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Anyone know how to prove the result that every closed hypersurface of ## \mathbb R^n ## , i.e., any closed (n-1)-submanifold of ## \mathbb R^n ## is orientable? Note that if we assume this is true, this shows ## \mathbb RP^n ## cannot be embedded in ## \mathbb R^{n+1} ##

EDIT: there is a result that every hypersurface can be represented as ## f^{-1}(0)## , where I think ##f## is at least an immersion. I heard this has to see too with Alexander duality but I don't see clearly where this duality comes into place, though.

EDIT: there is a result that every hypersurface can be represented as ## f^{-1}(0)## , where I think ##f## is at least an immersion. I heard this has to see too with Alexander duality but I don't see clearly where this duality comes into place, though.

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