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Separation of plane sets by equidistant simple closed curves

Aleksei Volkov, Mikhail Patrakeev

Abstract

We prove that if two subsets ${A}$ and ${B}$ of the plane are connected, ${A}$ is bounded, and the Euclidean distance $ρ({A},{B})$ between ${A}$ and ${B}$ is greater than zero, then for every positive $\varepsilon<ρ({A},{B})$, the sets ${A}$ and ${B}$ can be separated by a simple closed curve (also known as a Jordan curve) whose points all lie at distance $\varepsilon$ from the set ${A}$. We also prove that the $\varepsilon$-boundary of a connected bounded subset ${A}$ of the plane contains a simple closed curve bounding the domain containing the open $\varepsilon$-neighbourhood of ${A}$. It is shown that in both statements the connectivity condition can be significantly weakened. We also show that the $\varepsilon$-boundary of a nonempty bounded subset of the plane contains a simple closed curve. This result complements Morton Brown's statement that the $\varepsilon$-boundary of a nonempty compact subset of the plane is contained in the union of a finite number of simple closed curves.

Separation of plane sets by equidistant simple closed curves

Abstract

We prove that if two subsets and of the plane are connected, is bounded, and the Euclidean distance between and is greater than zero, then for every positive , the sets and can be separated by a simple closed curve (also known as a Jordan curve) whose points all lie at distance from the set . We also prove that the -boundary of a connected bounded subset of the plane contains a simple closed curve bounding the domain containing the open -neighbourhood of . It is shown that in both statements the connectivity condition can be significantly weakened. We also show that the -boundary of a nonempty bounded subset of the plane contains a simple closed curve. This result complements Morton Brown's statement that the -boundary of a nonempty compact subset of the plane is contained in the union of a finite number of simple closed curves.
Paper Structure (12 sections, 8 theorems, 21 equations)

This paper contains 12 sections, 8 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Terminology and notation
  3. Auxiliary lemmas
  4. The main result
  5. Corollaries of the theorem
  6. Necessity of conditions in the theorem and its corollaries
  7. Boundedness of the set ${A}$
  8. $2\varepsilon$-chainedness of the set ${A}$
  9. Simply connectedness of the closed $\varepsilon$-neighbourhood of the set ${A}$
  10. $2(\rho({A},{B})-\varepsilon)$-chainedness of the set ${B}$
  11. $\rho({A},{B})$-chainedness of the sets ${A}$ and ${B}$
  12. Similar questions in ${{\mathbb{R}}^3}$

Key Result

Lemma 3.1

Suppose that ${E}$ is a connected metric space, ${U}\subseteq{E}$ is open, ${p}\in{U}$, and ${U}\setminus\{{p}\}$ is connected. Then ${E}\setminus\{{p}\}$ is also connected.

Theorems & Definitions (19)

  • Remark 2.1
  • Remark 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 4.1
  • proof
  • Corollary 5.1
  • proof
  • ...and 9 more