= Real projective hyperplane is not a global regular zero set
{title2=$\mathbb{RP}^{n-1}\subset\mathbb{RP}^n\quad(n\ge1)$}
The coordinate hyperplane is an <embedded submanifold>, as standard projective charts flatten its defining homogeneous coordinate. Its complement is the affine chart $\mathbb R^n$ and is connected. The lemma <regular zero hypersurfaces have disconnected complements> therefore excludes its being the zero set of a smooth real function with zero a <regular value>.
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