Let $X$ be the space obtained by gluing two copies of $S^2$ together by identifying their north poles together, and separately their south poles together. Compute $\pi_1(X)$.
My thoughts so far: first of all, a quick sketch of the space -
Here, the red points are the north and south poles. I've tried applying van Kampen, but this doesn't seem to work for me since I keep getting intersections which are not path connected. Note that this space can be thought of as the torus where we collapse two distinct circles (one for each red point), does this help? What's a good way to compute $\pi_1(X)$ here? Deformation retractions maybe? Hints are appreciated.
For Van Kampen's Theorem, try letting one of the open sets $U$ be a neighborhood of that inner black circle $C$, where $U$ is chosen so that $C$ is a deformation retract of $U$.
The set $X-C$ is a union of a disjoint pair of open sets $V_{left}$ and $V_{right}$. Now just do a two step Van Kampen, first with $U \cup V_{left}$, and next with $(U \cup V_{left}) \cup V_{right}$.