1. ## Retraction, Fundamental group

Hello Math experts, i need some help
let $X$ be a topological space.
Let $A \subset X$ and let $r: X \rightarrow A$ be a retraction.
Given $a_0 \in A$ , show that
$r*: \Pi_1(X,a_0) \rightarrow \Pi_1(A,a_0)$ is surjective

thanks

2. In general, if $f\colon Y\rightarrow Z$ and if $g\colon Z\rightarrow Y$ are functions, and if $f\circ g = \text{id}_Z$, then g is injective, and f is surjective.

In your scenario, let $i\colon A\rightarrow X$ be the inclusion. Then $r\circ i = \text{id}_A$, and by going to the induced maps on fundamental groups, we get

$r^*\circ i^* = \text{id}_{\pi_1(A,a_0)},$

so the map induced by r is surjective.