PreprintCC BY 4.0arXiv
The topology of Gromov--Hausdorff space
Abstract
We prove that the space of isometry classes of nonempty compact metric spaces, equipped with the Gromov--Hausdorff distance, is homeomorphic to the real separable infinite-dimensional Hilbert space. We construct a continuous assignment of full-support probability measures that is equivariant under isometries and finite-dimensional local approximations that control all pairwise distances. These approximations yield the absolute retract property for all metrizable spaces. We also prove that any countable family of continuous maps from compact metrizable spaces can be approximated, with respect to a prescribed open cover, by maps whose images form a discrete family.
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BibTeX
@misc{ishiki2026topology,
title = {{The topology of Gromov--Hausdorff space}},
author = {Yoshito Ishiki},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09639},
eprint = {2609.09639},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09639},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The topology of Gromov--Hausdorff space
AU - Yoshito Ishiki
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.09639
UR - https://arxiv.org/abs/2609.09639
AB - We prove that the space of isometry classes of nonempty compact metric spaces, equipped with the Gromov--Hausdorff distance, is homeomorphic to the real separable infinite-dimensional Hilbert space. We construct a continuous assignment of full-support probability measures that is equivariant under isometries and finite-dimensional local approximations that control all pairwise distances. These approximations yield the absolute retract property for all metrizable spaces. We also prove that any countable family of continuous maps from compact metrizable spaces can be approximated, with respect to a prescribed open cover, by maps whose images form a discrete family.
ER -