Related papers: Around definable types in $p$-adically closed fiel…

We continue our earlier study of finite dimensional definable groups in models of the the model companion of an o-minimal L-theory T expanded by a generic derivation as in [F-K]. We generalize Buium's notion of an algebraic D-group to…

We introduce the notion of a definable category--a category equivalent to a full subcategory of a locally finitely presentable category that is closed under products, directed colimits and pure subobjects. Definable subcategories are…

Universality has been an important concept in computable structure theory. A class $\mathcal{C}$ of structures is universal if, informally, for any structure, of any kind, there is a structure in $\mathcal{C}$ with the same…

Let $K$ be a field. The \'etale open topology on the $K$-points $V(K)$ of a $K$-variety $V$ was introduced in our previous work. The \'etale open topology is non-discrete if and only if $K$ is large. If $K$ is separably, real, $p$-adically…

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…

Let G be a definably compact group in an o-minimal expansion of a real closed field. We prove that if dim(G X) < dim G for some definable X subset of G then X contains a torsion point of G. Along the way we develop a general theory for…

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 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…

We consider an arbitrary topological group $G$ definable in a structure $\mathcal M$, such that some basis for the topology of $G$ consists of sets definable in $\mathcal M$. To each such group $G$ we associate a compact $G$-space of…

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…

This paper has two parts. In the first one, we prove that an invariant dp-minimal type is either finitely satisfiable or definable. We also prove that a definable version of the (p,q)-theorem holds in dp-minimal theories of small or medium…

The notion of a topological Jordan decomposition of a compact element of a reductive p-adic group has proven useful in many contexts. In this paper, we generalise it to groups defined over fairly general discretely-valued fields and prove…

We continue the study of a class of topological $\mathcal{L}$-fields endowed with a generic derivation $\delta$, focussing on describing definable groups. We show that one can associate to an $\mathcal{L}_{\delta}$ definable group a type…

We are concerned with topology of Hensel minimal structures on non-trivially valued fields $K$, whose axiomatic theory was introduced in a recent paper by Cluckers-Halupczok-Rideau. We additionally require that every definable subset in the…

Recently Paw\l{}ucki showed that compact sets that are definable in some o-minimal structure admit triangulations of class $\mathcal{C}^p$ for each integer $p\geq 1$. In this work, we make use of these new techniques of triangulation to…

In this paper it is shown how to construct a finite topological space $X$ for a given finitely presentable group $G$ such that $\pi_1(X)\cong G$. Our construction is not optimal in the sense that the cardinality of the space $X$ might not…

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.

We prove that definable ring topologies on NIP fields are closely connected to NIP integral domains. More precisely, we show that up to elementary equivalence, any NIP topological field arises from an NIP integral domain. As an application,…

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…

Peterzil and Steinhorn proved that if a group $G$ definable in an $o$-minimal structure is not definably compact, then $G$ contains a definable torsion-free subgroup of dimension one. We prove here a $p$-adic analogue of the…