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My question is connected (pun intended) to the Exercise 1.3.14 of Allen Hatcher's Algebraic Topology.

The question is to find all connected covering spaces of $\mathbb RP^2\vee \mathbb RP^2.$ I understand how to solve the problem by first finding the fundamental group of $\mathbb RP^2\vee \mathbb RP^2$ and then computing the covers corresponding to the subgroups of the fundamental group. This provides a detailed solution as of how to find the connected covering spaces.

My question is this: how do we find the covering spaces of $\mathbb RP^2\vee \mathbb RP^2.$ that are not connected? Are there any general techniques or methods that can be used to answer this question?

Thank you!

Susan Bradely
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  • if you dont add the path conected requisite, i belive the covers are just disjoint union of path conected covers. – Fran Aguayo Dec 03 '23 at 03:03
  • @Fran Aguayo Could you please elaborate? What did you mean by "if you don't add path connected requisite"? – Susan Bradely Dec 03 '23 at 03:08
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    i am not sure all covers will be like that, but there will be a family of not conected covers that will be the disjoint union of conected covers, with the map define by parts in each of them. like for a example, for a space X the union of any number of disjoint X, will be a cover to X, with the map "identity". and in just that case you have infinitly many covers, and this can be done for any cover $p:Y \to X$ (the disjoint union of any number of Y will cover X, with the map p, p(y) = p(y). – Fran Aguayo Dec 03 '23 at 03:17
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    so that is why i think when they say find all covers, they mean conected ones (i made the mistake of saying path connected insted of conected in the first comment) – Fran Aguayo Dec 03 '23 at 03:19
  • @Fran Aguayo Ohh that makes sense. Thank you so much! – Susan Bradely Dec 03 '23 at 03:40
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    Our space $X$ is locally path-connected, so any covering space of it will be locally path-connected, hence topologically the disjoint union of its components and each of those components is necessarily a connected covering space. So what @FranAguaygo describes are indeed all covers. – Thorgott Dec 03 '23 at 11:52

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