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Is there a nonprincipal ultrafilter $\omega$ on $\mathbb N$ such that for any metric space $M$ there is an isometry $$(M^\omega)^\omega\to M^\omega?$$ (In other words, the $\omega$-power of $\omega$-power of $M$ is isometric to the $\omega$-power of $M$.)

Comments.

  • You may assume that $M$ has finite diameter, otherwise $M^\omega$ is an $\infty$-metric space; that is, distance between points might take infinite value.

  • I want to thank Will Brian for pointing to this post of Andreas Blass. It solves my problem negatively, at least if one assumes some natural condition on the isometry.

Anton Petrunin
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    I presume $M^\omega$ is the "standard part" of the usually-defined ultrapower, right (namely, take the subset of the full ultrapower consisting of points a finite distance from the image of some element of $M$, and then quotient out by the infinitesimal distance relation)? – Noah Schweber Jul 05 '22 at 16:53
  • @NoahSchweber, I added a comment about it. – Anton Petrunin Jul 05 '22 at 20:55
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    You still need to mod out by infinitesimal distance - otherwise, it's not a genuine metric space. For example, consider (the points corresponding to) the constant sequence $(0,0,0,...)$ and the harmonic sequence $(1,{1\over 2},{1\over 3},{1\over 4},...)$ in the ultrapower of $[0,1]$ (with the usual metric). – Noah Schweber Jul 05 '22 at 21:06
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    I don't know the answer to this question, but I think this thread is relevant: https://mathoverflow.net/questions/324254/selective-ultrafilter-and-bijective-mapping. If I understand correctly, Andreas' answer there blocks a plausible way to proving a positive answer to your question. – Will Brian Jul 05 '22 at 21:16
  • @NoahSchweber ultralimit is usually defined as the corresponding metric space to the constructed pseudometric space; that is, it is a metric on equivalence classes defined by $x\sim y$ iff $|x-y|=0$. – Anton Petrunin Jul 06 '22 at 11:06
  • Is this possible for arbitrary first-order structures (in a countable language)? You can code arbitrary first-order structures in metric spaces in a way that is compatible with ultraproducts. – James Hanson Jul 10 '22 at 16:33

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