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