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...

For A ⊂ I = [0,1], let $L_A$ be the set of continuous real-valued functions on I which vanish on a neighborhood of A. We prove that if A is an analytic subset which is not an $F_{\sigma }$ and whose closure has an empty interior, then $L_A$ is homeomorphic to the space of differentiable functions from I into ℝ.

In this paper we consider a number of sequence and function spaces that are known to be homeomorphic to the countable product of the linear space $\sigma$. The spaces we are interested in have a canonical imbedding in both a topological Hilbert space and a Hilbert cube. It turns out that when we consider these spaces as subsets of a Hilbert cube then there is only one topological type. For imbeddings in the countable product of lines there are two types depending on whether the space is contained in a...

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 J(n) be the hyperspace of all centrally symmetric compact convex bodies $A\subseteq {ℝ}^n$, n ≥ 2, for which the ordinary Euclidean unit ball is the ellipsoid of maximal volume contained in A (the John ellipsoid). Let $J_0\left(n\right)$ be the complement of the unique O(n)-fixed point in J(n). We prove that: (1) the Banach-Mazur compactum BM(n) is homeomorphic to the orbit space J(n)/O(n) of the natural action of the orthogonal group O(n) on J(n); (2) J(n) is an O(n)-AR; (3) $J_0\left(2\right)/SO\left(2\right)$ is an Eilenberg-MacLane space $𝐊(\mathbb{Q},2)$; (4) $B{M}_0\left(2\right)=J_0\left(2\right)/O\left(2\right)$ is noncontractible;...

Let X be a topological group or a convex set in a linear metric space. We prove that X is homeomorphic to (a manifold modeled on) an infinite-dimensional Hilbert space if and only if X is a completely metrizable absolute (neighborhood) retract with ω-LFAP, the countable locally finite approximation property. The latter means that for any open cover $$𝒰$$ of X there is a sequence of maps (f n: X → X)nεgw such that each f n is $$𝒰$$ -near to the identity map of X and the family f n(X)n∈ω is locally finite...

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...

In each manifold $M$ modeled on a finite or infinite dimensional cube ${[0,1]}^n$, $n\le \omega$, we construct a meager $F_{\sigma }$-subset $X\subset M$ which is universal meager in the sense that for each meager subset $A\subset M$ there is a homeomorphism $h:M\to M$ such that $h\left(A\right)\subset X$. We also prove that any two universal meager $F_{\sigma }$-sets in $M$ are ambiently homeomorphic.