Tame properties of sets and functions definable in weakly o-minimal structures

Abstract

Let \({{\mathcal{M}}=(M, <, \ldots )}\) be a weakly o-minimal expansion of a dense linear order without endpoints. Some tame properties of sets and functions definable in \({{\mathcal{M}}}\) which hold in o-minimal structures, are examined. One of them is the intermediate value property, say IVP. It is shown that strongly continuous definable functions in \({{\mathcal{M}}}\) satisfy an extended version of IVP. After introducing a weak version of definable connectedness in \({{\mathcal{M}}}\) , we prove that strong cells in \({{\mathcal{M}}}\) are weakly definably connected, so every set definable in \({{\mathcal{M}}}\) is a finite union of its weakly definably connected components, provided that \({{\mathcal{M}}}\) has the strong cell decomposition property. Then, we consider a local continuity property for definable functions in \({{\mathcal{M}}}\) and conclude some results on cell decomposition regarding that property. Finally, we extend the notion of having no dense graph (NDG) which was examined for definable functions in (Dolich et al. in Trans. Am. Math. Soc. 362:1371–1411, 2010) and related to uniform finiteness, definable completeness, and others. We show that every weakly o-minimal structure \({{\mathcal{M}}}\) having cell decomposition, satisfies NDG, i.e. every definable function in \({{\mathcal{M}}}\) has no dense graph.