Orientability of real projective space (${\mathbb{RP}}^n\text{ orientable}⟺n=0\text{ or }n\text{ odd}$)

The outward-normal-first volume form on the unit sphere pulls back under the antipodal map by $(-1{)}^{n+1}$. For $n\ge 1$, the sphere is connected, and its orientation descends to the twofold quotient exactly when this deck transformation preserves it. Thus Real projective space is orientable precisely in odd positive dimensions; the zero-dimensional case is one point and is also orientable.