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

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.