Orientation on manifolds I am trying to understand the definitions here. In many books (say Tu or even Guillemin-Pollack) an orientation on a manifold is an assignment to affix $+1$ and $-1$ to classes of (tangent) basis. It is proven in nearly every differential geometry book that connected orientable manifolds admits exactly two orientations. 
Here is what I don't get, since you are assigning only $\{\pm 1\}$, how can it make sense to talk about anything but more than or fewer than two orientations? It only otherwise make senses to talk about not oriented right?
Removing the connectedness, can you even talk about, say three orientations? 
 A: The concept of orientation arises in linear algebra by taking equivalence classes of ordered bases of real vector spaces, two such bases $\{b_i \}$ and $\{b'_i \}$ being equivalent if the linear automorphism sending $b_i$ to $b'_i$ has positive determinant. There are exactly two orientations of a vector space with dimension $> 0$.
Here you assign an orientation $\omega_p$ to each tangent space $T_pM$ and require that these pointwise orientations are "locally compatible". An orientation of $M$ is any such compatible family of pointwise orientations $\omega_p$, $p \in M$. It is not all clear that $M$ has an orientation, and in fact there exist non-orientable manifolds. Moreover, there are uncountably many families of pointwise orientations $\omega_p$, but most of them are not compatible. Moreover, in my opinion there is no a-priori reason why there should exist not more than two orientations. It is true for connected manifolds, but it requires a proof. For a manifold with $n$ connected components you get $2^n$ orientions.
