Expansions of Dense Linear Orders with The Intermediate Value Property

Let R be an expansion of a dense linear order (R,<) without endpoints having the intermediate value property, that is, for all a, b ∈ R, every continuous (parametrically) definable function f : [a, b] → R takes on all values in R between f(a) and f(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 1.9). 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 endomorphisms 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: If G is a proper nontrivial subgroup of (R, ∗) definable in R, then either G is dense and codense in R, or G contains an element u such that (R,<, ∗, e, u,G) is elementarily equivalent to (Q, <,+, 0, 1,Z), where e denotes 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).