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Let $H \subset \mathbb{R}^n$ be a closed subset homeomorphic to $\mathbb{R}^{n - 1}$. Could you give me an idea of how to prove that $\mathbb{R}^n - H$ has exactly two path components. I am expecting an answer using singular homology.

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    This theorem in Hatcher is a special case of Alexander duality.2012-12-14

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