Let $X=\{(x,y) \in RP^n\times RP^n;x=x_0, or, y=x_0\}$ where $x_0$ is some fixed point of $RP^n$. In other words, $X$ is two copies of $RP^n$ with one point $x_0$ in common. find $\pi_1( X,x_0)$?
I think the fundamental group of $RP^n$ is $Z_2$, so the join $\pi_1( RP^n ∨ RP^n)$ should be $Z_2*Z_2$. but can I confirm this use the Seifert–van Kampen theorem. I am not sure what will the $U$ and $V$ be as in the theorem, so that I can write down a formal proof. Thanks in advanced, any help is appreciated!
Take $U$ to be $\mathbf{RP}^n_1$ union a small neighborhood of $x_0$ in $\mathbf{RP}^n_2$, and $V$ to be $\mathbf{RP}^n_2$ union a small neighborhood of $x_0$ in $\mathbf{RP}^n_1$. We can take the small open neighborhoods in question to be contractible. Then $U \cap V$ is also contractible.
Exercise: in what greater generality does this argument apply?