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In his book Introduction to smooth manifolds, Lee defines a smooth covering map as follows:

a map $\pi: E \rightarrow M$ between connected smooth manifolds with or without boundary is an open covering map if it is smooth and surjective, and every point in $M$ has a neighbourhood $U$ such that the components of $\pi ^{-1}(U)$ are mapped diffeomorphically onto $U$ by $\pi$.

Comparing it with Munkres's definition of a "topological" covering map, the latter doesn't assume that the slices are precisely the components of the inverse image, namely, it seems that Lee's definition is not the perfect analogue (one would expect) to the smooth world of the topological version. Let's define a smooth covering map as follows:

a map $\pi: E \rightarrow M$ between connected smooth manifolds with or without boundary is an open covering map if it is smooth and surjective, and every point in $M$ has a neighbourhood $U$ such that $\pi ^{-1}(U)$ can be decomposed in a disjoint union of open sets, which are mapped diffeomorphically onto $U$ by $\pi$.

Are both definitions equivalent? If they are not, could you give an example to illustrate that?

Thank you in advance for any help.

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The two definitions are equivalent:

Since $M$ is locally connected, $U$ may be assumed to be connected, and in a disjoint union of open connected sets, these sets are the (connected) components.

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