Regular zero hypersurfaces have disconnected complements

A nonempty regular level set $f^{-1}(0)$ for a real smooth function has both positive and negative values immediately beside any of its points, by the inverse function theorem. On its complement the function never vanishes, so the positive and negative regions are disjoint nonempty open sets partitioning that complement. Hence the complement is disconnected. This is a global necessary condition beyond local embeddedness or coorientation.

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.