Let us consider $A$ and $B$ to be two copies of the the $2$-torus $T^2 = S^1 \times S^1$. Now, consider the following space: $$X = (A \sqcup B)/R,$$ endowed with the quotient topology, where $R$ is the following relation on $A \sqcup B: xRy \iff x \in A, y \in B$ and $ x = y = (z,0), \text{ with } z \in \mathbb{C}. $ What is the fundamental group of $X$?
I don't even know how to start. I know I need to apply the theorem of Seifert-van Kampen somehow, but I don't know how, because I don't know how to select an open cover $\{U,V\}$ of $X$ such that we know the fundamental groups of $U, V$ and of $U \cap V$ (it feels to me that the disjoint union and the quotient make the space more complicated).
Sometimes a picture can be elucidating. Here’s a hint that might help; consider the disjoint union as being two nested tori, with the quotient gluing a pair of corresponding curves together.