Trivial covering map

Hello.

I'm reading Munkres Topology, and I'm in chapter 9, section 53. Right on the first exercise I'm missing something important and I don't know what. The exercise says: Let Y have the discrete topology. Show that if p: XxY --> X is projection on the first coordinate, then p is a covering map.

I've worked on it and, from my prespective, the only thing that fails is the homeomorphism between p^{-1}(U) and U. It can't be a homeomorphism because it's not injective (as long as Y has more than one point). What am I doing wrong?

Thanks in advance.

Ok, I got it.