Mathematical Logic Quarterly 62 (3):241-248 (2016)

In o-minimal structures, every cell is definably connected and every definable set is a finite union of its definably connected components. In this note, we introduce pseudo definably connected definable sets in weakly o-minimal structures having strong cell decomposition, and prove that every strong cell in those structures is pseudo definably connected. It follows that every definable set can be written as a finite union of its pseudo definably connected components. We also show that the projections of pseudo definably connected definable sets are pseudo definably connected. Finally, we compare pseudo definable connectedness with weak definable connectedness of definable sets in weakly o-minimal structures.
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DOI 10.1002/malq.201400034
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Weakly o-Minimal Nonvaluational Structures.Roman Wencel - 2008 - Annals of Pure and Applied Logic 154 (3):139-162.
Elimination of Quantifiers for Ordered Valuation Rings.M. A. Dickmann - 1987 - Journal of Symbolic Logic 52 (1):116-128.

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