I have a $1 \frac{1}{2}$questions about two examples introduced in Szamuely's "Galois Groups and Fundamental Groups" in the excerpt below (see page 122):
Let $X$ be an integral proper normal $k$-curve with function field $K$, and $U ⊂ X$ a nonempty open subset. The construction below describes how to obtain a Galois extension $K_U \vert K$:
Then we define the algebraic fundamental group $π_1(U)$ of $U$ to be the Galois group $Gal (K_U |K)$.
QUESTION:
I don't understand how 4.6.7 and 4.6.10 imply for algebraic fundamental groups $\pi_1(\mathbb{P}^1_k)=\pi_1(\mathbb{A}^1_k)$ in example 1.
4.6.10 describes invariance of $\pi_1$ under base change by field extensions. 4.6.7 tells that is essentially the profinite completion of the topological fundamental group of the Riemann surface associated with $U$.
Why does this provide $\pi_1(\mathbb{P}^1_k)=\pi_1(\mathbb{A}^1_k)$?
Futhermore regarding example 2: I know that $\mathbb{P}^1_{\mathbb{C}} \cong S^2$ so $\mathbb{P}^1_{\mathbb{C}}\backslash \{0, \infty\} \cong S^1$ but what about $\mathbb{P}^1_{k}$ for arbitrary algebraically closed $k$ of char $0$?
Here we have thm 4.6.7:
