I am a little confused about how to understand the path connected intersection requirement of Van Kampen's Theorem.
Most examples I have seen for $S^1 \vee S^1$ take the following decomposition:
However, why not take the point of contact to be $U\cap V$ and $U,V$ to be the circles. $U\cap V$ would still be path connected, since it is a single point.
Am I correct and it just so happens that I haven't come across a point intersection example, or is there a reason why point intersection are not taken in applying the Van Kampen Theorem?
In the standard vesion of the Seifert - van Kampen theorem both $U,V$ are required to be open.
However, we easily get the following corollary:
Let $(Z,*) = (X_1,x_1) \vee (X_2,x_2)$ be the one-point union of the two based spaces $(X_i,x_i)$. If $\{x_i\}$ is a strong deformation retract of an open neighborhood $U_i \subset X_i$, then $\pi_1(Z,*) \approx \pi_1(X_1,x_1) * \pi_1(X_2,x_2)$.
This clearly applies to $S^1 \vee S^1$.
Proof. Let $V_1 = X_1 \vee U_2$ and $V_2 = U_1 \vee X_2$. These are open subsets of $Z$ which cover $Z$. The intersection $V_1 \cap V_2$ is contractible. Now apply Seifert - van Kampen, noting that $(V_i,*) \simeq (X_i,x_i)$.