Abstract 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-minimal structures on $(\mathbb {R},<)$ have the property, as do all expansions of $(\mathbb {R},+,\cdot ,\mathbb {N})$ . Our main analytic-geometric result is that any such expansion of $(\mathbb {R},<,+)$ by Boolean combinations of open sets (of any arities) either is o-minimal or defines an isomorph of $(\mathbb N,+,\cdot )$ . We also show that any given expansion of $(\mathbb {R}, <, +,\mathbb {N})$ by subsets of $\mathbb {N}^n$ (n allowed to vary) has the property if and only if it defines all arithmetic sets. Variations arise by considering connected components or quasicomponents instead of path components.

Given a theoryTextending that of dense linear orders without endpoints (DLO), in a language ℒ ⊇ {<}, we are interested in extensionsT′ ofTin languages extending ℒ by unary relation symbols that are each interpreted in models ofT′ as sets that are both dense and codense in the underlying sets of the models. There is a canonically “wild” example, namelyT= Th(〈ℝ, <, +, ·〉) andT′ = Th(〈ℝ, <, +, · ℚ 〉). Recall thatTis o-minimal, and so every open set definable in any model ofThas only finitely many definably connected components. But it is well known that 〈ℝ, <, +, · ℚ 〉 defines every real Borel set, in particular, every open subset of any finite cartesian power of ℝ and every subset of any finite cartesian power of ℚ. To put this another way, the definable open sets in models ofTare essentially as simple as possible, whileT′ has a model where the definable open sets are as complicated as possible, as is the structure induced on the new predicate. In contrast to the preceding example, if ℝalgis the set of real algebraic numbers andT′ Th(〈ℝ, <, +, ·, 〈alg〉), then no model ofT′ defines any open set (of any arity) that is not definable in the underlying model ofT.

Abstract An openU⊆ ℝ is produced such that (ℝ, +, ·,U) defines a Borel isomorph of (ℝ, +, ·, ℕ) but does not define ℕ. It follows that (ℝ, +, ·,U) defines sets in every level of the projective hierarchy but does not define all projective sets. This result is elaborated in various ways that involve geometric measure theory and working over o-minimal expansions of (ℝ, +, ·). In particular, there is a Cantor setE⊆ ℝ such that (ℝ, +, ·, ℕ) defines a Borel isomorph of (ℝ, +, ·, ℕ) and, for every exponentially bounded o-minimal expansion of (ℝ, +, ·), every subset of ℝ definable in ( ,E) either has interior or is Hausdorff null.

Abstract Let ℝ be an o-minimal expansion of (ℝ, <, +) and (ϕk)kЄℕbe a sequence of positive real numbers such that limtk→+∞f(ϕk)/ϕk+1 = 0 for everyf: ℝ → ℝ definable in ℜ (Such sequences always exist under some reasonable extra assumptions on ℜ, in particular, if ℜ is exponentially bounded or if the language is countable.) Then (ℜ, (S)) is d-minimal. whereSranges over all subsets of cartesian powers of the range ofϕ.

In this paper, we continue investigations into the asymptotic behavior of solutions of differential equations over o-minimal structures. Let ℜ be an expansion of the real field (ℝ, +, ·). A differentiable mapF= (F1,…,F1): (a, b) → ℝiisℜ-Pfaffianif there existsG: ℝ1+l→ ℝldefinable in ℜ such thatF′(t) =G(t, F(t)) for allt∈ (a, b) and each component functionGi: ℝ1+l→ ℝ is independent of the lastl−ivariables (i= 1, …,l). If ℜ is o-minimal andF: (a, b) → ℝlis ℜ-Pfaffian, then (ℜ,F) is o-minimal (Proposition 7). We say thatF: ℝ → ℝlis ultimately ℜ-Pfaffian if there existsr∈ ℝ such that the restrictionF↾(r, ∞) is ℜ-Pfaffian. (In general,ultimatelyabbreviates “for all sufficiently large positive arguments”.) The structure ℜ isclosed under asymptotic integrationif for each ultimately non-zero unary (that is, ℝ → ℝ) functionfdefinable in ℜ there is an ultimately differentiable unary functiongdefinable in ℜ such that limt→+∞[g′(t)/f(t)] = 1- If ℜ is closed under asymptotic integration, then ℜ is o-minimal and definesex: ℝ → ℝ (Proposition 2). Note that the above definitions make sense for expansions of arbitrary ordered fields.

Let ℜ be an expansion of a dense linear order (R, <) without endpoints having theintermediate value property, that is, for alla, b∈R, every continuous (parametrically) definable functionf: [a, b] →Rtakes on all values inRbetweenf(a) andf(b). Every expansion of the real line (ℝ, <), as well as every o-minimal expansion of (R, <), has the intermediate value property. Conversely, some nice properties, often associated with expansions of (ℝ, <) or with o-minimal structures, hold for sets and functions definable in ℜ. For example, images of closed bounded definable sets under continuous definable maps are closed and bounded (Proposition 1.10). Of particular interest is the case that ℜ expands an ordered group, that is, ℜ defines a binary operation * such that (R, <, *) is an ordered group. Then (R, *) is abelian and divisible (Proposition 2.2). Continuous nontrivial definable endo-morphisms of (R, *) are surjective and strictly monotone, and monotone nontrivial definable endomorphisms of (R, *) are strictly monotone, continuous and surjective (Proposition 2.4). There is a generalization of the familiar result that every proper noncyclic subgroup of (ℝ, +) is dense and codense in ℝ: IfGis a proper nontrivial subgroup of (R, *) definable in ℜ, then eitherGis dense and codense inR, orGcontains an elementusuch that (R, <, *,e, u, G) is elementarily equivalent to (ℚ, <, +, 0, 1, ℤ), whereedenotes the identity element of (R, *) (Theorem 2.3). Here is an outline of this paper. First, we deal with some basic topological results. We then assume that ℜ expands an ordered group and establish the results mentioned in the preceding paragraph. Some examples are then given, followed by a brief discussion of analytic results and possible limitations. In an appendix, an explicit axiomatization (used in the proof of Theorem 2.3) is given for the complete theory of the structure (ℚ, <, +, 0, 1, ℤ).