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Prove that a simply connected covering space of X is also a covering space for any other covering space of X.

Actually I don't have an idea how to start with. But if X has a universal cover, then the covering space also have universal cover, then we need to prove that they are isomorphic?

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?

A simply connected covering space is called a universal covering space and the reason for calling it so is the fact you are asking a proof of.

Anyway, for a proof, use the lifting criterion. In this case it tells you that any map from a simply connected space to another space lifts to any covering space of the target space. Check that the lifting is a covering space. (You might need a locally path connected assumption on the base space )

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    $\begingroup$ This is a wrong (concept) definition what people usually use. A Universal cover is a covering s.t given any other covering it is covering of that covering space too. In general universal covering space are not simple connected. But for nice spaces it is simple connected. Even there are spaces for which there does not exists universal cover. $\endgroup$ May 14, 2016 at 19:22

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