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I have followed this. question and this. one. Since a homeomorphism is automatically a bijection, shouldn't that mean it should be surjective? Am I correct in assuming that the domain is only homeomoprhic with the range, and would the result be any different if it was compact?

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    Why don't you tell us what definition you're using for isometry? Also: what's the actual question you're wondering about? The two related questions are quite different. So why not state, right here, the thing you're wondering about and the associated definitions? – John Hughes Sep 15 '16 at 12:32
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    I believe, the questions you linked use two different definitions of isometry. In the first, it is simply a distance preserving map ($d'(f(x),f(y)) = d(x,y)$), in the second, it is a bijection which preserves distances. – Matthias Klupsch Sep 15 '16 at 12:35

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