Related papers: PL and differential topology in o-minimal structur…
We initiate an investigation of structures on the set of real numbers having the property that path components of definable sets are definable. All o\nobreakdash-\hspace{0pt}minimal structures on $(\mathbb{R},<)$ have the property, as do…
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 define a discrete closure operation for definably complete locally o-minimal structures $\mathcal M$. The pair of the underlying set of $\mathcal M$ and the discrete closure operation forms a pregeometry. We define the rank of a…
Hilbert initiated the standpoint in foundations of mathematics. From this standpoint, we allow only a finite number of repetitions of elementary operations when we construct objects and morphisms. When we start from a subset of a Euclidean…
In this paper we work in an arbitrary o-minimal structure with definable Skolem functions and we prove that definably connected, locally definable manifolds are uniformly definably path connected, have an admissible cover by definably…
We consider locally o-minimal structures possessing tame topological properties shared by models of DCTC and uniformly locally o-minimal expansions of the second kind of densely linearly ordered abelian groups. We derive basic properties of…
We demonstrate that And\'ujar Guerrero, Thomas and Walsberg's results on definable compactness in o-minimal structures still hold true in definably complete locally o-minimal structures. As an application, we show that a definably simple…
In this paper we prove tight bounds on the combinatorial and topological complexity of sets defined in terms of $n$ definable sets belonging to some fixed definable family of sets in an o-minimal structure. This generalizes the…
We study groups and rings definable in d-minimal expansions of ordered fields. We generalize to such objects some known results from o-minimality. In particular, we prove that we can endow a definable group with a definable topology making…
These notes focus on the Lipschitz geometry of sets that are definable in o-minimal structures (expanding the real field). We show that every set which is definable in a polynomially bounded o-minimal structure admits a stratification which…
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…
Every o-minimal expansion R-tilde of the real field has an o-minimal expansion P(R-tilde) in which the solutions to Pfaffian equations with definable C^1 coefficients are definable.
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 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…
A subset of a topological space is constructible if it is a finite Boolean combination of closed sets. We prove that every NTP$_2$ expansion of $(\mathbb{R},<,+)$ by constructible sets defines only constructible sets, and that definable…
This paper addresses some questions about dimension theory for P-minimal structures. We show that, for any definable set A, the dimension of the frontier of A is strictly smaller than the dimension of A itself, and that A has a…
Let $T$ be a theory which is t-minimal, meaning that with respect to some definable topology, a unary definable set $D \subseteq M$ has non-empty interior iff it is infinite. If $K$ is a definable field in $T$, then $K$ is finite or "large"…
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…
Dense pairs of geometric topological fields have tame open core, that is, every definable open subset in the pair is already definable in the reduct. We fix a minor gap in the published version of van den Dries's seminal work on dense pairs…
In this paper, we study definably compact semigroups in o-minimal structures, aiming to extend the theory of definable groups to a broader algebraic setting. We show that any definably compact semigroup contains idempotents and admits a…