I would like to apply Seifert van Kampen to a simple example taken from Wikipedia: I have $X = S^2$ and $A = S^2 - n$, where $n$ is the north pole and $B = S^2 - s$, where $s$ is the south pole.
According to my understanding, which might be wrong, Seifert van Kampen tells me that $\pi_1(X) = \pi_1(A) *_{\pi_1(A \cap B)} \pi_1(B)$, where the right hand side is the free product with amalgamation.
$A \cap B$, the sphere minus the two points has a fundamental group isomorphic to $\mathbb{Z}$.
The free product with amalgamation of two groups $G, H$ is $ G * H / N$, where $N$ is the smallest normal subgroup in $G * H$ (according to the Wikipedia entry about free product with amalgamation).
Note : I am aware that in the sphere example both $G$ and $H$ are trivial and so quotienting them with anything will be trivial again. This question is not about actually computing the fundamental group of $S^2$!
In the sphere example, this means I have to find the smallest normal subgroup of $\mathbb{Z}$.
Question 1: Is my understanding of Seifert van Kampen correct?
Question 2: What is the smallest normal subgroup of $\mathbb{Z}$?
As for question 2, what do you think about $\lim\limits_{k \rightarrow \infty} k\mathbb{Z}$?
Thanks for your help.