The topological structure of fuzzy sets with sendograph metric

摘要

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.