The Urysohn universal metric space U is characterized up to isometry by the following properties: (1) U is complete and separable; (2) U contains an isometric copy of every separable metric space; (3) every isometry between two finite subsets of U can be extended to an isometry of U onto itself. We show that U is homeomorphic to the Hilbert space l_2 (or to the countable power of the real line).
Suppose G is a topological group containing a (closed) topological copy of the Frechet-Urysohn fan. If G is a perfectly normal sequential space (a normal k-space) then every closed metrizable subset in $G$ is locally compact. Applying this result to topological groups whose underlying topological space can be written as a direct limit of a sequence of closed metrizable subsets, we get that every such a group either is metrizable or is homeomorphic to the product of a $k_ω$-space and a discrete space.
In this paper a generalization of Urysohn's metrization theorem is given for higher cardinals. Namely, it is shown that a topological space with a basis of cardinality at most $|ω_μ|$ or smaller is $ω_μ$-metrizable if and only if it is $ω_μ$-additive and regular, or, equivalently, $ω_μ$-additive, zero-dimensional, and T\textsubscript{0}. Furthermore, all such spaces are shown to be embeddable in a suitable generalization of Hilbert's cube.
Let $X$ be a metrizable space and ${\rm Comp}(X)$ be the hyperspace consisting of non-empty compact subsets of $X$ endowed with the Vietoris topology. In this paper, we give a necessary and sufficient condition on $X$ for ${\rm Comp}(X)$ to be homeomorphic to a non-separable Hilbert space. Moreover, we consider the topological structure of pair $({\rm Comp}(\overline{X}),{\rm Fin}(X))$ of hyperspaces of $X$ and its completion $\overline{X}$, where ${\rm Fin}(X)$ is the hyperspace of non-empty finite sets in $X$.
Let $X$ be a universal (Urysohn) space. We prove that every topological fractal is homeomorphic (isometric) to the attractor $A_{\mathcal F}$ of a function system ${\mathcal F}$ on $X$ consisting of Rakotch contractions.
For a metrizable space $X$ of density $κ$, let $PM(X)$ be the space of continuous bounded pseudometrics on $X$ endowed with the uniform convergence topology. In this paper, its topology shall be classified as follows: (i) If $X$ is finite, then $PM(X)$ is homeomorphic to $\{0\}$ when $X$ is a singleton, and then $PM(X)$ is homeomorphic to $[0,1]^{κ(κ- 1)/2 - 1} \times [0,1)$ when $κ> 1$; (ii) If $X$ is infinite and generalized compact, then $PM(X)$ is homeomorphic to the Hilbert space $\ell_2(2^{< κ})$ of density $2^{< κ}$; (iii) If $X$ is not generalized compact, then $PM(X)$ is homeomorphic …
The counterparts of the Urysohn universal space in category of metric spaces and the Gurarii space in category of Banach spaces are constructed for separable valued Abelian groups of fixed (finite) exponents (and for valued groups of similar type) and their uniqueness is established. Geometry of these groups, denoted by G_r(N), is investigated and it is shown that each of G_r(N)'s is homeomorphic to the Hilbert space l^2. Those of G_r(N)'s which are Urysohn as metric spaces are recognized. `Linear-like' structures on G_r(N) are studied and it is proved that every separable metrizable topologic…
In this paper we further study links between concentration of measure in topological transformation groups, existence of fixed points, and Ramsey-type theorems for metric spaces. We prove that whenever the group $\Iso(\U)$ of isometries of Urysohn's universal complete separable metric space $\mathbb U$, equipped with the compact-open topology, acts upon an arbitrary compact space, it has a fixed point. The same is true if $\U$ is replaced with any generalized Urysohn metric space $U$ that is sufficiently homogeneous. Modulo a recent theorem by Uspenskij that every topological group embeds into…
It is shown that the hyperspace of all nonempty closed subsets $\Cld_{AW}(X)$ of a separable metric space $X$ endowed with the Attouch-Wets topology is homeomorphic to a separable Hilbert space if and only if the completion of $X$ is proper, locally connected and contains no bounded connected component, $X$ is topologically complete and not locally compact at infinity.
We establish that every second countable completely regularly preordered space (E,T,\leq) is quasi-pseudo-metrizable, in the sense that there is a quasi-pseudo-metric p on E for which the pseudo-metric p\veep^-1 induces T and the graph of \leq is exactly the set {(x,y): p(x,y)=0}. In the ordered case it is proved that these spaces can be characterized as being order homeomorphic to subspaces of the ordered Hilbert cube. The connection with quasi-pseudo-metrization results obtained in bitopology is clarified. In particular, strictly quasi-pseudometrizable ordered spaces are characterized as bei…
We prove that for k an uncountable cardinal, there exist 2^k many non homeomorphic weakly compact convex subsets of weight k in the Hilbert space of density k.
Let $X, Y$ be separable metrizable spaces, where $X$ is noncompact and $Y$ is equipped with an admissible complete metric $d$. We show that the space $C(X,Y)$ of continuous maps from $X$ into $Y$ equipped with the uniform topology is locally homeomorphic to the Hilbert space of weight $2^{\aleph_0}$ if (1) $(Y, d)$ is an ANRU, a uniform version of ANR and (2) the diameters of components of $Y$ is bounded away from zero. The same conclusion holds for the subspace $C_B(X,Y)$ of bounded maps if $Y$ is a connected complete Riemannian manifold.
