The Whitney Property of a Fiber of a Definable Mapping

摘要

We prove the following theorem Let S be a polynomially bounded o-minimal structure on (R,+,·) and let f:A → R~n be a continuous, definable function on a compact definable set A is contained in R~m. Then there exist a positive real number α ∈ R_+ and a definable function C:f(A) → R_+ such that for any x ∈ f(A) and any two points p and q in the same connected component of f~(-1)(x) there exists a piece-wise analytic curve γ joinning p and q in f~(-1)(x) with length γ ≤ C(x)|p - q|~α. As a consequence we obtain the regular separation with parameter for definable sets.