Example of a non-orientable 3-manifold

Question

I was reading a paper and it was affirmed in there that $\mathbb{R}\mathbb{P}^2\times\mathbb{S}^1$ was a non-orientable 3-manifold.

Does anyone knows how to prove it? if not, is there another (simple) example of a non-orientable 3-manifold?

Answer 1

It has a connected orientable double cover, which is $S^2\times S^1$. Thus the space itself must be non-orientable.

Or you could use the fact that the product of manifolds $X$ and $Y$ is orientable if and only if both $X$ and $Y$ are orientable.