While reading some notes about index formula I found the expression which involved $[T^*M]$-the fundamental class of cotangent bundle. As far as I remember in order to speak about fundamental class $[M]$ two conditions must be satisfied:
1. $M$ has to be orientable
2. $M$ has to be closed.
These two conditions guarantee that $H_n(M,\mathbb{Z})=\mathbb{Z}$ and in this circumstances $[M]$ is defined. In the case of cotangent bundle I would therefore like to ask:
Why $T^*M$ is always orientable (as a manifold) (even if $M$ is not)?
Also appareantly $T^*M$ is never compact (therefore not closed) so
How do we understand the fundamental class of $T^*M$?