Real projective hyperplane is not a global regular zero set (${\mathbb{RP}}^{n-1}\subset {\mathbb{RP}}^n(n\ge 1)$)

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.