TITLE: Topology on the space of paths
QUESTION [18 upvotes]: Let $X$ be a topological space, and define a path as a continuous map $\gamma : [a,b] \rightarrow X$. Two paths $\gamma : [a,b] \rightarrow X$ and $\phi : [c,d] \rightarrow X$ are equivalent ($\gamma \sim \phi$) iff there exists an increasing homeomorphism $\psi: [a,b] \rightarrow [c,d]$ such that $\phi \circ \psi = \gamma$. The equivalence class of a path is denoted by $[\gamma ]$.
Now define the space of paths $P(X) = \lbrace [\gamma]\ \vert\ \gamma : [a,b] \rightarrow X\ \text{is a path} \rbrace$.
I am wondering: is there is a useful or a natural topology that can be put on $P(X)$, generated by $X$?
Usually topologies are chosen to make a certain type of function continuous, but I can't think of anything in particular that would be a natural type of function on paths.

REPLY [3 votes]: Your equivalence relation seems well-suited to studying just the image of the path, so in a metric space, you could use the Hausdorff metric on the collection of images of the paths. The Hausdorff metric is where the distance between two compact sets $A,B$ is the supremum of all distances $d(a,B)$ and $d(b,A)$ for points $a$ in $A$ and $B$ in $B$.
In this topology, two equivalence slashes are close if their images are almost the same.
However, this doesn't work completely well since some paths have the same image without being equivalent.