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Why do homeomorphisms map interiors to interiors and boundaries to boundaries? I cannot find a good proof for it that does not involve algebraic topology. I only need it for spaces in $\mathbb{R}^n.$

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    (related: http://math.stackexchange.com/q/46353/)2011-06-29
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    It's fine. I don't understand the proof in that link either. I just need a basic proof that doesn't involve knowledge of homotopy or fundamental topological groups.2011-06-29
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    There are proofs without algebraic topology techniques, but they need dimension theory and Brouwer's fixed point theorem (which can be proved elementarily). It's non-trivial, and you won't find a really short proof. What's the purpose of having such a proof? Can't you just refer to it, using a reference?2011-06-29
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    Some proofs of the change of variables theorem in analysis such as the one in Buck's advanced calculus text.2011-06-29
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    They might use it, but then you can just assume it's true, right?2011-06-29

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