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It is asserted in A Course in Metric Geometry by Burago, Burago, Ivanov that

there can be no more than continuum of mutually nonisometric compact spaces

How is this proven?

Its clear that there must be at least a continuum of mutually nonisometric compact spaces, i.e. $([0,\alpha], d_{\mathbb{R}})$ for $\alpha>0$ are a family of nonisometric metric spaces, but I don't know enough set theory to have any ideas how to bound the cardinality from above. A first guess was that the fact that compact metric spaces are totally bounded should be useful?

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    $\begingroup$ Here are some steps - Each (infinite) compact contains a countable dense set. Order it -- $(x_1,...)$. Consider an infinite matrix $d(x_i,x_j)$. One can recover the metric $d$ from such a matrix. $\endgroup$
    – Petya
    Commented Dec 10, 2010 at 18:24

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I think "compact" can be even weakened here to "separable and complete" (and regarding your first guess, total boundedness is essentially used to prove that compact implies separable). Here's a sketch: any such space is determined, up to isometry, by the restriction of the metric to a countable dense subset. Thus the number of such isometry classes is bounded above by the cardinality of $\mathbb{R}^{\mathbb{N}\times \mathbb{N}}$, which is the same as the cardinality of $\mathbb{R}$, which is the continuum.

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Since a compact metric space is in particular separable, its type of isometry is determined by a dense countable subspace. There are continuum many distances on , say $\mathbb{N}$.

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Extend Qiaochu's deleted comment. Compact metric spaces are separable, can be isometrically embedded as closed subsets of the separable metric space $C[0,1]$. This space has continuum many closed subsets.

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  • $\begingroup$ Very nice ! $\endgroup$
    – Mark Meckes
    Commented Dec 10, 2010 at 18:35
  • $\begingroup$ Sorry, could someone provide more details? How does the embedding work? Also how do you know that there are at most a continuum of closed subsets? $\endgroup$
    – Otis Chodosh
    Commented Dec 10, 2010 at 22:37
  • $\begingroup$ @OtisChodosh See Wikipedia article Urysohn universal space or Corollary 5.18 in Functional Analysis and Infinite-Dimensional Geometry by Fabian et al. $\endgroup$
    – Martin Sleziak
    Commented Jan 9, 2018 at 7:37

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