A Cover of an Orientable Manifold is Orientable The following question comes from Introduction to Smooth Manifolds by Lee:
Suppose $\widetilde{M}$ smoothly covers $M$ where $M$ is orientable. Show that $\widetilde{M}$ is orientable. 
I think the following proof works:
Orientability is equivalent to the existence of a nowhere vanishing continuous top form on $M$, so let $\Omega$ be any such form on $M$. Then the pullback $\pi^*\Omega$ is a top form on $\widetilde{M}$ which cannot vanish ($\pi$ here denotes the smooth covering map). For if it did that would imply that $\Omega$ vanished somewhere.
This proof concerns me since I am nowhere using the fact that $\widetilde{M}$ is covering space other than to know that $\pi^*\Omega$ is a top form on $\widetilde{M}$ since $\pi$ is a local diffeomorphism. Have I proven too much here?
 A: What you say is perfectly correct.
Indeed, if $\pi:\tilde M\to M$ is a local diffeomorphism between $n$-dimensional differential manifolds and $M$ is oriented by the nowhere zero top form $\omega\in \Omega^n(M)$, then you obtain an orientation on $\tilde M$  by lifting $\omega$ to the nowhere zero top form $\pi^*\omega\in \Omega^n(\tilde M)$ .
In other words, the stronger  assumption  that $\pi$  be a covering map is irrelevant.  
Remark 1
The converse is  false: in the universal covering $\pi:S^2\to \mathbb P^2(\mathbb R)$ of the projective plane by the 2-sphere, the sphere is orientable but the projective plane is not.
Actually, any manifold admits of  an orientable covering map: its universal cover, but this says nothing about the orientability of the manifold.
Remark 2
The same result holds more generally for local homeomorphisms between topological manifolds. The above proof obviously doesn't work since you cannot talk about differential forms on topological manifolds: even the definition of orientability must be changed.
The key to the solution of these problems is the Algebraic Topology concept of relative homology: see for example Greenberg-Harper's Algebraic Topology, Chapter 22, page 157. 
