There is an old construction, apparently due to Frechet's PhD thesis (which is unfortunately written in French and in ancient notation), which turns the set of curves in a metric space modulo reparametrizations into a metric space itself. The idea is that if $f$ and $g$ are curves in a metric space $X$ (meaning continuous maps from $[0,1]$ to $X$), then their distance is defined as $$d(f,g)\equiv \mathrm{Inf}_{\phi,\psi}\mathrm{Sup}_{t\in [0,1]} d(f(\phi(t)),g(\phi(t)),$$ where $\phi$ and $\psi$ are orientation-preserving homeomorphisms from $[0,1]$ to itself. Most of the properties of a distance follow trivially from the definition, but I am having some trouble showing that $d(f,g)=0$ implies that $f$ and $g$ differ only by a homeomorphism. Potentially we could merely have a sequence $(\phi_n,\psi_n)$ of homeomorphisms for which the the supremum converged to zero, but which did not converge to a pair of homeomorphisms $(\phi,\psi)$ for which it vanishes. Why must the infimum be realized? I believe it should be possible to show this using some kind of uniform convergence, and also possibly the compactness of $[0,1]$, but haven't succeeded. For my own purposes I am especially interested in whether or not it is necessary to use the compactness of $[0,1]$ (I am trying to understand to what extent $[0,1]$ can be generalized to an arbitrary topological space in this definition).