In [R. D. Anderson, Bull. Am. Math. Soc. 72, 515–519 (1966; Zbl 0137.09703)], it is shown that the Hilbert space \(\ell^2\) (the space of all sequences \((x_i)_{i\in\mathbb{N}}\) such that \(\sum_{i\in\mathbb{N}}x_i^2<\infty\)) is homeomorphic to the infinite product \((0,1)^{\mathbb{N}}\), where \((0,1)\) is the interior of the unit interval \(\mathbb{I}=[0,1]\), and \(\mathbb{N}\) is the set of natural numbers. Moreover, Z. Yang and L. Zhang [Fuzzy Sets Syst. 160, No. 20, 2937–2946 (2009; Zbl 1183.03057)] proved that, given a subset \(Y\) of the \(n\)-dimensional Euclidian space \(\mathbb{R}^n\), the set \(\mathbb{F}_e(Y)\) of fuzzy sets in \(\mathbb{R}^n\) (maps \(f:\mathbb{R}^n\rightarrow\mathbb{I}\)), which additionally are (1) normal (\(f(x)=1\) for some \(x\in\mathbb{R}^n\)); (2) fuzzy convex (\(\min\{f(x),f(y)\}\leqslant f(rx+(1-r)y)\) for every \(x\), \(y\in\mathbb{R}^n\) and every \(r\in\mathbb{I}\)); (3) upper semicontinuous (the set \(\{x\in\mathbb{R}^n\mid f(x)<r\}\) is open for every \(r\in\mathbb{I}\)); and (4) have the compact support \(\operatorname{supp}(f)\) (the closure of the set \(\{x\in\mathbb{R}^n \mid 0<f(x)\}\)) contained in \(Y\); which (the set \(\mathbb{F}_e(Y)\)) is equipped with the endograph metric (given \(f\in\mathbb{F}_e(Y)\), the endograph \(\operatorname{end}(f)\) of \(f\) is defined by \(\{(x,t)\in\mathbb{R}^n\times\mathbb{I}\mid t\leqslant f(x)\}\), and the distance \(D(f,g)\) is calculated then as the standard distance in \(\mathbb{R}^n\times\mathbb{I}\) between the sets \(\operatorname{end}(f)\) and \(\operatorname{end}(g)\)), is homeomorphic to the Hilbert cube \([0,1]^{\mathbb{N}}\) if and only if \(Y\) is compact; and is homeomorphic to \([0,1]^{\mathbb{N}}\backslash(0,1)^{\mathbb{N}}\) if and only if \(Y\) is non-compact locally compact. The current paper shows that the set \(\mathbb{F}_s(Y)\) of the above fuzzy sets, equipped with the sendograph metric (given \(f\in\mathbb{R}^n\), the sendograph \(\operatorname{send}(f)\) of \(f\) is defined by \(\{(x,t)\in\mathbb{R}^n\times\mathbb{I}\mid t\leqslant f(x)\) and \(x\in\operatorname{supp}(f)\}\), and the distance \(D^{\prime}(f,g)\) employs then the sendographs \(\operatorname{send}(f)\) and \(\operatorname{send}(g)\) as above), is homeomorphic to \(\ell^2\) if and only if \(Y\) is a non-degenerate compact convex subset of \(\mathbb{R}^n\) (Theorem 1). Moreover, \(\mathbb{F}_s(\mathbb{R}^n)\) is homeomorphic to \(\ell^2\) (Theorem 2). The proofs of the just mentioned two homeomorphism theorems, which are assembled from a long sequence of preliminary definitions and results, take the whole (but short) paper.
The paper is well written, and moderately provides the required preliminaries. Its proofs, however, are rather technical and sometimes difficult to follow.