Wieslaw Kubiś, Matatyahu Rubin
We prove some extension theorems involving uniformly continuous maps of the universal Urysohn space. We also prove reconstruction theorems for certain groups of autohomeomorphisms of this space and of its open subsets.
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In a paper published posthumously, P.S. Urysohn constructed a complete, separable metric space that contains an isometric copy of every complete separable metric space, nowadays referred to as the Urysohn universal space. Here we study various convexity properties of the Urysohn universal space and show that it has a finite ball intersection property. We also note that Urysohn universal space is not hyperconvex.
We establish universality and ultra-homogeneity of $(\mathcal{U},u_\mathrm{GH})$, the collection of all compact ultrametric spaces endowed with the so-called Gromov-Hausdorff ultrametric. This result also gives rise to a novel construction of the so-called $R$-Uryoshn universal ultrametric space for each countable subset $R\subset\mathbb{R}_{\geq 0}$ containing $0$.
Recently, much interest was devoted to the Urysohn universal metric space U and its isometries; this paper is a contribution to this field of research. In particular, we study some properties of isometries of U, and prove the following result, which answers a question of Urysohn: if X is a subset of U is such that, for any X' which is isometric to X there exists an isometry of U which maps X to X', then X is compact (the converse was already well-known). We also answer in the negative a question of Clemens, proving that any Polish metric space is isometric to the set of fixed points of some is…
In this paper, we give new constructions of Urysohn universal ultrametric spaces. We first characterize a Urysohn universal ultrametric subspace of the space of all continuous functions whose images contain the zero, from a zero-dimensional compact Hausdorff space without isolated points into the space of non-negative real numbers equipped with the nearly discrete topology. As a consequence, the whole function space is Urysohn universal, which can be considered as a non-Archimedean analog of Banach--Mazur theorem. As a more application, we prove that the space of all continuous pseudo-ultramet…
The Urysohn space is a complete separable metric space, universal among separable metric spaces for extending finite partial isometries into it. We present an alternative construction of the Urysohn space which enables us to show that extending isometries can be done in a canonical and continuous way, and allows us to equip the Urysohn space with algebraic structure. This is achieved in a constructive setting without assuming any choice principles.
The Urysohn space is a separable complete metric space with two fundamental properties: (a) universality: every separable metric space can be isometrically embedded in it; (b) ultrahomogeneity: every finite isometry between two finite subspaces can be extended to an auto-isometry of the whole space. The Urysohn space is uniquely determined up to isometry within separable metric spaces by these two properties. We introduce an analogue of the Urysohn space for diversities, a recently developed variant of the concept of a metric space. In a diversity any finite set of points is assigned a non-neg…
Characterizations of Urysohn universal ultrametric spaces
本研究通过闭球无穷等距子集刻画超度量空间的有限内射性与Urysohn万有超度量空间，证明笛卡尔积、超空间、对称积保持该空间结构，且完备有限内射超度量空间必含其等距子空间。
A metric space $\mathrm{M}=(M;\de)$ is {\em homogeneous} if for every isometry $f$ of a finite subspace of $\mathrm{M}$ to a subspace of $\mathrm{M}$ there exists an isometry of $\mathrm{M}$ onto $\mathrm{M}$ extending $f$. A metric space $\boldsymbol{U}$ is an {\em Urysohn} metric space if it is homogeneous and separable and complete and if it isometrically embeds every separable metric space $\mathrm{M}$ with $\dist(\mathrm{M})\subseteq \dist(\boldsymbol{U})$. (With $\dist(\mathrm{M})$ being the set of distances between points in $\mathrm{M}$.) The main results are: (1) A characterization of…
Uniqueness and homogeneity of non-separable Urysohn universal ultrametric spaces
本文研究了非可分Urysohn泛超度量空间的等距问题，引入petaloid超度量空间概念，证明了所有petaloid空间彼此等距且对有限及紧子空间具有齐次性，并验证了四类空间满足该性质。
Answering a question of Gao and Kechris, we show that, given any polish group G, there exists a closed subset F of Urysohn's universal metric space U such that G is (topologically) isomorphic to the subgroup of isometries of U which map F onto itself.
We constract various subgroups of the group of isometries of universal Urysohn spaces (unique complete separable metric space which is iniversal and homogeneous) including abelian groups which act transitively, and free groups which are dense in the full isometry group.
