Scoring metrics on separable metric spaces
Kerry M. Soileau
我们定义评分度量可分离的度量空间，并表明，他们总是没有粗糙的度量，他们的春天。
We define scoring metrics on separable metric spaces and show that they are always no coarser than the metrics from which they spring.
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Transfinite sequences of continuous and Baire 1 functions on separable metric spaces
研究了可分度量空间上Baire 1函数良序序列的存在性.
Given a metric space $\langle X,ρ\rangle$, consider its hyperspace of closed sets $CL(X)$ with the Wijsman topology $τ_{W(ρ)}$. It is known that $\langle{CL(X),τ_{W(ρ)}}\rangle$ is metrizable if and only if $X$ is separable and it is an open question by Di Maio and Meccariello whether this is equivalent to $\langle{CL(X),τ_{W(ρ)}}\rangle$ being normal. In this paper we prove that if the weight of $X$ is a regular uncountable cardinal and $X$ is locally separable, then $\langle{CL(X),τ_{W(ρ)}}\rangle$ is not normal. We also solve some questions by Cao, Junnilla and Moors regarding isolated poin…
Generic separable metric structures
我们比较了三个概念的可分度量结构的泛型。我们的分析提供了一个一般的模型理论技术，表明结构是通用的描述集理论（拓扑）的意义上，在测度论意义上。特别是，它给出了一个新的视角Vershik定理的一般性和Urysohn的空间之间的可分度量空间的随机性。
Let X be a Hausdorff quotient of a standard space (that is of a locally compact separable metric space). It is shown that the following are equivalent: (i) X is the image of an irreducible quotient map from a standard space; (ii) X has a sequentially dense subset satisfying two technical conditions involving double sequences; (iii) whenever q : Y\to X is a quotient map from a standard space Y , the restriction q_{\st}|V is an irreducible quotient map from V onto X (where q_{\st} : Y_{\st}\to X is the pure quotient derived from q, and V is the closure of the set of singleton fibres of Y_{\st}).…
Let X be a separable metric space and let βbe the strict topology on the space of bounded continuous functions on X, which has the space of τ-additive Borel measures as a continuous dual space. We prove a Banach-Dieudonneé type result for the space of bounded continuous functions equipped with β. As a consequence, this space is hypercomplete and a Pták space. Additionally, the closed graph, inverse mapping and open mapping theorems holds for linear maps between space of this type.
The Urysohn universal metric space is homeomorphic to a Hilbert space
Urysohn泛度量空间U在等距上具有如下性质：（1）U是完备的和可分的;（2）U包含每个可分的度量空间的等距副本;（3）U的两个有限子集之间的每个等距都可以推广为U到自身上的等距。我们证明了U同胚于希尔伯特空间l_2（或实直线的可数幂）。
Denote by Q(k) a σ-discrete metric weight-homogeneous space of weight k. We give an internal description of the space Q(k)^ω. We prove that the Baire space B(k) is densely homogeneous with respect to Q(k)^ωif k > ω. Properties of some non-separable h-homogeneous absolute F_{σδ}-sets and G_{δσ}-sets are investigated.
Connected, not separably connected complete metric spaces
在可分连通空间中，任意两点都包含在可分连通子集中。我们展示了一种机制，采取一个连接的有界度量空间，并产生一个完整的连接度量空间，其可分离的组件形成一个商空间等距的原始空间。我们反复应用这种机制来构造，作为一个逆极限，一个完整的连接度量空间的每个可分子集是零维的。
We prove that for any constant $K>0$ there exists a separable group equipped with a complete bi-invariant metric bounded by $K$, isometric to the Urysohn sphere of diameter $K$, that is of `almost-universal disposition'. It is thus an object in the category of separable groups with bi-invariant metric analogous in its properties to the Gurarij space from the category of separable Banach spaces. We show that this group contains an isometric copy of any separable group equipped with bi-invariant metric bounded by $K$. As a consequence, we get that it is a universal Polish group admitting compati…
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.
