Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A simplified proof of the o-minimal Whitney Extension Theorem

Beata Kocel-Cynk, Wiesław Pawłucki, Anna Valette

Abstract

We give a proof of the o-minimal version of the Whitney Extension Theorem simplified as compared to the original ones. A new simplifying ingredient is a definable variant of Urysohn's lemma for class $\mathcal{C}^q$ (see Section 3).

A simplified proof of the o-minimal Whitney Extension Theorem

Abstract

We give a proof of the o-minimal version of the Whitney Extension Theorem simplified as compared to the original ones. A new simplifying ingredient is a definable variant of Urysohn's lemma for class (see Section 3).
Paper Structure (8 sections, 7 theorems, 35 equations)

This paper contains 8 sections, 7 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. $\varLambda_p$-regular maps and stratifications into $\varLambda_p$-regular cells
  3. Definable $\mathcal{C}^q$-Urysohn lemma
  4. $\mathcal{C}^p$-Whitney fields
  5. The generic case: Whitney field flat except on one $\varLambda_q$-regular cell
  6. Proof of Theorem 1.1
  7. Final remarks
  8. Statements and Declarations

Key Result

Theorem 1.1

Given an o-minimal extension of the field of real numbers $\mathbb R$, let $E$ be a definable closed subset of $\mathbb R^n$ and let $p$ and $q$ be positive integers such that $p\leq q$. Let be a definable $\mathcal{C}^p$-Whitney field on $E$. $($Definability of $F$ means that all $F^{\alpha}$ are definable functions.$)$ Then there exists a definable $\mathcal{C}^p$-function $f: \mathbb R^n\lon

Theorems & Definitions (23)

  • Theorem 1.1
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Remark 2.5
  • Definition 2.6
  • Theorem 2.7: [ P1, Proposition 4 or P2, Theorem 3]
  • Remark 2.8
  • Definition 3.1
  • ...and 13 more