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

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Sakai K., Uehara S.

Fundamenta Mathematicae

1999

tom 160

nr 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 $ℓ_2(2^ℕ)$.

Ograniczanie wyników

1 Fundamenta Mathematicae

1 Sakai K.

1 Uehara S.

1 1999

Wyniki wyszukiwania

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