Spaces of upper semicontinuous multi-valued functions on complete metric spaces

Let X = (X,d) be a metric space and let the product space X × ℝ be endowed with the metric ϱ ((x,t),(x',t')) = max{d(x,x'), |t - t'|}. We denote by ${\displaystyle US{ℂ}_B\left(X\right)}$ the space of bounded upper semicontinuous multi-valued functions φ : X → ℝ such that each φ(x) is a closed interval. We identify ${\displaystyle \phi \in US{ℂ}_B\left(X\right)}$ with its graph which is a closed subset of X × ℝ. The space ${\displaystyle US{ℂ}_B\left(X\right)}$ admits the Hausdorff metric induced by ϱ. It is proved that if X = (X,d) is uniformly locally connected, non-compact and complete, then ${\displaystyle US{ℂ}_B\left(X\right)}$ is homeomorphic to a non-separable Hilbert space. In case X is separable, it is homeomorphic to ${\displaystyle {\ell }_2\left(2^{\mathbb{N}}\right)}$.