In the definition of the integral of a 1-form over a path, in a Riemann surface, we use the "fact" that any partially smooth path can be written as concatenation of partially smooth paths $\gamma_i$ with image in a single coordinate neighborhood $U_i$. I'm trying to understand why this is true.
The way I see it, it should be just a topological proposition. Since $\{{U_\alpha}\}_\alpha$ is an open cover of $X$, $\{\gamma^{-1}(U_\alpha)\}_\alpha$ is an open cover of $[0,1]$ and then by compactness it has a finite subcover, that is the image of $\gamma$ is contained in a finite number of coordinate neighborhoods $U_1,\ldots,U_n$. So far so good.
I would like to take $\gamma^{-1}(U_i)$ and set $\gamma=\gamma_1*\gamma_2*\cdots*\gamma_n$ for $\gamma_i=\gamma|_{\gamma^{-1}(U_i)}$, but this shouldn't be true in general. I need some way to devide $[0,1]$ to intervals $[t_i,t_j]$ such that the image of each interval would lie in one of this $U_i$. I thought it could be done by some combinatorical argument using the finite collection $U_i$, but I couldn't make it work.