arXiv2026
Let $\mathfrak{GH}$ 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 $\mathfrak{GH}$. More precisely, for every nonempty extended metric space $X$, all image points may be represented by metrics on a common set of cardinality $\max\{\operatorname{dens} X,\aleph_0\}$. 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 $\max\{|I|,\aleph_0\}$. 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 $\mathfrak{GH}$ and for spheres in the associated zero-distance quotients.