Every Extended Metric Space Is Isometrically Embeddable into the Gromov--Hausdorff Class

Let $\mathfrak{G}\mathfrak{H}$ denote the Gromov-Hausdorff class of isometry classes of nonempty metric spaces, equipped with the Gromov-Hausdorff distance, which is an extended pseudometric. We prove that every nonempty extended metric space is isometrically embeddable into $\mathfrak{G}\mathfrak{H}$. More precisely, for every nonempty extended metric space $X$, the embedding can be chosen so that all image points are represented by metrics on a common underlying set of cardinality $\max \{\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{s}X,{\mathrm{\aleph }}_0\}$, where $\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{s}X$ denotes the density character of $X$. To establish this result, for every nonempty set $I$, we realize the space of all real-valued functions on $I$, equipped with the uniform extended metric, as a family of metrics on a fixed underlying set. Under this realization, the Gromov-Hausdorff distance agrees with the uniform extended distance between functions, including when this distance is infinite, and each resulting metric space is complete, uniformly discrete, and unbounded, with cardinality $\max \{\mathrm{\mid }I\mathrm{\mid },{\mathrm{\aleph }}_0\}$. When $I$ is countable, the representing spaces can also be chosen proper and of bounded geometry. We also show that, for every infinite cardinal $\kappa$, there are exactly $2^{\kappa }$ clouds represented by metric spaces of cardinality at most $\kappa$, and that $2^{\kappa }$ of their cardinality-restricted parts are isometrically universal for all metric spaces of density character at most $\kappa$. Furthermore, every sphere of positive radius centered at a point of the constructed copies of ${\mathrm{\ell }}^{\mathrm{\infty }}(\kappa )$ in the zero-distance quotients admits isometric embeddings of all metric spaces of density character at most $\kappa$ and diameter at most twice the radius.