= Orientability of real projective space
{title2=$\mathbb{RP}^n\text{ orientable}\iff n=0\text{ or }n\text{ odd}$}
The outward-normal-first <volume form> on the unit sphere pulls back under the <antipodal map> by $(-1)^{n+1}$. For $n\ge1$, 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.
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