
Covering space
I realize this may sound like a simple question, but I'm having trouble understanding.
This regards the universal cover for RP2 v RP2 (that is, the wedge sum of two copies of the projective plane). It makes the most sense to me that it would be an infinite "string" (in both directions) of copies of S^2. I'm told, however, that the actual universal cover is such a "string" with alternating copies of S^2 and RP^2. (so, an infinitely long "necklace" with alternating colors of beads, so to speak) I don't have a problem seeing this as a covering space, but I also don't understand how this can be simply connected. If we take a typical path entirely contained in one of the RP^2's that represents the nontrivial class of pi_1(RP^2), from its northern pole to its southern pole (which are the same point), I don't see how this path is any more nullhomotopic in this space than it is in RP^2 alone. ... I'd like to understand what I'm missing here, so if anyone can help explain it to me, I'd much appreciate it.
Thanks.