Wyniki wyszukiwania

1

A Hilbert cube compactification of the function space with the compact-open topology

100%

Kogasaka A., Sakai K.

Open Mathematics

|

2009

|

tom 7

|

nr 4

670-682

EN

Let X be an infinite, locally connected, locally compact separable metrizable space. The space C(X) of real-valued continuous functions defined on X with the compact-open topology is a separable Fréchet space, so it is homeomorphic to the psuedo-interior s = (−1, 1)ℕ of the Hilbert cube Q = [−1, 1]ℕ. In this paper, generalizing the Sakai-Uehara’s result to the non-compact case, we construct a natural compactification $$ \bar C $$(X) of C(X) such that the pair ($$ \bar C $$(X), C(X)) is homeomorphic to (Q, s). In case X has no isolated points, this compactification $$ \bar C $$(X) coincides with the space USCCF(X,) of all upper semi-continuous set-valued functions φ: X → = [−∞, ∞] such that each φ(x) is a closed interval, where the topology for USCCF(X, ) is inherited from the Fell hyperspace Cld*F(X × ) of all closed sets in X × .

2

The AR-Property of the spaces of closed convex sets

100%

Sakai K., Yaguchi M.

Colloquium Mathematicum

|

2006

|

tom 106

|

nr 1

15-24

EN

Let $Conv_{H}(X)$, $Conv_{AW}(X)$ and $Conv_{W}(X)$ be the spaces of all non-empty closed convex sets in a normed linear space X admitting the Hausdorff metric topology, the Attouch-Wets topology and the Wijsman topology, respectively. We show that every component of $Conv_{H}(X)$ and the space $Conv_{AW}(X)$ are AR. In case X is separable, $Conv_{W}(X)$ is locally path-connected.

3

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

100%

Sakai K., Uehara S.

Fundamenta Mathematicae

|

1999

|

tom 160

|

nr 3

199-218

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 $ℓ_2(2^ℕ)$.

4

The Spaces of Closed Convex Sets in Euclidean Spaces with the Fell Topology

100%

Sakai K., Yang Z.

Bulletin of the Polish Academy of Sciences. Mathematics

|

2007

|

tom 55

|

nr 2

139-143

EN

Let $Conv_F(ℝ ⁿ)$ be the space of all non-empty closed convex sets in Euclidean space ℝ ⁿ endowed with the Fell topology. We prove that $Conv_F(ℝ ⁿ) ≈ ℝ ⁿ × Q$ for every n > 1 whereas $Conv_F(ℝ) ≈ ℝ × 𝕀$.

5

Open Subsets of LF-spaces

100%

Mine K., Sakai K.

Bulletin of the Polish Academy of Sciences. Mathematics

|

2008

|

tom 56

|

nr 1

25-37

EN

Let F = ind lim Fₙ be an infinite-dimensional LF-space with density dens F = τ ( ≥ ℵ ₀) such that some Fₙ is infinite-dimensional and dens Fₙ = τ. It is proved that every open subset of F is homeomorphic to the product of an ℓ₂(τ)-manifold and $ℝ^∞ = ind lim ℝ ⁿ$ (hence the product of an open subset of ℓ₂(τ) and $ℝ^∞$). As a consequence, any two open sets in F are homeomorphic if they have the same homotopy type.

6

Hyperspaces of CW-complexes

100%

Guo B.-L., Sakai K.

Fundamenta Mathematicae

|

1993

|

tom 143

|

nr 1

23-40

EN

It is shown that the hyperspace of a connected CW-complex is an absolute retract for stratifiable spaces, where the hyperspace is the space of non-empty compact (connected) sets with the Vietoris topology.

7

Probability measure functors preserving infinite-dimensional spaces

100%

Nhu N. T., Sakai K.

Colloquium Mathematicum

|

1996

|

tom 70

|

nr 2

291-304

8

Recognizing the topology of the space of closed convex subsets of a Banach space

81%

Banakh T., Hetman I., Sakai K.

