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

Sakai, Katsuro; Uehara, Shigenori

Fundamenta Mathematicae, Tome 159 (1999), p. 199-218 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $φ \in U S C C_{B} (X)$ 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 $ℓ_{2} (2^{ℕ})$ .

Publié le : 1999-01-01

Zbl 0944.54013

EUDML-ID : urn:eudml:doc:212389

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     author = {Katsuro Sakai and Shigenori Uehara},
     title = {Spaces of upper semicontinuous multi-valued functions on complete metric spaces},
     journal = {Fundamenta Mathematicae},
     volume = {159},
     year = {1999},
     pages = {199-218},
     zbl = {0944.54013},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv160i3p199bwm}
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Sakai, Katsuro; Uehara, Shigenori. Spaces of upper semicontinuous multi-valued functions on complete metric spaces. Fundamenta Mathematicae, Tome 159 (1999) pp. 199-218. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv160i3p199bwm/

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