Cardinality Homogeneous Metric Spaces

• Cardinality homogeneous metric spaces are defined by the property that every nonempty open subset has the same cardinality as the entire space.
• They feature strong extension properties and universal homogeneity, as seen in examples like Banach spaces, Hilbert spaces, and Urysohn spaces.
• Their study employs cardinal invariants and Gromov–Hausdorff theory to analyze approximation stability and coarse classification in metric geometry.

A cardinality homogeneous metric space is a topological space in which every nonempty open subset is equicardinal with the whole space. In the metric setting, this property interacts nontrivially with classical notions of homogeneity—where local isometries or automorphisms can be extended globally—as well as with group actions, universal constructions, and cardinality bounds. The study of cardinality homogeneous metric spaces is tightly connected to results on ultrahomogeneous spaces, cardinal invariants, and the fine structure of approximate isometries.

1. Definition and Fundamental Properties

Cardinality homogeneity is defined as follows: a topological space $X$ is cardinality homogeneous if every nonempty open subset $U \subset X$ satisfies $|U| = |X|$ (Bogatyi et al., 28 Dec 2025). The property extends naturally to metric spaces via their metric topology. This notion is distinct from classical topological homogeneity (where points are indistinguishable via homeomorphisms), yet frequently coincides for spaces with rich automorphism or isometry groups.

Every Banach space, including $\ell_p$, $c_0$, $\mathbb{R}^n$, and separable or nonseparable Hilbert spaces, is cardinality homogeneous [(Bogatyi et al., 28 Dec 2025), Proposition 6.1]. More generally, any metric space admitting a transitive continuous group action by a group of matching cardinality is cardinality homogeneous.

2. Cardinality Homogeneous Spaces and Universal Homogeneity

Cardinality homogeneous metric spaces exhibit strong extension phenomena. For infinite cardinals $m$, the existence of a unique ($m$-homogeneous, universal) metric space $U_m$ of weight $m$ is guaranteed when the cardinal arithmetic condition $m^n \leq m$ for all $n < m$ holds (Mbombo et al., 2010). Here, $m$-homogeneity asserts that every isometry between subspaces of density $< m$ extends to a global isometry.

The $U_m$ spaces constructed by Katětov generalize the Urysohn space $U_{\aleph_0}$. Their assembly uses iterative extension via Katětov functions, ensuring at each limit stage the density remains $m$. The universality property guarantees that any metric space of weight $\leq m$ isometrically embeds into $U_m$, while $m$-homogeneity ensures maximal global symmetry on density $< m$ subsets.

Critical attributes:

• $d(U_m) = w(U_m) = m$.
• Ultrahomogeneity: every isometry between subspaces of size $< m$ extends globally.
• One-point extension property: Katětov functions on small subspaces are always realized.
• Isometry group structure: $\mathrm{Iso}(U_m)$ has weight $m$, pointwise-convergence topology, and nontrivial subgroup dynamics for uncountable $m$ (Mbombo et al., 2010).
• Extreme amenability and Ramsey–Milman-type properties may hold under suitable set-theoretic hypotheses.

3. Cardinality Bounds and Metric Homogeneity

Cardinality homogeneous metric spaces are subject to well-developed cardinality bounds that rely on classical cardinal invariants:

• Density $d(X)$: minimal cardinality of a dense subset.
• Weight $w(X)$: minimal cardinality of a basis of open sets.
• Character $\chi(X)$, tightness $t(X)$, π-character, and cellularity $c(X)$.

For any homogeneous metric space, the strongest uniform bound is

$|X| \leq d(X)^{\aleph_0}$

where $\aleph_0$ reflects the countable nature of the base and network invariants in metric spaces (Carlson, 2020). This is sharper than $|X| \leq 2^{d(X)}$ and $|X| \leq 2^{\aleph_0} = \mathfrak{c}$ (the continuum), especially for nonseparable $X$. Homogeneity alone provides no strict improvement in this metric setting, because first-countable spaces already realize maximal cardinality for a given density.

