Universal Cover of wedges $S^{2} \vee S^{2}, \mathbb{R}P^{2} \vee S^{2}$ and $\mathbb{R}P^{2} \vee \mathbb{R}P^{2}$. We are asked to find the universal cover of the wedges $S^{2} \vee S^{2}, \mathbb{R}P^{2} \vee S^{2}$ and $\mathbb{R}P^{2} \vee \mathbb{R}P^{2}$.
I am second guessing myself on this problem because I came up with the wedge of two spheres as the universal cover for all three of these. Is this the correct universal cover for them all? 

Edit (last answer was wrong)
The universal covering space of $S^2\lor S^2$ is itself. However, once you introduce the projective plane, the wedge point splits, so $S^2\lor \Bbb RP^2$ has a chain of three spheres as universal cover, where the middle sphere is a two-sheeted cover of $\Bbb RP^2$, and the two other spheres each cover the $S^2$. $\Bbb RP^2\lor\Bbb RP^2$ is similarly an infinite chain of spheres, each sphere being a two-sheeted cover of one of the $\Bbb RP^2$'s.
