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Some definable properties of sets in non-valuational weakly o-minimal structures

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Let \({\mathcal {M}}=(M,<,+,\cdot ,\ldots )\) be a non-valuational weakly o-minimal expansion of a real closed field \((M,<,+,\cdot )\). In this paper, we prove that \({\mathcal {M}}\) has a \(C^r\)-strong cell decomposition property, for each positive integer r, a best analogous result from Tanaka and Kawakami (Far East J Math Sci (FJMS) 25(3):417–431, 2007). We also show that curve selection property holds in non-valuational weakly o-minimal expansions of ordered groups. Finally, we extend the notion of definable compactness suitable for weakly o-minimal structures which was examined for definable sets (Peterzil and Steinhorn in J Lond Math Soc 295:769–786, 1999), and prove that a definable set is definably compact if and only if it is closed and bounded.

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Correspondence to Somayyeh Tari.

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Tari, S. Some definable properties of sets in non-valuational weakly o-minimal structures. Arch. Math. Logic 56, 309–317 (2017). https://doi.org/10.1007/s00153-017-0523-1

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