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

Katsuro Sakai; Shigenori Uehara

Fundamenta Mathematicae (1999)

- Volume: 160, Issue: 3, page 199-218
- ISSN: 0016-2736

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topSakai, Katsuro, and Uehara, Shigenori. "Spaces of upper semicontinuous multi-valued functions on complete metric spaces." Fundamenta Mathematicae 160.3 (1999): 199-218. <http://eudml.org/doc/212389>.

@article{Sakai1999,

abstract = {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’)) = maxd(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 $ℓ_2(2^ℕ)$.},

author = {Sakai, Katsuro, Uehara, Shigenori},

journal = {Fundamenta Mathematicae},

keywords = {space of upper semicontinuous multi-valued functions,; hyperspace of non-empty closed sets,; Hausdorff metric,; Hilbert space,; uniformly locally connected},

language = {eng},

number = {3},

pages = {199-218},

title = {Spaces of upper semicontinuous multi-valued functions on complete metric spaces},

url = {http://eudml.org/doc/212389},

volume = {160},

year = {1999},

}

TY - JOUR

AU - Sakai, Katsuro

AU - Uehara, Shigenori

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

JO - Fundamenta Mathematicae

PY - 1999

VL - 160

IS - 3

SP - 199

EP - 218

AB - 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’)) = maxd(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 $ℓ_2(2^ℕ)$.

LA - eng

KW - space of upper semicontinuous multi-valued functions,; hyperspace of non-empty closed sets,; Hausdorff metric,; Hilbert space,; uniformly locally connected

UR - http://eudml.org/doc/212389

ER -

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