The following extension of the Jordan Curve Theorem is well known: every closed connected hypersurface of the sphere $\mathbb S^N$ separates $S^N$ into exactly two connected components. As a consequence, every compact connected hypersurface of the euclidean space $\mathbb R^N$ separates $\mathbb R^N$ into exactly two connected components.

Does this result keep its validity if the word `compact' is replaced by *closed*? With other words, is it true that every closed connected hypersurface of the euclidean space $\mathbb R^N$ separates $\mathbb R^N$ into exactly two connected components?