If one wanted to find all connected covering spaces of a product of two spaces, say $S^1\times RP(3)$, how would you go about it?
I'm thinking finding the fundamental group of $S^1\times RP(3)$, and then we know that to each subgroup of $\pi_1(S^1\times RP(3)$(there is s 1-1 correspondence between coverings of a space $X$ and conjugacy classes of subgroups of $\pi_1(X)$) corresponds a covering space of $S^1\times RP(3)$. Is this correct?
If yes, then how would I go about identifying the corresponding covering spaces? Would I simply try to identify which spaces have these subgroups as their fundamental groups? If I found all these subgroups, would this imply that there cannot be any more connected covering spaces because of this one-to-one correspondence?