44 research outputs found
Lipschitz functions on topometric spaces
We study functions on topometric spaces which are both (metrically) Lipschitz
and (topologically) continuous, using them in contexts where, in classical
topology, ordinary continuous functions are used. We study the relations of
such functions with topometric versions of classical separation axioms, namely,
normality and complete regularity, as well as with completions of topometric
spaces. We also recover a compact topometric space from the lattice of
continuous -Lipschitz functions on , in analogy with the recovery of a
compact topological space from the structure of (real or complex) functions
on
Metrizable universal minimal flows of Polish groups have a comeagre orbit
We prove that, whenever is a Polish group with metrizable universal
minimal flow , there exists a comeagre orbit in . It then follows
that there exists an extremely amenable, closed, coprecompact of such
that
Generic orbits and type isolation in the Gurarij space
We study the question of when the space of embeddings of a separable Banach
space into the separable Gurarij space admits a generic orbit
under the action of the linear isometry group of . The question is
recast in model-theoretic terms, namely type isolation and the existence of
prime models. We characterise isolated types over using tools from convex
analysis. We show that if the set of isolated types over is dense, then a
dense orbit exists, and otherwise all orbits are meagre. We then
study some (families of) examples with respect to this dichotomy. We also point
out that the class of Gurarij spaces is the class of models of an
-categorical theory with quantifier elimination, and calculate the
density character of the space of types over , answering a question of
Avil{\'e}s et al
Continuous first order logic and local stability
We develop continuous first order logic, a variant of the logic described in
\cite{Chang-Keisler:ContinuousModelTheory}. We show that this logic has the
same power of expression as the framework of open Hausdorff cats, and as such
extends Henson's logic for Banach space structures. We conclude with the
development of local stability, for which this logic is particularly
well-suited
Polish topometric groups
International audienceWe define and study the notion of \emph{ample metric generics} for a Polish topological group, which is a weakening of the notion of ample generics introduced by Kechris and Rosendal in \cite{Kechris-Rosendal:Turbulence}. Our work is based on the concept of a \emph{Polish topometric group}, defined in this article. Using Kechris and Rosendal's work as a guide, we explore consequences of ample metric generics (or, more generally, ample generics for Polish topometric groups). Then we provide examples of Polish groups with ample metric generics, such as the isometry group \Iso(\bU_1) of the bounded Urysohn space, the unitary group of a separable Hilbert space, and the automorphism group \Aut([0,1],\lambda) of the Lebesgue measure algebra on . We deduce from this and earlier work of Kittrell and Tsankov that this last group has the automatic continuity property, i.e., any morphism from \Aut([0,1],\lambda) into a separable topological group is continuous
Definability of groups in -stable metric structures
We prove that in a continuous -stable theory every type-definable
group is definable. The two main ingredients in the proof are:
\begin{enumerate} \item Results concerning Morley ranks (i.e., Cantor-Bendixson
ranks) from \cite{BenYaacov:TopometricSpacesAndPerturbations}, allowing us to
prove the theorem in case the metric is invariant under the group action; and
\item Results concerning the existence of translation-invariant definable
metrics on type-definable groups and the extension of partial definable metrics
to total ones. \end{enumerate