# A non-orientable surface $S$ such that $T_pS$ is time-type.

I'm currently taking a course in Semi-Euclidean (Semi-Riemannian) Geometry and am given the task to give an example of a non-orientable surface $$S$$ in $$\mathbb{R}^3_1$$ which is time-type.

With the Lorentzian dot product: $$\langle u,v\rangle = u_1v_1 + u_2v_2 - u_3v_3$$ We say that a vector is time-type whenever $$\langle v,v\rangle < 0$$. And we say that a surface is time-type iff the tangent plane to at any point in $$S$$ contains a time-type vector.

I cannot think of an example that is non-orientable and time-type. We know the classic example of the mobius band which is non-orientable but there is at least one point where the tangent plane does not contain one such vector.