In this paper we introduce a new class of metric actions on separable (not necessarily connected) metric spaces called "Cauchy-indivisible" actions. This new class coincides with that of proper actions on locally compact metric spaces and, as examples show, it may be different in general. The concept of "Cauchy-indivisibility" follows a more general research direction proposal in which we investigate the impact of basic notions in substantial results, like the impact of local compactness and connectivity in the theory of proper transformation groups. In order to provide some basic theory for t…
A Nonseparably Connected Metric Space as a Dense Connected Graph
给出了一个不含非平凡可分连通子空间的连通度量空间。我们的空间是一个稠密的连通图的功能，从真正的线满足柯西方程。
Countable dense homogeneity of definable spaces
研究了可定义的可分度量空间是可数稠密齐次空间的问题。证明了Borel CDH空间是完全度量化的，并给出了零维Borel CDH空间的完整列表。我们还证明了对于2 ^omega的Borel X子集，下列等价：（1）X是2 ^omega中的G_delta，（2）X^omega是CDH，（3）X^omega同胚于2 ^omega或omega^omega.在投影确定性公理的假设下，结果推广到所有的投射集，并在确定性公理下推广到所有的可分度量空间。特别地，模大基数假设所有CDH可分度量空间都是完全可度量化的，这与ZF是相对一致的。我们还回答了Steprans和Zhou的一个问题，证明了基数p = min{kappa：2^kappa不是CDH}。
Let f be a continuous map of a complete separable metric space E onto the irrationals. We show that if a complete separable metric space M contains isometric copies of every closed relatively discrete set in E, then M contains also an isometric copy of some fiber of f. We shall show also that if all fibers of f have positive dimension, then the collection of closed zero-dimensional sets in E is non-analytic in the Wijsman hyperspace of E.
How many miles to beta-X? -- d miles, or just one foot
已知可度量化空间X的Stone-Cech紧化可近似为X的Smirnov紧化的集合，其中X是X上所有相容度量的集合。如果我们把自己限制在局部紧的可分度量化空间，相应的陈述对Higson紧化也成立。我们研究了X上相容度量集D的最小基数，使得X的Stone-Cech紧化对D中的所有度量都可用Smirnov或Higson紧化来逼近.证明了对于局部紧可分度量空间，它要么是控制数，要么是1。
Plastic metric spaces and groups
本文研究了塑料度量空间的性质，证明了可数稠密子空间非塑料，以及可分度量空间和严格凸度量群中存在塑料稠密子空间。
We provide a very short elementary proof that every bounded separable metric group embeds into a monothetic bounded metric group, in such a way that the result of Morris and Pestov that every separable abelian topological group embeds into a monothetic group is an immediate corollary. We show that the boundedness assumption is essential.
Almost isometric embeddings of metric spaces
研究了度量空间之间的几乎等距嵌入和几乎等距的关系，证明了：（1）存在可数泛度量空间. (2)在aleph_1上可能存在少于连续统可分的度量空间，使得当连续统假设失效时，每个可分度量空间几乎等距嵌入其中一个。(3)不存在少于基数aleph_2的连续度量空间的集合，使得基数aleph_2的每个超度量空间几乎等距嵌入其中一个，如果aleph_2<2^{aleph_0}。我们还证明了各种空间X满足：如果空间X几乎等距于X，则Y等距于X。
Isometric shifts and metric spaces
设M是完全度量空间.证明了如果M上的空间或标值有界连续函数允许等距移位，则M是可分的。
We construct a universal action of a countable locally finite group (the Hall's group) on a separable metric space by isometries. This single action contains all actions of all countable locally finite groups on all separable metric spaces as subactions. The main ingredient is the amalgamation of actions by isometries. We show that an equivalence class of this universal action is generic. We show that the restriction to locally finite groups in our results is necessary as analogous results do not hold for infinite non-locally finite groups. We discuss the problem also for actions by linear iso…
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。
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.
We prove that many completeness properties coincide in metric spaces, precompact groups and dense subgroups of products of separable metric groups. We apply these results to function spaces C_p(X,G) of G-valued continuous functions on a space X with the topology of pointwise convergence, for a separable metric group G. Not only the results but also the proofs themselves are novel even in the classical case when G is the real line. A space X is weakly pseudocompact if it is G_delta-dense in at least one of its compactifications. A topological group G is precompact if it is topologically isomorp…
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