Universal Cover of wedges $S^{2} \vee S^{2}, \mathbb{R}P^{2} \vee S^{2}$ and $\mathbb{R}P^{2} \vee \mathbb{R}P^{2}$.

Question

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?

Answer 1

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.