The topological structures of the spaces of fuzzy numbers with the sendograph metric

For every non-degenerate convex subset Y of the n-dimensional Euclidean space $R^n$, let $K\left(Y\right)$ denote the set of all fuzzy numbers, which are fuzzy sets that are upper semi-continuous, fuzzy convex and normal with compact supports contained in Y. Consider the sendograph metric $D^{\prime }$ defined on $K\left(Y\right)$. Zhang has proven that $\left(K\right(Y),D^{\prime })$ is homeomorphic to the separable Hilbert space ${\ell }^2$ when Y is compact or $Y=R^n$. In this paper, we show that $\left(K\right(Y),D^{\prime })$ is homeomorphic to ${\ell }^2$ if and only if Y is topologically complete.