The following table summarizes key cardinality bounds in metric spaces:

Theorem (General)	Bound in Metric Case	Attainment Condition
de la Vega: $|X|\leq 2^{t(X)}$	$|X| \leq \mathfrak{c}$	Separable, compact
$d(X)^{\pi\chi(X)}$-type	$|X| \leq d(X)^{\aleph_0}$	General metric
$2^{c(X)\pi\chi(X)}$	$|X| \leq 2^{d(X)}$	General metric

For θⁿ-Urysohn spaces (a generalization of Urysohn spaces), it holds that every homogeneous metric θⁿ-Urysohn space has $|X| \leq \mathfrak{c}$, since all relevant invariants are countable (Basile et al., 2018). This suggests that cardinality homogeneous and ultrahomogeneous spaces share tight constraints on cardinality.

4. Approximate Isometry and Gromov–Hausdorff Theory

Cardinality homogeneity enables robust approximation results for metric spaces, notably in the context of the Hyers–Ulam stability problem and Gromov–Hausdorff distance computations. The Main Theorem [(Bogatyi et al., 28 Dec 2025), Thm 7.1] states:

Given cardinality homogeneous metric spaces $X$, $Y$ with $|X| = |Y|$, if $f : X \to Y$ is a $\delta$-surjective $d$-isometry, then there exists a bijective $(d+2\delta)$-isometry $\tilde f : X \to Y$.

This result reduces several classical stability theorems (Dilworth–Tabor, Gevirtz–Omladič–Šemrl) to considerations about cardinality homogeneous spaces. The proof leverages a transfinite Cantor–Schröder–Bernstein argument, made possible by the equicardinality of nonempty opens, to glue $f$ and its quasi-inverse.

For cardinality homogeneous spaces, the Gromov–Hausdorff distance $d_{GH}(X, Y)$ may be determined via bijections rather than arbitrary correspondences:

$$d_{GH}(X,Y) = \frac{1}{2} \inf \{ \mathrm{dis} f : f \colon X \to Y \text{ is a bijection} \}$$

where $\mathrm{dis} f$ is the distortion. This characterization streamlines computations in large discrete or regular settings.

5. Coarse Classification and Homogeneous Ultrametrics

The structure of isometrically homogeneous ultrametric spaces is classified coarsely using two cardinal invariants: $\mathrm{cov}^\flat(X)$ and $\mathrm{cov}^\sharp(X)$, denoting the local and global covering numbers of balls (Banakh et al., 2014). Their coarse invariance implies:

• Two isometrically homogeneous ultrametric spaces $X, Y$ are coarsely equivalent if and only if $\mathrm{cov}^\sharp(X) = \mathrm{cov}^\sharp(Y)$ (which equals $\mathrm{cov}^\flat(X) = \mathrm{cov}^\flat(Y)$).
• The coarse structure of any isometrically homogeneous ultrametric space is completely determined by the single cardinal $\mathrm{cov}^\sharp(X)$.

In particular, an ultrametric space $X$ is coarsely equivalent to an isometrically homogeneous ultrametric space if and only if $\mathrm{cov}^\flat(X) = \mathrm{cov}^\sharp(X)$. This means cardinality homogeneous ultrametric spaces are classified, up to coarse equivalence, by a single cardinal function.

6. Connections, Examples, and Open Questions

Cardinality homogeneous metric spaces subsume numerous classical examples, including Banach spaces, Hilbert spaces, and universal Urysohn spaces. These spaces often admit transitive group actions, large automorphism groups, and universal extension properties.

Despite the robustness of cardinality bounds and extension properties in metric spaces, the search for strictly tighter bounds using other invariants (spread, extent) has not yielded improved estimates to date (Carlson, 2020). The possibility of refining approximation theorems (e.g., replacing the $2\delta$ term in the bijection distortion bound by $\delta$ under additional structure) remains open for future exploration.

Broader implications include potential applications in algorithmic Gromov–Hausdorff distance calculation, asymptotic and coarse geometry, and extensions to noncommutative or quantum metric spaces, where suitable analogues of cardinality homogeneity may emerge (Bogatyi et al., 28 Dec 2025).