Let $CLB_H(X)$ denote the hyperspace of closed bounded subsets of a metric space $X$, endowed with the Hausdorff metric topology. We prove, among others, that natural dense subspaces of $CLB_H(R^m)$ of all nowhere dense closed sets, of all perfect sets, of all Cantor sets and of all Lebesgue measure zero sets are homeomorphic to the Hilbert space $\ell_2$. Moreover, we investigate the hyperspace $CL_H(R)$ of all nonempty closed subsets of the real line $R$ with the Hausdorff (infinite-valued) metric. We show that a nonseparable component of $CL_H(R)$ is homeomorphic to the Hilbert space $\ell_…
We investigate certain geometric properties of the spaces of idempotent measures. In particular, we prove that the space of idempotent measures on an infinite compact metric space is homeomorphic to the Hilbert cube.
Let ${\rm Fin}(X)$ be the hyperspace consisting of non-empty finite subsets of a space $X$ endowed with the Vietoris topology. In this paper, we characterize a metrizable space $X$ whose hyperspace ${\rm Fin}(X)$ is homeomorphic to the linear subspace spanned by the canonical orthonormal basis of a non-separable Hilbert space.
The aim of this work is to give a pointfree description of the Cantor set. It can be shown that the Cantor set is homeomorphic to the $p$-adic integers $\mathbb{Z}_{p}:=\{x\in\mathbb{Q}_{p}: |x|_p\leq 1\}$ for every prime number $p$. To give a pointfree description of the Cantor set, we specify the frame of $\mathbb{Z}_{p}$ by generators and relations. We use the fact that the open balls centered at integers generate the open subsets of $\mathbb{Z}_{p}$ and thus we think of them as the basic generators; on this poset we impose some relations and then the resulting quotient is the frame of the …
Let $X$ be a Banach space and $Conv_H(X)$ be the space of non-empty closed convex subsets of $X$, endowed with the Hausdorff metric $d_H$. We prove that each connected component of the space $Conv_H(X)$ is homeomorphic to one of the spaces: a singleton, the real line, a closed half-plane, the Hilbert cube multiplied by the half-line, the separable Hilbert space, or a Hilbert space of density not less than continuum.
CONTENTS: On Selective screenability and examples of R. Pol. Workshops and conferences: The Oxford Conference on Topology and Computer Science in Honour of Peter Collins and Mike Reed; Boise Extravaganza In Set Theory (BEST2006). Research announcements: The isometry group of the Urysohn space as a Levy group; Chasing Silver; Disjoint Non-Free Subgoups of Abelian Groups; Characterizing metric spaces whose hyperspaces are absolute neighborhood retracts; A Vitali set can be homeomorphic to its complement; o-Boundedness of free objects over a Tychonoff space; On the consistency strength of the Mil…
For an infinite cardinal $κ$ let $\ell_2(κ)$ be the linear hull of the standard othonormal base of the Hilbert space $\ell_2(κ)$ of density $κ$. We prove that a non-separable convex subset $X$ of density $κ$ in a locally convex linear metric space if homeomorphic to the space (i) $\ell_2^f(κ)$ if and only if $X$ can be written as countable union of finite-dimensional locally compact subspaces, (ii) $[0,1]^ω\times \ell_2^f(κ)$ if and only if $X$ contains a topological copy of the Hilbert cube and $X$ can be written as a countable union of locally compact subspaces.
By a metric fractal we understand a compact metric space $K$ endowed with a finite family $\mathcal F$ of contracting self-maps of $K$ such that $K=\bigcup_{f\in\mathcal F}f(K)$. If $K$ is a subset of a metric space $X$ and each $f\in\mathcal F$ extends to a contracting self-map of $X$, then we say that $(K,\mathcal F)$ is a fractal in $X$. We prove that each metric fractal $(K,\mathcal F)$ is $\bullet$ isometrically equivalent to a fractal in the Banach spaces $C[0,1]$ and $\ell_\infty$; $\bullet$ bi-Lipschitz equivalent to a fractal in the Banach space $c_0$; $\bullet$ isometrically equivale…
We prove that each non-separable completely metrizable convex subset of a Frechet space is homeomorphic to a Hilbert space. This resolves an old (more than 30 years) problem of infinite-dimensional topology. Combined with the topological classification of separable convex sets due to Klee, Dobrowoslki and Torunczyk, this result implies that each closed convex subset of a Frechet space is homemorphic to $[0,1]^n\times [0,1)^m\times l_2(k)$ for some cardinals $0\le n\leω$, $0\le m\le 1$ and $k\ge 0$.
Denote by $[0,ω_1)$ the locally compact Hausdorff space consisting of all countable ordinals, equipped with the order topology, and let $C_0[0,ω_1)$ be the Banach space of scalar-valued, continuous functions which are defined on $[0,ω_1)$ and vanish eventually. We show that a weakly$^*$ compact subset of the dual space of $C_0[0,ω_1)$ is either uniformly Eberlein compact, or it contains a homeomorphic copy of the ordinal interval $[0,ω_1]$. Using this result, we deduce that a Banach space which is a quotient of $C_0[0,ω_1)$ can either be embedded in a Hilbert-generated Banach space, or it is i…
