Need help to solve this problem (which is in "Algebraic Topology" by Hatcher):
Let $X$ be the quotient space of $S^2$ obtained by identifying the north and south poles to a single point. Put a cell complex structure on $X$ and use it to compute $\Pi_1(X)$.
In this problem, $D^3$ is the unit disk in $\mathbb{R}^3$ centered in the origin; $S^2 \subseteq \mathbb{R}^3$ the sphere; and ${\partial}(D^3)$ is the boundary of $D^3$.
I tried to think like this: if I'm identifying the north and south poles, then my space is something like $X = S^2/\{ \mathrm{North} \sim \mathrm{South} \}$, so it's like two balloons which has a single point in common. So, if I put a cell complex structure on $X$, the natural try is consider $D^3/{\partial}(D^3) \vee D^3/{\partial}(D^3)$. But $\partial(D^3) = S^2$, so my space is $X = S^2 \vee S^2$.
Using this, the fundamental group of $X$ should be
$$ \Pi_1(X) = \Pi_1(S^2 \vee S^2) = \{ 0 \} $$
because $\Pi_1(S^2) = \{ 0 \}$. So, $X$ is simply connected.
Is my reasoning right? Thanks for your help!
I can't post this as a comment, but this question has some discussion of a similar space and pictures of the space in question that might help: Are these two spaces homotopy equivalent? .