= 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.
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