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

摘要

LetM= (M,<,...) be a weakly o-minimal expansion of a dense linear order without endpoints. Some tame properties of sets and functions definable inM 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 Msatisfy an extended version of IVP. After introducing a weak version of definable connectedness inM, we prove that strong cells inMare weakly definably connected, so every set definable inMis a finite union of its weakly definably connected components, provided thatMhas the strong cell decomposition property. Then, we consider a local continuity property for definable functions in M and conclude some resultson 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 Mhaving cell decomposition, satisfies NDG, i.e. every definable function inMhas no dense graph.