Q:

Fundamental group S^2/~

Let $X = \mathbb{S}^2/\sim$  where $(cos(\theta), sin(\theta), 0 ) \sim (cos(\theta + \pi), sin(\theta + \pi), 0 )$ for all $\theta \in [0, 2\pi]$
Calculate fundamental group of $X$
I try use Seifert - Van Kampen theorem but and i find that the group is $\mathbb{Z} *_{\mathbb{Z}} \mathbb{Z}$ but 
i'm not sure it's correct
any help to take neighborhoods?

A:

Let $U=X-\{\overline{1,0,0}\}$ and $V=X-\{\overline{(-1,0,0)}\}.$ You should prove that $\pi_1(U)=\mathbb{Z_2}$ and $\pi_1(V)\cong\mathbb{Z}_2.$ The intersection $U\cap V$ has fundamental group $\pi_1(U\cap V)\cong \mathbb{Z}.$ Use Seifert-van Kampen to conclude that $\pi_1(X)\cong \mathbb{Z}_2.$

