Rough path metric ($d_p(\mathbf{X},\mathbf{Y})={\max }_{1\le k\le \lfloor p\rfloor }{\left({\sup }_D{\sum }_{[u,v]\in D}|X_{u,v}^{(k)}-Y_{u,v}^{(k)}{|}^{p/k}\right)}^{k/p}$)

This inhomogeneous variation metric compares the tensor increments level by level. A common alternative is the Hölder metric ${\rho }_{\alpha }={\max }_k{\sup }_{s<t}|X_{s,t}^{(k)}-Y_{s,t}^{(k)}|/|t-s{|}^{k\alpha }$. On a finite interval convergence in this metric implies $p$-variation convergence whenever $p\alpha \ge 1$. Both topologies control the iterated integrals needed by the universal limit theorem.