On definable open continuous mappings

TL;DR

This paper characterizes when definable continuous mappings are open in terms of fibers and Jacobian sign, and confirms Whyburn's conjecture for such mappings, showing they are homeomorphisms under certain boundary conditions.

Contribution

It provides new equivalences for openness of definable mappings and proves Whyburn's conjecture within the definable setting.

Findings

Openness characterized by finite fibers and Jacobian sign stability.

Definable open maps with boundary homeomorphisms are homeomorphisms.

Validation of Whyburn's conjecture for definable 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.