On page 64 of Hatcher's book on Algebraic Topology, he writes the following:
...if the map $\pi_1(U) \to \pi_1 (X)$ is trivial for one choice of basepoint in $U$, it is trivial for all choices of basepoint since $U$ is path-connected.
I interpret this as the following claim:
Let $\iota : U \to X$ be the canonical embedding, let $x \in U$, and let $\iota_{\ast,x} : \pi_1(U,x) \to \pi_1(X,x)$ be the induced homomorphism. If $\iota_{\ast ,x}$ is trivial, then $\iota_{\ast, y}$ is trivial for every $y \in U$.
Is this a correct interpretation? How does one prove this? I am having trouble proving it. Let $\alpha$ be a path in $U$ from $x$ to $y$. Then $\hat{\alpha} : \pi_1(U,x) \to \pi_1(U,y)$ given by $\hat{\alpha}([\gamma]) = [\overline{\alpha} \ast \gamma \ast \alpha]$ is an isomorphism. Initially, I thought that $i_{\ast ,y} = \iota_{\ast ,x} \hat{\alpha}^{-1}$, but this doesn't appear to be true (domains/codomains don't match up). I have tried other things, but to no avail...