A special fuzzy star-shaped number space with the sendograph metric

As a natural generalization of fuzzy numbers, fuzzy star-shaped numbers play a crucial role in fuzzy mathematics. Let Y be a non-degenerate compact convex subset of the n-dimensional Euclidean space, and let ${\mathrm{FS}}_{z_0}\left(Y\right)$ be the family consisting of all fuzzy star-shaped numbers with respect to $z_0$ whose supports contained in Y. Using methods from infinite-dimensional topology, we mainly show that the space ${\mathrm{FS}}_{z_0}\left(Y\right)$ with the topology induced by the sendograph metric is homeomorphic to the separable Hilbert space ${\ell }^2$.