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1999 | 160 | 3 | 199-218
Tytuł artykułu

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

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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 $ℓ_2(2^ℕ)$.
Rocznik
Tom
160
Numer
3
Strony
199-218
Opis fizyczny
Daty
wydano
1999
otrzymano
1997-04-22
poprawiono
1998-08-05
poprawiono
1999-01-28
Twórcy
  • Institute of Mathematics, University of Tsukuba Tsukuba, 305-8571 Japan, uehara@math.tsukuba.ac.jp
  • Takamatsu National College of Technology, Takamatsu, 761-8085 Japan
Bibliografia
  • [BP] C. Bessaga and A. Pełczyński, Selected Topics in Infinite-Dimensional Topology, Monograf. Mat. 58, Polish Sci. Publ., Warszawa, 1975.
  • [Bo] C. R. Borges, A study of absolute extensor spaces, Pacific J. Math. 31 (1969), 609-617; Absolute extensor spaces: a correction and an answer, ibid. 50 (1974), 29-30.
  • [Ca] R. Cauty, Rétractions dans les espaces stratifiables, Bull. Soc. Math. France 102 (1974), 129-149.
  • [Cu] W. H. Cutler, Negligible subsets of infinite-dimensional Fréchet manifolds, Proc. Amer. Math. Soc. 23 (1969), 668-675.
  • [Fe1] V. V. Fedorchuk, On certain topological properties of completions of function spaces with respect to Hausdorff uniformity, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 1991, no. 4, 77-80 (in Russian); English transl.: Moscow Univ. Math. Bull. 46 (1991), 56-58.
  • [Fe2] V. V. Fedorchuk, Completions of spaces of functions on compact spaces with respect to the Hausdorff uniformity, Trudy Sem. Petrovsk. 18 (1995), 213-235 (in Russian); English transl.: J. Math. Sci. 80 (1996), 2118-2129.
  • [FK] V. V. Fedorchuk and H.-P. A. Künzi, Uniformly open mappings and uniform embeddings of function spaces, Topology Appl. 61 (1995), 61-84.
  • [Ku] K. Kuratowski, Topology, I, Polish Sci. Publ., Warszawa, 1966.
  • [Mi] E. Michael, Continuous selections, I, Ann. of Math. 63 (1956), 361-382.
  • [SU] K. Sakai and S. Uehara, A Hilbert cube compactification of the Banach space of continuous functions, Topology Appl. 92 (1999), 107-118.
  • [Sc] R. M. Schori, Topological stability for infinite-dimensional manifolds, Compositio Math. 23 (1971), 87-100.
  • [To1] H. Toruńczyk, Concerning locally homotopy negligible sets and characterization of $l_2$-manifolds, Fund. Math. 101 (1978), 93-110.
  • [To2] H. Toruńczyk, On Cartesian factors and the topological classification of linear metric spaces, ibid. 88 (1975), 71-86.
  • [To3] H. Toruńczyk, Characterizing Hilbert space topology, ibid. 111 (1981), 247-262.
  • [To4] H. Toruńczyk, A correction of two papers concerning Hilbert manifolds, ibid. 125 (1985), 89-93.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-fmv160i3p199bwm
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