I have just learned about the Seifert-Van Kampen theorem and I find it hard to get my head around. The version of this theorem that I know is the following (given in Hatcher):
If $X$ is the union of path - connected open sets $A_\alpha$ each containing the basepoint $x_0 \in X$ and if each intersection $A_\alpha \cap A_\beta$ is path connected, then the homomorphism $$\Phi:\ast_\alpha \pi_1(A_\alpha) \to \pi_1(X)$$ is surjective. If in addition each intersection triple intersection $A_\alpha \cap A_\beta \cap A_\gamma$ is path-connected, then $\ker \Phi$ is the normal subgroup $N$ generated by all elements of the form $i_{\alpha\beta}(\omega)i_{\beta\alpha}(\omega^{-1})$, and so $\Phi$ induces an isomorphism $$\pi_1(X) \cong \ast_\alpha \pi_1(A_\alpha)/N.$$
$i_{\alpha\beta}$ is the homomorphism $\pi_1(A_\alpha \cap A_\beta) \to \pi_1(A_\alpha)$ induced from the inclusion $A_\alpha \cap A_\beta \hookrightarrow A_\alpha$ and $\omega$ is an element of $\pi_1(A_\alpha \cap A_\beta)$.
Now I tried to get my head round this theorem by trying to understand the example in Hatcher on the computation of the fundamental group of a wedge sum. Suppose for the moment we look at $X = X_1 \vee X_2$. I cannot just apply the theorem blindly because $X_i$ is not open in $X$. So we need to look at
$$A_1 = X_1 \vee W_2, \hspace{3mm} A_2 = X_2 \vee W_1$$
where $W_i$ is a neighbourhood about the basepoint $x_1$ of $X_1$ that deformation retracts onto $\{x_1\}$, similarly for $W_2$. I believe each of these is open in $X_1 \vee X_2$ because each $A_i$ is the union of equivalence classes that is open in $X_1 \sqcup X_2$. Now how do I see rigorously that $A_1 \cap A_2$ deformation retracts onto the point $p_0$ (that I got from identifying $x_1 \sim x_2$) in $X$? If I can see that, then I know by Proposition 1.17 (Hatcher) that
$$\pi_1(A_1 \cap A_2) \cong \pi_1(p_0) \cong 0$$
from which it follows that $N= 0$ and the Seifert-Van Kampen Theorem tells me that
$$\pi_1(X_1\vee X_2) \cong \pi_1(X_1) \ast \pi_1(X_2).$$
1) Is my understanding of this correct?
2) What other useful exercises/ examples/applications are there to illustrate the power of the Seifert-Van Kampen Theorem? I have also seen that you can use it to prove that $\pi_1(S^n) = 0$ for $n \geq 2$.
I have had a look at the examples section of Hatcher after the proof the theorem, but unfortunately I don't get much out of it. The only example I sort of got was the computation of $\pi_1(\Bbb{R}^3 - S^1)$.
I would appreciate it very much if I could see some other examples to illustrate this theorem. In particular, I heard that you can use it to compute group presentations for the fundamental group - it would be good if I could see examples like that.
Thanks.
Edit: Is there a way to rigorously prove that $A_1 \cap A_2$ deformation retracts onto the point $\{p_0\}$?