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Follow up to this question. Motivated by wanting to extend a metric space into one that intuitively "extends indefinitely and has no gaps". And also motivated by the famous result that every metrizable space embeds into some $\mathbb R^\kappa$:

Question: Given a metric space $M$ that topologically embeds into some $\mathbb R^n$ for finite $n$, is there always a metric space $N$ that is topologically $\mathbb R^n$ for some finite $n$ (perhaps the same $n$?) such that $M$ isometrically embeds into $N$?

An answer for $n = \aleph_0$ would also be interesting, but I'm mostly interested in the finite $n$ case.

Carla_
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    $\mathbb{R}^\kappa$ is only metrizable if $\kappa$ is countable, so this is definitely impossible unless $M$ is separable. – Eric Wofsey Jan 31 '23 at 16:28
  • @EricWofsey Edited the question to no longer care about those cases. – Carla_ Jan 31 '23 at 16:38
  • If $M$ is locally compact, then yes (for a different $n$). But I do not understand the motivation. – Moishe Kohan Jan 31 '23 at 19:41
  • @MoisheKohan Do you have an example of such a (locally compact) metric space that satisfies this but only for different $n$? – Carla_ Jan 31 '23 at 22:02
  • No: I do not see why such examples would exist but this is what a proof requires. The issue is actually purely topological: If $X$ is a locally compact space which embeds in $R^n$, does it also embed properly in $R^n$? This is true if one increases the dimension. – Moishe Kohan Jan 31 '23 at 22:27
  • @MoisheKohan I don't understand. What do you mean by "embed properly"? – Carla_ Jan 31 '23 at 22:34
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    See here for the definition of a proper map. A proper embedding is an embedding which is also a proper map. I actually realize that I do have an example where the dimension has to go up Take the complete graph on 5 vertices $K_5$ and let $M=K_5\times (0,1)$. Then $M$ does embed in $R^3$ but does not properly embed in $R^3$ (which is a nontrivial result). One can use this to prove that the dimension for an isometric embedding has to go up from 3 to 4. – Moishe Kohan Jan 31 '23 at 22:45

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