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.

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.

Is there an ultrafilter $\omega$ 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 $M$.)

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.