TITLE: Compact manifolds and orientability
QUESTION [5 upvotes]: I've a doubt about compact manifolds and orientability.
I know that Compact Manifolds in $\mathbb{R^3}$ are orientable.
My questions is:
The statement above is valid only for compact manifolds without boundary (in this case, closed manifolds)?
I'm asking this because I'd read that the Möbius-Strip with its boundary is a compact manifold.
Can someone explain me this?
Thanks.

REPLY [5 votes]: It is pretty simple, actually. For some mathematicians the definition of manifold covers only those without boundary, while others distinguish between manifolds with or without boundary.
As you correctly observed, the Möbius strip is a non-orientable manifold with boundary which can be embedded in $\Bbb R^3$. Moreover it is a compact topological space, since for example it is a closed subspace of $\Bbb P^2_{\Bbb R}$.