Convergence of metric measure spaces via embeddings in the Urysohn universal space
本文通过将度量测度空间等距嵌入Urysohn万有空间，证明了Gromov盒拓扑与弱拓扑的商拓扑一致，并构造了完备可分的截断Wasserstein距离，与盒距离Hölder等价。
We prove the equivalence of the two important facts about finite metric spaces and universal Urysohn metric spaces $\Bbb U$, namely theorem A and theorem B below: Theorem A (Approximation): The group of isometry $ISO(\Bbb U)$ contains everywhere dense locally finite subgroup; Theorem G(Globalization): For each finite metric space $F$ there exists another finite metric space $\bar F$ and isometric imbedding $j$ of $F$ to $\bar F$ such that isometry $j$ induces the imbedding of the group monomorphism of the group of isometries of the space $F$ to the group of isometries of space $\bar F$ and eac…
A metric space $\mathrm{M}=(M;\de)$ is {\em homogeneous} if for every isometry $α$ of a finite subspace of $\mathrm{M}$ to a subspace of $\mathrm{M}$ there exists an isometry of $\mathrm{M}$ onto $\mathrm{M}$ extending $α$. The metric space $\mathrm{M}$ is {\em universal} if it isometrically embeds every finite metric space $\mathrm{F}$ with $\dist(\mathrm{F})\subseteq \dist(\mathrm{M})$. ($\dist(\mathrm{M})$ being the set of distances between points of $\mathrm{M}$.) A metric space $\mathrm{M}$ is {\em oscillation stable} if for every $ε>0$ and every uniformly continuous and bounded function …
Algebraic structures on non-Archimedean Urysohn universal metric spaces
研究乌里松通用超度量空间上的值域结构，引入\(p\)-adic Levi–Civita 域等，证明具有无限剩余域的完备值域对可分超度量空间通用，还给出可分情况下值域与乌里松空间等距及特定剩余域时全 Hahn 型值域成为乌里松通用超度量空间的条件。
We solve the oscillation stability problem for the Urysohn sphere, an analog of the distortion problem for the Hilbert space in the context of the Urysohn universal metric space. This is achieved by solving a purely combinatorial problem involving a family of countable homogeneous metric spaces with finitely many distances.
A theorem proved by Hrushovski for graphs and extended by Solecki and Vershik (independently from each other) to metric spaces leads to a stronger version of ultrahomogeneity of the infinite random graph $R$, the universal Urysohn metric space $\Ur$, and other related objects. We show how the result can be used to average out uniform and coarse embeddings of $\Ur$ (and its various counterparts) into normed spaces. Sometimes this leads to new embeddings of the same kind that are metric transforms and besides extend to affine representations of various isometry groups. As an application of this …
WWe define the notion of a random metric space and prove that with probability one such a space is isometricto the Urysohn universal metric space. The main technique is the study of universal and random distance matrices; we relate the properties of metric (in particulary universal) space to the properties of distance matrices. We show the link between those questions and classification of the Polish spaces with measure (Gromov or metric triples) and with the problem about S_{\infty}-invariant measures in the space of symmetric matrices. One of the new effects -exsitence in Urysohn space so ca…
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.
We introduce a model of the set of all Polish (=separable complete metric) spaces: the cone $\cal R$ of distance matrices, and consider geometric and probabilistic problems connected with this object. The notion of the universal distance matrix is defined and we proved that the set of such matrices is everywhere dense $G_δ$ set in weak topology in the cone $\cal R$. Universality of distance matrix is the necessary and sufficient condition on the distance matrix of the countable everywhere dense set of so called universal Urysohn space which he had defined in 1924 in his last paper. This means …
We use Fra\" issé theoretic methods to construct several universal and ultrahomogeneous Polish metric structures. Namely, universal and ultrahomogeneous Polish metric space equipped with countably many closed subsets of its powers, universal and ultrahomogeneous Polish metric space equipped with a closed subset of the product of itself and some fixed compact metric space, and universal and ultrahomogeneous Polish metric space equipped with an L-Lipschitz function, for an arbitrary positive L, to some fixed Polish metric space. These results are direct generalization of the classical result of …
We consider the problem of isometric embedding of metric spaces to the Banach spaces; and introduce and study the remarkable class of so-called linearly rigid metric spaces: these are the spaces that admit a unique, up to isometry, linearly dense isometric embedding into a Banach space. The first nontrivial example of such a space was given by R. Holmes; he proved that the universal Urysohn space has this property. We give a criterion of linear rigidity of a metric space, which allows us to give a simple proof of the linear rigidity of the Urysohn space and some other metric spaces. The variou…
On Metrics for Analysis of Functional Data on Geometric Domains
本文利用度量几何与最优传输理论，为度量空间上的函数数据（场）引入Gromov-Hausdorff等距离的场类比，建立离散模型与经验估计，并证明相关滤流的稳定性。
On universal-homogeneous hyperbolic graphs and spaces and their isometry groups
本文研究了δ-超几何空间的通用-均匀性，构造了满足特定条件的δ-超几何空间，并证明其等距群的性质。
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