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 
$$d(f,g)\equiv \mathrm{Inf}\_{\phi,\psi}\mathrm{Sup}\_{t\in [0,1]} d(f(\phi(t)),g(\psi(t))),$$
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). 



Properties of Frechet distance
In fact it is not true in this generality: the simple counterexample, inspired by a famous Aesop's fable, is: $X=\mathbb{R}$, $f(x)=x$, and $g(x)=\max(0, 2x-1)$ for $x\in I:=[0,1]$. The Fréchet distance is clearly zero: for $0<\epsilon< 1$ the homeomorphism $ \phi\_\epsilon :I\ni x \mapsto \max(\epsilon x, 2x-1)\in I$ gives
$\|f\circ \phi\_\epsilon-g\|\_\infty<\epsilon$. But of course for any homeomorphism $\phi$ from $I$ to itself we have $f\circ\phi=\phi\neq g$, as the former is injective and the latter is not.
$$\*$$
On the positive side, note that if for continuous $f$ and $g$ from $I$ to $X$ one has $f(I)\neq g(I)$, that is w.l.o.g. $f(t\_0)\notin g(I)$ for some $t\_0\in I$ , then for any homeomorphism $\phi:I\to I$, $d\_\infty(f,g\circ\phi)\ge \min\_{t\in I} d\_X(f(t\_0), g(t))>0$, so that the Fréchet distance $\tilde d(f,g)$ is non-zero.


Therefore, if $f$ and $g$ are *injective* and have zero Fréchet distance, then they are both homeomorphisms with the compact $f(I)=g(I)$ so $\phi:=f^{-1}\circ g$ is a homeomorphism $I\to I$ such that $g=f\circ\phi$. 
(rmk: At this point one also sees that $\phi:I\to I$ is increasing, because a decreasing homeomorphism would always give a positive Fréchet distance between $f$ and $f\circ\phi$, for any non-constant curve $f$).


(Nothing changes if we replace $I$ with a compact space $K$, where in the definition of the analogous Fréchet distance the quotient is over all homeomorphisms $K\to K$.)


