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.
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We are very grateful to the referee of the paper for his suggestions and for a comprehensive list of corrections. The first author is partially supported by a Grant from IPM (No. 89030061).
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Eivazloo, J.S., Tari, S. Tame properties of sets and functions definable in weakly o-minimal structures. Arch. Math. Logic 53, 433–447 (2014). https://doi.org/10.1007/s00153-014-0372-0
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DOI: https://doi.org/10.1007/s00153-014-0372-0
Keywords
- Weakly o-minimal
- Intermediate value property
- Weakly definably connected
- Having no dense graph
- Cell decomposition