Studia Mathematica

|

2013

|

tom 216

|

nr 1

17-33

9

Hyperspaces of Finite Sets in Universal Spaces for Absolute Borel Classes

81%

Mine K., Sakai K., Yaguchi M.

Bulletin of the Polish Academy of Sciences. Mathematics

|

2005

|

tom 53

|

nr 4

409-419

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 $ℓ₂^{f}(τ)$ the linear span of the canonical orthonormal basis of ℓ₂(τ). It is shown that if $E = ℓ₂^{f}(τ)$ 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.

10

A function space from a compact metrizable space to a dendrite with the hypo-graph topology

81%

Yang H., Sakai K., Koshino K.

Open Mathematics

|

2015

|

tom 13

|

nr 1

EN

Let X be an infinite compact metrizable space having only a finite number of isolated points and Y be a non-degenerate dendrite with a distinguished end point v. For each continuous map ƒ : X → Y , we define the hypo-graph ↓vƒ = ∪ x∈X {x} × [v, ƒ (x)], where [v, ƒ (x)] is the unique arc from v to ƒ (x) in Y . Then we can regard ↓v C(X, Y ) = {↓vƒ | ƒ : X → Y is continuous} as the subspace of the hyperspace Cld(X × Y ) of nonempty closed sets in X × Y endowed with the Vietoris topology. Let [...] be the closure of ↓v C(X, Y ) in Cld(X ×Y ). In this paper, we shall prove that the pair [...] , ↓v C(X, Y )) is homeomorphic to (Q, c0), where Q = Iℕ is the Hilbert cube and c0 = {(xi )i∈ℕ ∈ Q | limi→∞xi = 0}.

11

The hyperspace of finite subsets of a stratifiable space

81%

Cauty R., Guo B.-L., Sakai K.

Fundamenta Mathematicae

|

1995

|

tom 147

|

nr 1

1-9

EN

It is shown that the hyperspace of non-empty finite subsets of a space X is an ANR (an AR) for stratifiable spaces if and only if X is a 2-hyper-locally-connected (and connected) stratifiable space.

12

Spaces of retractions which are homeomorphic to Hilbert space

80%

Nhu N. T., Sakai K., Wong R. Y.

Fundamenta Mathematicae

|

1990

|

tom 136

|

nr 1

45-52

13

The space of Lipschitz maps from a compactum to an absolute neighborhood LIP extensor

79%

Sakai K.

Fundamenta Mathematicae

|

1991

|

tom 138

|

nr 1

27-34

14

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A mapping theorem for infinite-dimensional manifolds and its generalizations

79%

Sakai K.

Colloquium Mathematicum

|

1988

|

tom 56

|

nr 2

319-332

15

An embedding theorem of infinite-dimensional manifold pairs in the model space

79%

Sakai K.

Fundamenta Mathematicae

|

1978

|

tom 100

|

nr 1

83-87

Ograniczanie wyników

A Hilbert cube compactification of the function space with the compact-open topology

The AR-Property of the spaces of closed convex sets

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

The Spaces of Closed Convex Sets in Euclidean Spaces with the Fell Topology

Open Subsets of LF-spaces

Hyperspaces of CW-complexes

Probability measure functors preserving infinite-dimensional spaces

Recognizing the topology of the space of closed convex subsets of a Banach space

Hyperspaces of Finite Sets in Universal Spaces for Absolute Borel Classes

A function space from a compact metrizable space to a dendrite with the hypo-graph topology

The hyperspace of finite subsets of a stratifiable space

Spaces of retractions which are homeomorphic to Hilbert space

The space of Lipschitz maps from a compactum to an absolute neighborhood LIP extensor

A mapping theorem for infinite-dimensional manifolds and its generalizations

An embedding theorem of infinite-dimensional manifold pairs in the model space