Given a space $X$ and a path-connected subspace $A$ containing the basepoint $x_0$, show that the map $\pi_1(A,x_0) \to \pi_1(X,x_0)$ induced by the inclusion $A \to X$ is surjective iff every path in $X$ with endpoints in $A$ is homotopic to a path in $A$.
I am reading a solution in which for the "$\Longleftarrow$" direction they state that if $[f] \in \pi_1(X,x_0)$ is a loop based at $x_0$, then by assumption $f$ is homotopic to a path say $g$ in $A$. Now $g \in \pi_1(A,x_0)$ and so if $\iota_*: \pi_1(A,x_0) \to \pi_1(X,x_0)$ is the map induced by the inclusion we have that $\iota_*([g])=[\iota \circ g]= [g]=[f]$.
There are few questions which I arose here.
- I think they assume without stating that $g$ is actually a loop based at $x_0$ and not just arbitary path?
- If $[f] \in \pi_1(X,x_0)$, then if we are talking about the loop $f$ being homotopic to some other loop say $g$ we must have that $g$ is also a loop based at $x_0$ as otherwise they wouldn't be homotopic?
