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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.

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    @Brian Ah, this is what$I$was afraid of. Thanks for figuring this out!2011-12-08

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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.

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Wouldn't the function that has to be continuous be the one that takes a path $\gamma$ to its equivalence class?

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    The question seems to me to be two-fold. What topology do I use and where should I look? I merely pointed out that when your space is an equivalence class the quotient map is a natural candidate.2013-04-04