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On completeness of Hausdorff hyperspaces

Ján Komara

Abstract

The Hausdorff hyperspace of a metric space consists of all its non-empty bounded closed sets and it is equipped with the Pompeiu--Hausdorff set distance. We present a simpler novel proof that the Hausdorff hyperspace of a complete space is complete as well. The Main Lemma is crucial in this demonstration and though it uses an induction argument -- the only one in our completeness proof -- it is stated purely in terms of neighborhoods.

On completeness of Hausdorff hyperspaces

Abstract

The Hausdorff hyperspace of a metric space consists of all its non-empty bounded closed sets and it is equipped with the Pompeiu--Hausdorff set distance. We present a simpler novel proof that the Hausdorff hyperspace of a complete space is complete as well. The Main Lemma is crucial in this demonstration and though it uses an induction argument -- the only one in our completeness proof -- it is stated purely in terms of neighborhoods.
Paper Structure (4 sections, 4 theorems, 18 equations)

This paper contains 4 sections, 4 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Prerequisites
  3. Completeness
  4. Conclusion

Key Result

Theorem 1

If $P$ is a complete metric space, then so is its Hausdorff hyperspace ${P}^{\star}$.

Theorems & Definitions (10)

  • Theorem
  • proof
  • proof
  • Remark
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof