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

Volume 160 / 1999

Katsuro Sakai, Shigenori Uehara Fundamenta Mathematicae 160 (1999), 199-218 DOI: 10.4064/fm-160-3-199-218


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 $USCC_B(X)$ the space of bounded upper semicontinuous multi-valued functions φ : X → ℝ such that each φ(x) is a closed interval. We identify $φ ∈ USCC_B(X)$ with its graph which is a closed subset of X × ℝ. The space $USCC_B(X)$ admits the Hausdorff metric induced by ϱ. It is proved that if X = (X,d) is uniformly locally connected, non-compact and complete, then $USCC_B(X)$ is homeomorphic to a non-separable Hilbert space. In case X is separable, it is homeomorphic to $ℓ_2(2^ℕ)$.


  • Katsuro Sakai
  • Shigenori Uehara

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