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On the Constructive Theory of Jordan Curves

Douglas S. Bridges

TL;DR

This paper develops a Bishop-style constructive theory of Jordan curves by showing the Dieudonné-style Jordan curve notion is equivalent to Berg et al.'s constructive notion and by establishing a robust index framework. It provides a detailed constructive analysis of how lines cross a Jordan curve at smooth points and proves a key local cone-based lemma that governs index behavior near such points. The main achievement is proving the constructive equality between the index and the winding number for piecewise smooth Jordan curves, bridging geometric and complex-analytic viewpoints within constructive mathematics. The results pave the way for a constructive treatment of zeros, poles, and meromorphic functions, with explicit expository material to aid comprehension and future work.

Abstract

Using a definition of Jordan curve similar to that of Dieudonné, we prove that our notion is equivalent to that used by Berg et al. in their constructive proof of the Jordan Curve Theorem. We then establish a number of properties of Jordan curves and their corresponding index functions, including the important Proposition 32 and its corollaries about lines crossing a Jordan curve at a smooth point. The final section is dedicated to proving that the index of a point with respect to a piecewise smooth Jordan curve in the complex plane is identical to the familiar winding number of the curve around that point. The paper is written within the framework of Bishop's constructive analysis. Although the work in Sections 3--5 is almost entirely new, the paper contains a substantial amount of expository material for the benefit of the reader.

On the Constructive Theory of Jordan Curves

TL;DR

This paper develops a Bishop-style constructive theory of Jordan curves by showing the Dieudonné-style Jordan curve notion is equivalent to Berg et al.'s constructive notion and by establishing a robust index framework. It provides a detailed constructive analysis of how lines cross a Jordan curve at smooth points and proves a key local cone-based lemma that governs index behavior near such points. The main achievement is proving the constructive equality between the index and the winding number for piecewise smooth Jordan curves, bridging geometric and complex-analytic viewpoints within constructive mathematics. The results pave the way for a constructive treatment of zeros, poles, and meromorphic functions, with explicit expository material to aid comprehension and future work.

Abstract

Using a definition of Jordan curve similar to that of Dieudonné, we prove that our notion is equivalent to that used by Berg et al. in their constructive proof of the Jordan Curve Theorem. We then establish a number of properties of Jordan curves and their corresponding index functions, including the important Proposition 32 and its corollaries about lines crossing a Jordan curve at a smooth point. The final section is dedicated to proving that the index of a point with respect to a piecewise smooth Jordan curve in the complex plane is identical to the familiar winding number of the curve around that point. The paper is written within the framework of Bishop's constructive analysis. Although the work in Sections 3--5 is almost entirely new, the paper contains a substantial amount of expository material for the benefit of the reader.

Paper Structure

This paper contains 6 sections, 51 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Jordan curves
  4. Crossing a Jordan curve
  5. Index and winding number
  6. Concluding remarks

Key Result

Lemma 1

Let $L$ be a line in $\mathbb{C}$, $\zeta_{0}\in L$, and $\zeta\in\mathbb{C}-\left\{ \zeta_{0}\right\}$. Then there exists $\zeta^{\prime}\in L$ with $\left\vert \zeta^{\prime}-\zeta_{0}\right\vert \leq3\left\vert \zeta-\zeta_{0}\right\vert$ such that if $\zeta\neq \zeta^{\prime}$, then $\rho(\zeta

Theorems & Definitions (51)

  • Lemma 1
  • Lemma 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Lemma 6
  • Proposition 7
  • Lemma 8
  • Proposition 9
  • Lemma 10
  • ...and 41 more