Open book embeddings of closed non-orientable 3-manifolds

In this note, we discuss open book embeddings of closed non-orientable 3-manifolds in 5-manifolds. We also show that a huge class of closed orientable 3-manifolds, namely the orientation double covers of certain closed non-orientable 3-manifolds, open book embeds in the 5-sphere S5. Finally, we give a new proof of a well-known theorem which states that every closed non-orientable 3-manifold smoothly embeds in S5.