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The Whitney property of a fiber of a definable mapping

Kocel-Cynk B.

Bulletin of the Polish Academy of Sciences. Mathematics

2000

Vol. 48, no 2

141-147

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 subset of R^m]. Then there exist a positive real number [alpha belongs to R+] and a definable function C : f(A) --> R+ such that for any x [belongs to] 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 gamma joinning p and q in [f^-1](x) with length [gamma is less than or equal to C(x)][...]. As a consequence we obtain the regular separation with parameter for definable sets.