Products of metric spaces, covering numbers, packing numbers and characterizations of ultrametric spaces

We describe some Cartesian products of metric spaces and find conditions under which products of ultrametric spaces are ultrametric.

💡 Research Summary

The paper investigates the behavior of covering numbers and packing numbers under Cartesian products of metric spaces, with a special focus on ultrametric spaces. After recalling the basic definitions—metric spaces ((X,d_X)), ultrametrics satisfying the strong triangle inequality (d(x,z)\le\max{d(x,y),d(y,z)}), the covering number (N_\varepsilon(X)) (the smallest number of closed balls of radius (\varepsilon) needed to cover (X)), and the packing number (M_\varepsilon(X)) (the largest cardinality of a set of points whose pairwise distances exceed (\varepsilon))—the authors set up the problem of how these quantities behave when two spaces are combined.

Two natural product metrics are considered on (X\times Y): the maximum metric (\rho_\infty\big((x_1,y_1),(x_2,y_2)\big)=\max{d_X(x_1,x_2),d_Y(y_1,y_2)}) and the sum metric (\rho_1\big((x_1,y_1),(x_2,y_2)\big)=d_X(x_1,x_2)+d_Y(y_1,y_2)). The authors first prove that if both factors are ultrametric, then the product equipped with (\rho_\infty) is again an ultrametric space. The proof is straightforward: the strong triangle inequality is preserved under the (\max) operation, because the largest of the two component distances dominates the combined distance. Consequently, all ultrametric properties (e.g., every ball is both open and closed, any two balls are either disjoint or one contains the other) transfer to the product.

In contrast, the sum metric (\rho_1) generally destroys ultrametricity. The paper provides explicit counter‑examples showing that even when both factors are ultrametric, the product may fail the strong triangle inequality. The authors identify the precise circumstances under which (\rho_1) can still yield an ultrametric: essentially one factor must be a singleton, or both factors must be highly degenerate (e.g., discrete with a single non‑zero distance). Moreover, they prove a converse statement: if the product under (\rho_1) satisfies the equality (N_\varepsilon(X\times Y)=M_\varepsilon(X\times Y)) for every (\varepsilon>0), then the product must be ultrametric. This links the combinatorial notion of covering/packing equality directly to the metric structure.

The next major contribution concerns exact formulas for covering and packing numbers of product spaces. For the maximum metric the authors establish the clean multiplicative identities \