Related papers: Universal and ultrahomogeneous Polish metric struc…
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…
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…
We show, following W. Holsztynski, that there exists a continuous metric d on the set of real numbers R such that any finite metric space is isometrically embeddable into (R,d).
This paper presents a study of generic elements in full isometry groups of Polish ultrametric spaces. We obtain a complete characterization of Polish ultrametric spaces X whose isometry group Iso(X) contains an open subgroup H with ample…
In this paper we develop a technique of constructing uni- formly continuous maps between function spaces Cp(X) endowed with the pointwise topology. We prove that if a space X is compact metrizable and strongly countable-dimensional, then…
Urysohn constructed a separable complete universal metric space homogeneous for all finite subspaces, which is today called the Urysohn universal metric space. Some authors have recently investigated an ultrametric analogue of this space.…
A Lusin space is a Hausdorff space being the image of a Polish space under a continuous bijection. Such spaces have multiple applications, in particular, as state spaces of various stochastic systems. In this work, we consider the spaces…
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.
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…
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…
The notion of the ultrametrics can be considered as a zero-dimensional analogue of ordinary metrics, and it is expected to prove ultrametric versions of theorems on metric spaces. In this paper, we provide ultrametric versions of the…
The paper is concerned with b-metric and generalized b-metric spaces. One proves the existence of the completion of a generalized b-metric space and some fixed point results. The behavior of Lipschitz functions on b-metric spaces of…
We use countable metric spaces to code Polish metric spaces and evaluate the complexity of some statements about these codes and of some relations that can be determined by the codes. Also, we propose a coding for continuous functions…
We compute characteristic functionals of Dirichlet-Ferguson measures over a locally compact Polish space and prove continuous dependence of the random measure on the parameter measure. In finite dimension, we identify the dynamical symmetry…
We show that, on any given finite Borel measure space with the ambient space being a Polish metric space, every Borel real-valued function is almost a bounded, uniformly continuous function in the sense that for every $\varepsilon > 0$…
In this paper we provide several \emph{metric universality} results. We exhibit for certain classes $\cC$ of metric spaces, families of metric spaces $(M_i, d_i)_{i\in I}$ which have the property that a metric space $(X,d_X)$ in $\cC$ is…
It is proved that any countable index, universally measurable subgroup of a Polish group is open. By consequence, any universally measurable homomorphism from a Polish group into the infinite symmetric group $S_\infty$ is continuous. It is…
In this paper, we investigate the set $\mathcal{U}(\mathbb{U})$ of universal and ultrahomogeneous $1$-Lipschitz retractions acting on the Urysohn space as the subspace of the space $\mathcal{R}(\mathbb{U})$ of all $1-$Lipschitz retractions…
In 2003, Kechris, Pestov and Todorcevic showed that the structure of certain separable metric spaces - called ultrahomogeneous - is closely related to the combinatorial behavior of the class of their finite metric spaces. The purpose of the…
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…