The question comes from Whitehead's book: Elements of Homotopy Theory, Chapter 6, Theorem 1.11(Page 263 below).
" Let $B$ be a connected topological space. For each $b\in B$, choose an element $\xi(b)\in \pi_1(B;b_0,b)$, (where $b_0$ is a fixed point and $\pi_1(B;b_0,b)$ denotes the homotopy classes of the paths from $b$ to $b_0$) we may assume that $\xi(b_0)=1\in \pi_1(B;b_0)$."
Q I do not follow:
What is the definition of $\xi(b_0)$? I guess it is the restriction of this homotopy class to the end point $b_0$.
Why the author assumes that $\xi(b_0)=1$?
$\xi(b)$ is a homotopy class of some path from $b$ to $b_0$ (it's not clear why would such path always exist, does the author actually assume that $B$ is path connected instead of just connected?).
When $b=b_0$ then there's a special path available: the constant path at $b_0$. And the author chooses it as $\xi(b_0)$. The "$1$" symbol typically corresponds to the identity element of the fundamental group, which in turn corresponds to the trivial constant loop.