Square Metric Spaces

Abstract

Product decompositions of metric spaces are built from coordinate maps, but these maps are not part of the resulting metric space. We recover this missing coordinate structure through equivalence relations whose classes are candidate coordinate fibers, and the resulting quotient metrics reconstruct the coordinate factors. This framework characterizes exactly when a metric space admits a finite product or power presentation. We prove an equivalence of categories showing that these equivalence-relation data preserve exactly the ordered coordinate information of power presentations. For spaces with suitable $\ell^\infty$-prime factorizations, we use prime multiplicities to determine the existence and classification of roots. We also study metric spaces satisfying $X\cong X\times_\infty X$, where repeated coordinate splitting gives a family of metric quotients indexed by infinite binary sequences. We prove that these binary tree structures exactly characterize metric spaces satisfying $X\cong X\times_\infty X$. As an application to persistent homology, we show how to recover filtration parameters whose products or powers form a given space of intervals.