Complete metric absolute neighborhood retracts

Kubiś, Wiesław

doi:10.1016/s0166-8641(03)00031-2

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Proof Of Theorem I1

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2005

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“…It has been proved in [14] that Cld W (X) is an AR for an infinite-dimensional separable Banach space X. To show Theorem I, it remains to verify the discrete approximation property in Theorem 2.3.…”

Section: Proof Of Theorem Imentioning

confidence: 99%

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Hyperspaces of separable Banach spaces with the Wijsman topology

Kubiś

Sakai

Yaguchi

2005

Topology and its Applications

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Let X be a separable metric space. By Cld W (X), we denote the hyperspace of non-empty closed subsets of X with the Wijsman topology. Let Fin W (X) and Bdd W (X) be the subspaces of Cld W (X) consisting of all non-empty finite sets and of all non-empty bounded closed sets, respectively. It is proved that if X is an infinite-dimensional separable Banach space then Cld W (X) is homeomorphic to (≈) the separable Hilbert space 2 and Fin W (Moreover, we show that if the complement of any finite union of open balls in X has only finitely many path-components, all of which are closed in X, then Fin W (X) and Cld W (X) are ANR's. We also give a sufficient condition under which Fin W (X) is homotopy dense in Cld W (X).

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Section: Proof Of Theorem Imentioning

confidence: 99%

“…In case X is an infinite-dimensional separable Banach space, it is shown in [14] that Cld W (X) is an AR. This is a direct consequence of Theorem III above.…”

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confidence: 99%