Metrization of the Fell Topology and the Wijsman Topology for the Box Metric on Closed Downward Families

Abstract

We study nonempty Box-closed families of metric measure spaces that are downward under the Lipschitz order. If a space is below a member in this order, it also belongs to the family. Such families need not be pyramids, because directedness is an additional condition. We define a metric on these families by taking the supremum, over all finite metric measure spaces, of the normalized absolute difference between the infimal Box distances from each finite space to the two families. We divide each difference by the sum of the support cardinality and diameter of the finite test space. The topology induced by this metric agrees on the full domain with the relative Fell topology and with the restriction of the Wijsman topology for the Box metric. On the subspace of pyramids, it induces the weak topology and makes the natural inclusion into the hyperspace of nonempty closed subsets a topological embedding for either hyperspace topology. Pyramids associated with metric measure spaces are dense in the pyramid space, which is the completion of metric measure spaces under the metric pulled back along the map that sends each space to its associated pyramid.

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BibTeX RIS

BibTeX

@misc{yokota2026metrization,
  title = {{Metrization of the Fell Topology and the Wijsman Topology for the Box Metric on Closed Downward Families}},
  author = {Shigeaki Yokota},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.27795},
  eprint = {2610.27795},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.27795},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Metrization of the Fell Topology and the Wijsman Topology for the Box Metric on Closed Downward Families
AU  - Shigeaki Yokota
PY  - 2026
DA  - 2026/10/20
PB  - arXiv
DO  - 10.48550/arxiv.2610.27795
UR  - https://arxiv.org/abs/2610.27795
AB  - We study nonempty Box-closed families of metric measure spaces that are downward under the Lipschitz order. If a space is below a member in this order, it also belongs to the family. Such families need not be pyramids, because directedness is an additional condition. We define a metric on these families by taking the supremum, over all finite metric measure spaces, of the normalized absolute difference between the infimal Box distances from each finite space to the two families. We divide each difference by the sum of the support cardinality and diameter of the finite test space. The topology induced by this metric agrees on the full domain with the relative Fell topology and with the restriction of the Wijsman topology for the Box metric. On the subspace of pyramids, it induces the weak topology and makes the natural inclusion into the hyperspace of nonempty closed subsets a topological embedding for either hyperspace topology. Pyramids associated with metric measure spaces are dense in the pyramid space, which is the completion of metric measure spaces under the metric pulled back along the map that sends each space to its associated pyramid.
ER  -

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