We prove that the boolean algebras of sets definable in elementarily equivalent o-minimal expansions of real closed fields are back-and-forth equivalent, and in particular elementarily equivalent, in the language of boolean algebras with new predicates indicating the dimension, Euler characteristic and open sets. We also show that the boolean algebra of semilinear subsets of [0,1]ⁿ definable in an o-minimal expansion of a real closed field is back-and-forth equivalent to the boolean algebra of definable...

Given an o-minimal expansion ℜ of the ordered additive group of real numbers and E ⊆ ℝ, we consider the extent to which basic metric and topological properties of subsets of ℝ definable in the expansion (ℜ,E) are inherited by the subsets of ℝ definable in certain expansions of (ℜ,E). In particular, suppose that $f\left(E^m\right)$ has no interior for each m ∈ ℕ and $f:{ℝ}^m\to ℝ$ definable in ℜ, and that every subset of ℝ definable in (ℜ,E) has interior or is nowhere dense. Then every subset of ℝ definable in (ℜ,(S)) has interior...

Let K be a subfield of the real field, D ⊆ K be a discrete set and f: Dⁿ → K be such that f(Dⁿ) is somewhere dense. Then (K,f) defines ℤ. We present several applications of this result. We show that K expanded by predicates for different cyclic multiplicative subgroups defines ℤ. Moreover, we prove that every definably complete expansion of a subfield of the real field satisfies an analogue of the Baire category theorem.

The open core of a structure ℜ := (ℝ,<,...) is defined to be the reduct (in the sense of definability) of ℜ generated by all of its definable open sets. If the open core of ℜ is o-minimal, then the topological closure of any definable set has finitely many connected components. We show that if every definable subset of ℝ is finite or uncountable, or if ℜ defines addition and multiplication and every definable open subset of ℝ has finitely many connected components, then the open core of ℜ is...

Letg:U→ℝ (U open in ℝn) be an analytic and K-subanalytic (i. e. definable in ℝanK, whereK, the field of exponents, is any subfield ofℝ) function. Then the set of points, denoted Σ, whereg does not admit an analytic extension is K-subanalytic andg can be extended analytically to a neighbourhood of Ū.

We investigate several extension properties of Fréchet differentiable functions defined on closed sets for o-minimal expansions of real closed fields.

We show that Conway's field of surreal numbers with its natural exponential function has the same elementary properties as the exponential field of real numbers. We obtain ordinal bounds on the length of products, reciprocals, exponentials and logarithms of surreal numbers in terms of the lengths of their inputs. It follows that the set of surreal numbers of length less than a given ordinal is a subfield of the field of all surreal numbers if and only if this ordinal is an ε-number. In that case,...

A subset X of a group G is called left genericif finitely many left translates of X cover G. Our main result is that if G is a definably compact group in an o-minimal structure and a definable X ⊆ G is not right generic then its complement is left generic. Among our additional results are (i) a new condition equivalent to definable compactness, (ii) the existence of a finitely additive invariant measure on definable sets in a definably compact group G in the case where G = *H...

It is shown that in an elementary extension of a compact complex manifold M, the K-analytic sets (where K is the algebraic closure of the underlying real closed field) agree with the ccm-analytic sets if and only if M is essentially saturated. In particular, this is the case for compact Kähler manifolds.

An o-minimal expansion ℳ = ⟨M,<,+,0, ...⟩ of an ordered group is called semi-bounded if it does not expand a real closed field. Possibly, it defines a real closed field with bounded domain I ⊆ M. Let us call a definable set short if it is in definable bijection with a definable subset of some Iⁿ, and long otherwise. Previous work by Edmundo and Peterzil provided structure theorems for definable sets with respect to the dichotomy ’bounded versus unbounded’. Peterzil (2009) conjectured a refined...

Let R be an o-minimal field and V a proper convex subring with residue field k and standard part (residue) map st: V → k. Let $k_{ind}$ be the expansion of k by the standard parts of the definable relations in R. We investigate the definable sets in $k_{ind}$ and conditions on (R,V) which imply o-minimality of $k_{ind}$. We also show that if R is ω-saturated and V is the convex hull of ℚ in R, then the sets definable in $k_{ind}$ are exactly the standard parts of the sets definable in (R,V).

This is a generalized and improved version of our earlier article [Studia Math. 124 (1997)] on the Whitney extension theorem for subanalytic ${}^p$-Whitney fields (with p finite). In this new version we consider Whitney fields definable in an arbitrary o-minimal structure on any real closed field R and obtain an extension which is a ${}^p$-function definable in the same o-minimal structure. The Whitney fields that we consider are defined on any locally closed definable subset of Rⁿ. In such a way, a local...

We define a notion of volume for sets definable in an o-minimal structure on an archimedean real closed field. We show that given a parametric family of continuous functions on the positive cone of an archimedean real closed field definable in an o-minimal structure, the set of parameters where the integral of the function converges is definable in the same structure.

In this paper we work in o-minimal structures with definable Skolem functions, and show that: (i) a Hausdorff definably compact definable space is definably normal; (ii) a continuous definable map between Hausdorff locally definably compact definable spaces is definably proper if and only if it is a proper morphism in the category of definable spaces. We give several other characterizations of definably proper, including one involving the existence of limits of definable types. We also prove the...

Let ℳ be an o-minimal expansion of a real closed field. It is known that a definably connected abelian group is divisible. We show that a definably compact definably connected group is divisible.

We prove that if ℳ = (M,≤,+,...) is a weakly o-minimal non-valuational structure expanding an ordered group (M,≤,+), then its expansion by a family of "non-valuational" unary predicates remains non-valuational. The paper is based on the author's earlier work on strong cell decomposition for weakly o-minimal non-valuational expansions of ordered groups.

We study analogues of the notions from Lie theory of Levi subgroup and Levi decomposition, in the case of groups G definable in an o-minimal expansion of a real closed field. With a rather strong definition of ind-definable semisimple subgroup, we prove that G has a unique maximal ind-definable semisimple subgroup S, up to conjugacy, and that G = R· S where R is the solvable radical of G. We also prove that any semisimple subalgebra of the Lie algebra of G corresponds to a unique ind-definable semisimple...