On definable open continuous mappings

Abstract

For a definable continuous mapping $f$ from a definable connected open subset $\Omega$ of ${\mathbb{R}}^n$ into ${\mathbb{R}}^n,$ we show that the following statements are equivalent: (i) The mapping $f$ is open. (ii) The fibers of $f$ are finite and the Jacobian of $f$ does not change sign on the set of points at which $f$ is differentiable. (iii) The fibers of $f$ are finite and the set of points at which $f$ is not a local homeomorphism has dimension at most $n-2.$ As an application, we prove that Whyburn's conjecture is true for definable mappings: A definable open continuous mapping of one closed ball into another which maps boundary homeomorphically onto boundary is necessarily a homeomorphism.

References in corpus (1)

• Some classical analysis results for continuous definable mappings