Hyperspaces of non-compact metrizable spaces which are homeomorphic to the Hilbert cube

Sakai, Katsuro; Yang, Zhongqiang

doi:10.1016/s0166-8641(02)00097-4

Cited by 17 publications

(9 citation statements)

References 11 publications

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“…Indeed, it will be enough to show that CLC F (X) is not σ-compact since it is a subset of CL F (X), which is second countable by Proposition 2.3.5 (in fact, it is homeomorphic to the complement of a point in the Hilbert cube; see [32,Theorem 1]).…”

Section: B)mentioning

confidence: 99%

Homotopical properties of hyperspaces of generalized continua: the proper and ordinary cases

Charatonik

Fernández-Bayort²,

Quintero

2021

Dissertationes Math.

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Section: B)mentioning

confidence: 99%

Homotopical properties of hyperspaces of generalized continua: the proper and ordinary cases

Charatonik

Fernández-Bayort²,

Quintero

2021

Dissertationes Math.

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“…Hence, applying Theorem 3, we get the following. It has been proved by Sakai & Yang [8] that the Wijsman hyperspace of R n is homeomorphic to the Hilbert cube minus a point (the authors of [8] consider hyperspaces with the Fell topology which, in the case of locally compact metric spaces, is equivalent to the Wijsman one). So the Wijsman hyperspace of every separable Banach space is an AR.…”

Section: 2mentioning

confidence: 99%

Complete metric absolute neighborhood retracts

Kubiś

2003

Topology and its Applications

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We characterize complete metric absolute (neighborhood) retracts in terms of existence of certain maps of CW-polytopes. Using our result, we prove that a compact metric space with a convex and locally convex simplicial structure is an AR. This answers a question of Kulpa from [5]. As another application, we prove that the hyperspace of closed subsets of a separable Banach space endowed with the Wijsman topology is an absolute retract.

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“…Many scholars generalized this theorem and gave some similar results. Among these, Sakai and Yang in [18,22] discussed, for a non-compact space Y , the hyperspace Cld F (Y ) endowed with the so-called Fell topology and its some natural subspaces, for example, the subspace consisting of all compact subsets. Particularly, they proved a result similar to the Curtis-Schori-West Hyperspace Theorem.…”

Section: Introductionmentioning

confidence: 99%

The hyperspace of the regions below of continuous maps is homeomorphic to c0

Yang¹

2006

Topology and its Applications

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For a compact metric space (X, d), we use ↓USC(X) and ↓C(X) to denote the families of the regions below of all upper semicontinuous maps and the regions below of all continuous maps from X to I = [0, 1], respectively. In this paper, we consider the two spaces topologized as subspaces of the hyperspace Cld(X × I) consisting of all non-empty closed sets in X × I endowed with the Vietoris topology. We shall show that ↓C(X) is Baire if and only if the set of isolated points is dense in X, but ↓C(X) is not a G δσ -set in ↓USC(X) unless X is finite. As the main result, we shall prove that if X is an infinite locally connected compact metric space then (↓USC(X), ↓C(X)) ≈ (Q, c 0 ), where Q = [−1, 1] ω is the Hilbert cube and c 0 = {(x n ) ∈ Q: lim n→∞ x n = 0}.

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Hyperspaces of non-compact metrizable spaces which are homeomorphic to the Hilbert cube

Cited by 17 publications

References 11 publications

Homotopical properties of hyperspaces of generalized continua: the proper and ordinary cases

Homotopical properties of hyperspaces of generalized continua: the proper and ordinary cases

Complete metric absolute neighborhood retracts

The hyperspace of the regions below of continuous maps is homeomorphic to c0

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