The following is an exercise I was assigned in homotopy theory.
Defined $X = \mathbb{RP}^2\vee \mathbb{RP}^2$.
a) Find $\pi_1(X)$.
b) Find the universal cover of $X$.
c) Find all of its connected $2$-sheeted covers.
I am trying to use van Kampen's theorem for (a). I decompose $X$ into two open sets $U_1$ and $U_2$, where each $U_i$ contains one copy of $\mathbb{RP}^2$ as well as a neighborhood around the point of attachment small enough so that each $U_i$ deformation retracts onto the copy of $\mathbb{RP}^2$ that it contains. Then van Kampen's theorem gives that $\pi_1(X)$ is a quotient of $\mathbb{Z}_2*\mathbb{Z}_2$, where we quotient out by the subgroup normally generated by the elements $\iota_{1, 2}(\omega)\iota_{2, 1}(\omega)^{-1}$, where $\iota_{1, 2}: \pi_1(U_1 \cap U_2) \to \pi_1(U_1)$ is the inclusion map, and similarly for $\iota_{2, 1}$, and the $\omega$'s are arbitrary elements of $\pi_1(U_1 \cap U_2)$.
I want to say that $U_1 \cap U_2$ is simply connected, so each $\iota_{1, 2}(\omega)\iota_{2, 1}(\omega)^{-1}$ is in fact trivial, so then $\pi_1(X)$ is isomorphic to $\mathbb{Z}_2*\mathbb{Z}_2$. Is this true, or did my reasoning go awry somewhere?
For (b), I know that the universal cover is just an infinite chain of copies of $S^2$, where the covering map is just ``locally'' the quotient map.
For (c), I know that the only covering spaces of $\mathbb{RP}^2$ is itself and $S^2$, but I don't really know how to go from there to finding all the connected $2$-sheeted covers.
To summarize: is in fact $\pi_1(X)$ isomorphic to the free product of two copies of $\mathbb{Z}_2$, and how do I find all connected $2$-sheeted covers of $X$?