Hyperspaces of finite sets in universal spaces for absolute Borel classes

• Autorzy: Mine, K. , Sakai, K. , Yaguchi, M.
• Języki publikacji: EN

Abstrakty

EN

By Fin(X) (resp. Fin^k (X)), we denote the hyperspace of all non-empty finite subsets of X (resp. consisting of at most k points) with the Vietoris topology. Let ℓ₂ (τ) be the Hilbert space with weight τ and ℓ^f₂ (τ) the linear span of the canonical orthonormal basis of ℓ₂ (τ). It is shown that if E = ℓ^f₂ (τ) or E is an absorbing set in ℓ₂ (τ) for one of the absolute Borel classes a_α (τ) and M_α (τ) of weight ≤ τ (α > 0) then Fin(E) and each Fin^k (E) are homeomorphic to E. More generally, if X is a connected E-manifold then Fin(X) is homeomorphic to E and each Fin^k (X) is a connected E-manifold.

Słowa kluczowe

PL

topologia Victorisa; miara Hausdorffa; przestrzeń Hilberta; zbiór pochłaniający

EN

hyperspace of finite subsets; Vietoris topology; Hausdorff metric; universal spaces; absolute Borel classes; Hilbert space; absorbing set

• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2005
• Tom: Vol. 53, no 4
• Strony: 409-419
• Opis fizyczny: Bibliogr. 19 poz.

Twórcy

autor

Mine, K.

• Institute of Mathematics, University of Tsukuba, Tsukuba, 305-8571 Japan,

autor

Sakai, K.

• Institute of Mathematics, University of Tsukuba, Tsukuba, 305-8571 Japan,

autor

Yaguchi, M.

• Institute of Mathematics, University of Tsukuba, Tsukuba, 305-8571 Japan,

Bibliografia

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• [3] D. Curtis and Nguyen To Nhu, Hyperspaces of finite subsets which are homeomorphic to No-dimensional linear metric spaces, ibid. 19 (1985), 251-260.
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• [9] Nguyen To Nhu, Investigating the ANR-property of metric spaces, Fund. Math. 124 (1984), 243-254.
• [10] —, Hyperspaces of compact sets in metric linear spaces, Topology Appl. 22 (1986), 109-122.
• [11] Nguyen To Nhu and K. Sakai, Probability measure functions preserving infinite-dimensional spaces, Colloq. Math. 70 (1996), 291-304.
• [12] K. Sakai, The completion of metric ANR’s and homotopy dense subsets, J. Math. Soc. Japan 52 (2000), 835-846.
• [13] K. Sakai and M. Yaguchi, Characterizing manifolds modeled on certain dense subspaces of non-separable Hilbert spaces, Tsukuba J. Math. 27 (2003), 143-159.
• [14] —, -, Hyperspaces of Banach spaces with the Attouch-Wets topology, Set-Valued Anal. 12 (2004), 329-344.
• [15] A. H. Stone, Absolute Fσ spaces, Proc. Amer. Math. Soc. 13 (1962), 495-499.
• [16] H. Toruńczyk, Concerning locally homotopy negligible sets and characterization of ℓ2-manifolds, Fund. Math. 101 (1978), 93-110.
• [17] -, Characterizing Hilbert space topology, ibid. 111 (1981), 247-262.
• [18] —, A correction of two papers concerning Hilbert manifolds, ibid. 125 (1985), 89-93.
• [19] M. Yaguchi, Hyperspaces of finite subsets of non-separable Hilbert spaces, Tsukuba J. Math., to appear.

Identyfikatory

DOI

10.4064/ba53-4-6