Suppose that X is a topological space and $x_0$, $x_1$ are points of $X$. What is meant by saying that two paths in $X$ from $x_0$ to $x_1$ are equivalent?
I presume it's enough to just say the path is a continuous function $\sigma: [0,1] \rightarrow X $ such that $\sigma(0)=\sigma(1)$?