On completeness of Hausdorff hyperspaces

TL;DR

This paper provides a simplified proof that the Hausdorff hyperspace of a complete metric space remains complete, emphasizing a neighborhood-based induction argument.

Contribution

It introduces a novel, simpler proof of the completeness of Hausdorff hyperspaces for complete metric spaces, utilizing a new Main Lemma.

Findings

Hausdorff hyperspace of a complete space is complete

Main Lemma is key to the proof

Proof uses neighborhood-based induction

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.