Let $h: \mathbb R^n \to \mathbb R^n$ be a homeomorphism with the image. Does it follow that the image of $h$ has an interior point?
Homeomorphic embeddings of $\mathbb R^n$ in itself
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algebraic-topology
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2This is the Invariance of Domain theorem https://en.wikipedia.org/wiki/Invariance_of_domain – 2017-02-15