I'm trying to really understand retractions and the fundamental group, and I came across this corollary that stumped me in trying to prove:
$\textbf{Corollary:}$ A retract of a simply connected space is simply connected.
Now, what I know about retractions is that if A is a retract of X, then I have a retraction map \begin{equation}r:X\rightarrow A\end{equation} And the homomorphism induced by it, \begin{equation} \iota_{A*}:\pi_1(A,p)\rightarrow \pi_1(X,p)\end{equation} For any $p\in A$ is injective.
I have a simply connected space, so it's fundamental group $\pi_1(X)$ is trivial. I want to show that $\pi_1(A)$ is also trivial. So, examine what happens in the above induced mapping.....
And this is where I can't figure out how to finish the proof. Can someone explain to me how the injectivity of $\iota_{A*}$ makes $\pi_1(A,p)$ trivial? I feel like it's something really easy but I am not quite seeing it.
I really want to say that $\iota_{A*}$ is also surjective. Why can I not say that? Or can I?
Thanks!