Is there a bijection between $\pi(X,x_0)$ and the set "classes of paths from $x_0$ at $x_1$?

Question

Let $x_0$ and $x_1$ two points of the same arc component of $X$. Is there a bijection between $\pi(X,x_0)$ and the set "classes of paths from $x_0$ at $x_1$"? How can I construct it?

Answer 1

Yes, there is. Hint: fix an arbitrary path $p:x_0\leadsto x_1$, and use that $p\cdot p^{-1}:x_0\leadsto x_0$ is a null-homotopic loop, where $p^{-1}:x_1\leadsto x_0$ is $p$ backwards.