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arXiv · 2505.06571

On completeness of Hausdorff hyperspaces

Ján Komara
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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.

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Ján Komara. 2025-05-10. On completeness of Hausdorff hyperspaces. https://arxiv.org/abs/2505.06571

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