Given a local diffeomorphism $f: N \to M$ with $M$ orientable. Why is $N$ orientable? My professor wrote this in class without giving a proof and said "you should try to prove this for fun :)". I am clueless, please help :(
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Do you know about the connection between differential forms and orientability? That $M$ is orientable means precisely that $M$ has a non-vanishing volume form $\omega$. To show $N$ is orientable, we must show $N$ also admits a non-vanishing volume form. It seems natural to consider the pullback of $\omega$, and you can check directly that it is non-vanishing at each point because $f$ is a local diffeomorphism.