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

Abstract

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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 U S C C 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 ∈ U S C C B ( X ) with its graph which is a closed subset of X × ℝ. The space U S C C 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 U S C C B ( X ) is homeomorphic to a non-separable Hilbert space. In case X is separable, it is homeomorphic to ℓ 2 ( 2 ℕ ) .

How to cite

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Sakai, 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 -

References

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