If a hypersurface in a manifold separates the ambient space into two disconnected pieces, is the surface necessarily orientable? This seems to be true when one considers the Jordan Brouwer theorem which implies the sphere $S^n$ embedded in $\mathbb{R}^n$ separates space into two disconnected components. But does the requirement of orientability extend to hypersurfaces in any manifold? A counter-example would show a non-orientable hypersurface separating the ambient space into two disconnected regions.
(edit: statement on Jordan Brouwer theorem refined per George Lowther's comment)