Number of Orientations of Disconnected Manifold

Question

This seems like a stupid question, but the number of orientations of a smooth manifold with $n$ maximal connected components would be $2^n$, right? Since each connected component $U\subset M$ is open $\Rightarrow$ $U$ is a (connected) manifold and hence has two orientations.

Answer 1

Correct. Another way to see this is (if closed) that an orientation corresponds to a choice of a generator of the $\mathbb Z$ summand for a connected manifold and the top homology of a manifold with $n$ components is isomorphic to $\mathbb Z^n$. So if you fix an orientation, all other orientations will correspond to words of length $n$ on $\{\pm 1\}$.

Answer 2

This is not quite correct. The number of orientations of a smooth orientable manifold with $n$ connected components is $2^n$.

However, a non-orientable manifold has no orientations --- and the existence of a single non-orientable component implies that the manifold as a whole is non-orientable.