of RnRn which are upper semicontinuous, fuzzy convex and normal with compact supports contained in Y  . We show that the space F(Y)F(Y) with the topology of sendograph metric is homeomorphic to the separable Hilbert space ℓ2ℓ2 if Y   is compact; and the space F(Rn)F(Rn) is also homeomorphic to ℓ2ℓ2.