Related papers: Types, transversals and definable compactness in o…
Arguments on PL,(=piecewise linear) topology work over any ordered field in the same way as over the real field, and those on differential topology do over a real closed field R in an o-minimal structure that expands (R,<,0,1,+,cdot). One…
We prove several structural results on definably compact groups G in o-minimal expansions of real closed fields, such as (i) G is definably an almost direct product of a semisimple group and a commutative group, and (ii) the group (G, .) is…
We settle some open problems in the special case of groups in o-minimal structures, such as the equality of G^00 and G^000 and the equivalence of definable amenability and existence of a type with bounded orbit. We prove almost exactness of…
In this paper we work in o-minimal structures with definable Skolem functions and show that a continuous definable map between Hausdorff locally definably compact definable spaces is definably proper if and only if it is proper morphism in…
We develop in this paper the theory of covers for Hausdorff properly $\bigvee $-definable manifolds with definable choice in an o-minimal structure $\N$. In particular, we show that given an $\N$-definably connected $\N$-definable group $G$…
We give an example of a dense o-minimal structure in which there is a definable quotient that cannot be eliminated, even after naming parameters. Equivalently, there is an interpretable set which cannot be put in parametrically definable…
Decomposable ordered structures were introduced in \cite{OnSt} to develop a general framework to study `finite-dimensional' totally ordered structures. This paper continues this work to include decomposable structures on which a ordered…
We show pro-definability of spaces of definable types in various classical complete first order theories, including complete o-minimal theories, Presburger arithmetic, $p$-adically closed fields, real closed and algebraically closed valued…
We establish the first global results for groups definable in tame expansions of o-minimal structures. Let $\mathcal N$ be an expansion of an o-minimal structure $\mathcal M$ that admits a good dimension theory. The setting includes dense…
We show that separability and second-countability are first-order properties among topological spaces definable in o-minimal expansions of $(\mathbb{R},<)$. We do so by introducing first-order characterizations -- definable separability and…
In this note we show that every definably connected, definably compact abelian definable group in an o-minimal expansion of a real closed field of dimension not 4 is definably homeomorphic to a torus of the same dimension. Moreover, in the…
We propose new structures called almost o-minimal structures and $\mathfrak X$-structures. The former is a first-order expansion of a dense linear order without endpoints such that the intersection of a definable set with a bounded open…
We study definably compact definably connected groups definable in a sufficiently saturated real closed field $R$. We introduce the notion of group-generic point for $\bigvee$-definable groups and show the existence of group-generic points…
We develop tame topology over dp-minimal structures equipped with definable uniformities satisfying certain assumptions. Our assumptions are enough to ensure that definable sets are tame: there is a good notion of dimension on definable…
We prove that groups definable in o-minimal structures have Cartan subgroups, and only finitely many conjugacy classes of such subgroups. We also delineate with precision how these subgroups cover the ambient group, in general very largely…
We prove in particular that, in a large class of dp-minimal theories including the p-adics, definable types are dense amongst non-forking types.
Let X be a definable sub-set of some o-minimal structure. We study the spectrum of X, in relation with the definability of types.
We study the topology of metric spaces which are definable in o-minimal expansions of ordered fields. We show that a definable metric space either contains an infinite definable discrete set or is definably homeomorphic to a definable set…
We study the model theory of covers of groups definable in o-minimal structures. This includes the case of covers of compact real Lie groups. In particular we study categoricity questions, pointing out some notable differences with the case…
We consider interpretable topological spaces and topological groups in a $p$-adically closed field $K$. We identify a special class of "admissible topologies" with topological tameness properties like generic continuity, similar to the…