 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.