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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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    $\begingroup$ I don't know if it will help in your application, but the compact-open topology is a useful topology to put on spaces of maps. $\endgroup$ Dec 8 '11 at 4:44
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    $\begingroup$ Useful in what way. Intuitively for me the "obvious" choice would be to topologize the set of maps $[a,b]\to X$ with the compact-open topology, and then let $P(X)$ just have the quotient topology. Is this meaningful? I'm not quite sure, but it's the first thing that "makes sense". $\endgroup$ Dec 8 '11 at 4:45
  • $\begingroup$ Yeah using the compact open topology with a quotient map would certainly give a topology on $P(X)$. I haven't really thought about what that topology is like to be honest, but it sounds interesting. $\endgroup$ Dec 8 '11 at 4:52
  • $\begingroup$ Another idea I had would be to maybe come up with a topology that makes a certain type of map continuous. For example, one which makes linear functionals on smooth paths in $\mathbb{C}$ into continuous maps $X \rightarrow \mathbb{R}$. That didn't seem to be all that interesting of a topology though when I worked it out. $\endgroup$ Dec 8 '11 at 4:53
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    $\begingroup$ The compact-open topology doesn’t seem to play well with $\sim$. Clearly we may as well consider only maps from $I=[0,1]$ to $X$. Take one of the simplest cases, the point $[1_I]\in C(I,I)/\sim$, the set of all strictly monotone increasing maps from $I$ onto $I$: if $\alpha\in C(I,I)\setminus[1_I]$ is such that $\alpha(0)=0$ and $\alpha(1)=1$, $\alpha$ cannot be separated from $[1_I]$ by an open set in the compact-open topology, and therefore $C(I,I)/\sim\;$ isn’t $T_1$. $\endgroup$ Dec 8 '11 at 5:16
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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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