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Verdier and strict Thom stratifications in o-minimal structures. (English) Zbl 0909.32008
We prove the existence of Verdier stratifications for sets definable in any \(o\)-minimal structure on \((\mathbb{R},+, \cdot)\) [for the definition of \(o\)-minimal structures see L. van den Dries, ‘Tame topology and \(O\)-minimal structures, Cambridge University Press (1997)]. It is also shown that the Verdier condition \((w)\) implies the Whitney condition (b) in \(o\)-minimal structures on \((\mathbb{R},+, \cdot)\). We establish the existence of \((w_f)\)-stratification of functions definable in polynomially bounded \(o\)-minimal structures [for the definition, see C. Miller, Ann. Pure Appl. Logic 68, No. 1, 79-84 (1994; Zbl 0823.03018)].
Reviewer: Ta Lê Loi (Dalat)

32S60 Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects)
14P10 Semialgebraic sets and related spaces
14B05 Singularities in algebraic geometry