Every Extended Metric Space Is Isometrically Embeddable into the Gromov-Hausdorff Class
Abstract
Let GH denote the Gromov--Hausdorff class of isometry classes of nonempty metric spaces, equipped with the Gromov--Hausdorff extended pseudometric. We prove that every nonempty extended metric space is isometrically embeddable into GH. More precisely, for every nonempty extended metric space X, all image points may be represented by metrics on a common set of cardinality max{dens X,ℵ0}. The main ingredient is a realization theorem for function spaces: for every nonempty set I, the space of all real-valued functions on I, endowed with the uniform extended metric, can be realized as a family of metrics on a fixed underlying set. Under this realization, the Gromov--Hausdorff distance agrees exactly with the uniform extended distance, including the case of infinite distance. The resulting metric spaces are complete, uniformly discrete, and unbounded, and have cardinality max{|I|,ℵ0}. If I is countable, they may moreover be chosen proper and of bounded geometry. As further consequences, we obtain sharp cardinality results for clouds and universality results for cardinality-restricted parts of GH and for spheres in the associated zero-distance quotients.
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BibTeX
@misc{fukano2026every,
title = {{Every Extended Metric Space Is Isometrically Embeddable into the Gromov-Hausdorff Class}},
author = {Ryo Fukano},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24279},
eprint = {2610.24279},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24279},
}
RIS
TY - UNPB
TI - Every Extended Metric Space Is Isometrically Embeddable into the Gromov-Hausdorff Class
AU - Ryo Fukano
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.24279
UR - https://arxiv.org/abs/2610.24279
AB - Let GH denote the Gromov--Hausdorff class of isometry classes of nonempty metric spaces, equipped with the Gromov--Hausdorff extended pseudometric. We prove that every nonempty extended metric space is isometrically embeddable into GH. More precisely, for every nonempty extended metric space X, all image points may be represented by metrics on a common set of cardinality max{dens X,ℵ_0}. The main ingredient is a realization theorem for function spaces: for every nonempty set I, the space of all real-valued functions on I, endowed with the uniform extended metric, can be realized as a family of metrics on a fixed underlying set. Under this realization, the Gromov--Hausdorff distance agrees exactly with the uniform extended distance, including the case of infinite distance. The resulting metric spaces are complete, uniformly discrete, and unbounded, and have cardinality max{|I|,ℵ_0}. If I is countable, they may moreover be chosen proper and of bounded geometry. As further consequences, we obtain sharp cardinality results for clouds and universality results for cardinality-restricted parts of GH and for spheres in the associated zero-distance quotients.
ER -