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.