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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’)) = maxd(x,x’), |t - t’|. We denote by $USC{C}_B\left(X\right)$ the space of bounded upper semicontinuous multi-valued functions φ : X → ℝ such that each φ(x) is a closed interval. We identify $\phi \in USC{C}_B\left(X\right)$ with its graph which is a closed subset of X × ℝ. The space $USC{C}_B\left(X\right)$ admits the Hausdorff metric induced by ϱ. It is proved that if X = (X,d) is uniformly locally connected, non-compact and complete, then $USC{C}_B\left(X\right)$ is homeomorphic to a...

Let $Con{v}_H\left(X\right)$, $Con{v}_{AW}\left(X\right)$ and $Con{v}_W\left(X\right)$ 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 $Con{v}_H\left(X\right)$ and the space $Con{v}_{AW}\left(X\right)$ are AR. In case X is separable, $Con{v}_W\left(X\right)$ is locally path-connected.

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 $$\overline{C}$$ (X) of C(X) such that the pair ($$\overline{C}$$ (X), C(X)) is homeomorphic to (Q, s). In case...

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 ${ℝ}^{\infty }=indlimℝⁿ$ (hence the product of an open subset of ℓ₂(τ) and ${ℝ}^{\infty }$). As a consequence, any two open sets in F are homeomorphic if they have the same homotopy type.

Let $Con{v}_F\left(ℝⁿ\right)$ be the space of all non-empty closed convex sets in Euclidean space ℝ ⁿ endowed with the Fell topology. We prove that $Con{v}_F\left(ℝⁿ\right)\approx ℝⁿ\times Q$ for every n > 1 whereas $Con{v}_F\left(ℝ\right)\approx ℝ\times$.

Let $L\left(X\right)$ be the space of all lower semi-continuous extended real-valued functions on a Hausdorff space $X$, where, by identifying each $f$ with the epi-graph $epi\left(f\right)$, $L\left(X\right)$ is regarded the subspace of the space ${Cld}_F^{*}(X\times ℝ)$ of all closed sets in $X\times ℝ$ with the Fell topology. Let $$LSC\left(X\right)=\{f\in L\left(X\right)\mid f\left(X\right)\cap ℝ\ne \varnothing ,f\left(X\right)\subset (-\infty ,\infty ]\}\text{and}{LSC}_B\left(X\right)=\{f\in L\left(X\right)\mid f\left(X\right)\text{is}\text{a}\text{bounded}\text{subset}\text{of}ℝ\}.$$ We show that $L\left(X\right)$ is homeomorphic to the Hilbert cube $Q={[-1,1]}^{\mathbb{N}}$ if and only if $X$ is second countable, locally compact and infinite. In this case, it is proved that $(L\left(X\right),LSC\left(X\right),{LSC}_B\left(X\right))$ is homeomorphic to $(ConeQ,Q\times (0,1),\Sigma \times (0,1\left)\right)$ (resp. $(Q,s,\Sigma )$) if $X$ is compact (resp. $X$ is non-compact), where $ConeQ=(Q\times 𝐈)/(Q\times \{1\left\}\right)$ is the cone over...

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.

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 [...]...

By Fin(X) (resp. $Fi{n}^k\left(X\right)$), 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 $\ell {₂}^f\left(\tau \right)$ the linear span of the canonical orthonormal basis of ℓ₂(τ). It is shown that if $E=\ell {₂}^f\left(\tau \right)$ or E is an absorbing set in ℓ₂(τ) for one of the absolute Borel classes ${}_{\alpha }\left(\tau \right)$ and ${}_{\alpha }\left(\tau \right)$ of weight ≤ τ (α > 0) then Fin(E) and each $Fi{n}^k\left(E\right)$ are homeomorphic to E. More generally, if X is a connected E-manifold then Fin(X) is homeomorphic...

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.